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Merge branch 'master' into gcl_cleanup
[maxima/cygwin.git] / src / ellipt.lisp
blob7286a32fe05baf5c2ca5c219f0210ad38c01f162
1 ;;; -*- Mode: Lisp; Package: Maxima; Syntax: Common-Lisp; Base: 10-*- ;;;;
2 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
3 ;;; The data in this file contains enhancements. ;;;;;
4 ;;; ;;;;;
5 ;;; Copyright (c) 1984,1987 by William Schelter,University of Texas ;;;;;
6 ;;; All rights reserved ;;;;;
7 ;;; ;;;;;
8 ;;; Copyright (c) 2001 by Raymond Toy. Replaced everything and added ;;;;;
9 ;;; support for symbolic manipulation of all 12 Jacobian elliptic ;;;;;
10 ;;; functions and the complete and incomplete elliptic integral ;;;;;
11 ;;; of the first, second and third kinds. ;;;;;
12 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
13
14 (in-package :maxima)
15 ;;(macsyma-module ellipt)
16
17 ;;;
18 ;;; Jacobian elliptic functions and elliptic integrals.
19 ;;;
20 ;;; References:
21 ;;;
22 ;;; [1] Abramowitz and Stegun
23 ;;; [2] Lawden, Elliptic Functions and Applications, Springer-Verlag, 1989
24 ;;; [3] Whittaker & Watson, A Course of Modern Analysis
25 ;;;
26 ;;; We use the definitions from Abramowitz and Stegun where our
27 ;;; sn(u,m) is sn(u|m). That is, the second arg is the parameter,
28 ;;; instead of the modulus k or modular angle alpha.
29 ;;;
30 ;;; Note that m = k^2 and k = sin(alpha).
31 ;;;
32
33 ;;
34 ;; Routines for computing the basic elliptic functions sn, cn, and dn.
35 ;;
36 ;;
37 ;; A&S gives several methods for computing elliptic functions
38 ;; including the AGM method (16.4) and ascending and descending Landen
39 ;; transformations (16.12 and 16.14). The latter are actually quite
40 ;; fast, only requiring simple arithmetic and square roots for the
41 ;; transformation until the last step. The AGM requires evaluation of
42 ;; several trigonometric functions at each stage.
43 ;;
44 ;; However, the Landen transformations appear to have some round-off
45 ;; issues. For example, using the ascending transform to compute cn,
46 ;; cn(100,.7) > 1e10. This is clearly not right since |cn| <= 1.
47 ;;
48
49 (in-package #:bigfloat)
50
51 (declaim (inline descending-transform ascending-transform))
52
53 (defun ascending-transform (u m)
54 ;; A&S 16.14.1
55 ;;
56 ;; Take care in computing this transform. For the case where
57 ;; m is complex, we should compute sqrt(mu1) first as
58 ;; (1-sqrt(m))/(1+sqrt(m)), and then square this to get mu1.
59 ;; If not, we may choose the wrong branch when computing
60 ;; sqrt(mu1).
61 (let* ((root-m (sqrt m))
62 (mu (/ (* 4 root-m)
63 (expt (1+ root-m) 2)))
64 (root-mu1 (/ (- 1 root-m) (+ 1 root-m)))
65 (v (/ u (1+ root-mu1))))
66 (values v mu root-mu1)))
67
68 (defun descending-transform (u m)
69 ;; Note: Don't calculate mu first, as given in 16.12.1. We
70 ;; should calculate sqrt(mu) = (1-sqrt(m1)/(1+sqrt(m1)), and
71 ;; then compute mu = sqrt(mu)^2. If we calculate mu first,
72 ;; sqrt(mu) loses information when m or m1 is complex.
73 (let* ((root-m1 (sqrt (- 1 m)))
74 (root-mu (/ (- 1 root-m1) (+ 1 root-m1)))
75 (mu (* root-mu root-mu))
76 (v (/ u (1+ root-mu))))
77 (values v mu root-mu)))
78
79 ;;
80 ;; This appears to work quite well for both real and complex values
81 ;; of u.
82 (defun elliptic-sn-descending (u m)
83 (cond ((= m 1)
84 ;; A&S 16.6.1
85 (tanh u))
86 ((< (abs m) (epsilon u))
87 ;; A&S 16.6.1
88 (sin u))
89 (t
90 (multiple-value-bind (v mu root-mu)
91 (descending-transform u m)
92 (let* ((new-sn (elliptic-sn-descending v mu)))
93 (/ (* (1+ root-mu) new-sn)
94 (1+ (* root-mu new-sn new-sn))))))))
95
96 ;; AGM scale. See A&S 17.6
97 ;;
98 ;; The AGM scale is
99 ;;
100 ;; a[n] = (a[n-1]+b[n-1])/2, b[n] = sqrt(a[n-1]*b[n-1]), c[n] = (a[n-1]-b[n-1])/2.
101 ;;
102 ;; We stop when abs(c[n]) <= 10*eps
103 ;;
104 ;; A list of (n a[n] b[n] c[n]) is returned.
105 (defun agm-scale (a b c)
106 (loop for n from 0
107 while (> (abs c) (* 10 (epsilon c)))
108 collect (list n a b c)
109 do (psetf a (/ (+ a b) 2)
110 b (sqrt (* a b))
111 c (/ (- a b) 2))))
112
113 ;; WARNING: This seems to have accuracy problems when u is complex. I
114 ;; (rtoy) do not know why. For example (jacobi-agm #c(1e0 1e0) .7e0)
115 ;; returns
116 ;;
117 ;; #C(1.134045970915582 0.3522523454566013)
118 ;; #C(0.57149659007575 -0.6989899153338323)
119 ;; #C(0.6229715431044184 -0.4488635962149656)
120 ;;
121 ;; But the actual value of sn(1+%i, .7) is .3522523469224946 %i +
122 ;; 1.134045971912365. We've lost about 7 digits of accuracy!
123 (defun jacobi-agm (u m)
124 ;; A&S 16.4.
125 ;;
126 ;; Compute the AGM scale with a = 1, b = sqrt(1-m), c = sqrt(m).
127 ;;
128 ;; Then phi[N] = 2^N*a[N]*u and compute phi[n] from
129 ;;
130 ;; sin(2*phi[n-1] - phi[n]) = c[n]/a[n]*sin(phi[n])
131 ;;
132 ;; Finally,
133 ;;
134 ;; sn(u|m) = sin(phi[0]), cn(u|m) = cos(phi[0])
135 ;; dn(u|m) = cos(phi[0])/cos(phi[1]-phi[0])
136 ;;
137 ;; Returns the three values sn, cn, dn.
138 (let* ((agm-data (nreverse (rest (agm-scale 1 (sqrt (- 1 m)) (sqrt m)))))
139 (phi (destructuring-bind (n a b c)
140 (first agm-data)
141 (declare (ignore b c))
142 (* a u (ash 1 n))))
143 (phi1 0e0))
144 (dolist (agm agm-data)
145 (destructuring-bind (n a b c)
146 agm
147 (declare (ignore n b))
148 (setf phi1 phi
149 phi (/ (+ phi (asin (* (/ c a) (sin phi)))) 2))))
150 (values (sin phi) (cos phi) (/ (cos phi) (cos (- phi1 phi))))))
151
152 (defun sn (u m)
153 (cond ((zerop m)
154 ;; jacobi_sn(u,0) = sin(u). Should we use A&S 16.13.1 if m
155 ;; is small enough?
156 ;;
157 ;; sn(u,m) = sin(u) - 1/4*m(u-sin(u)*cos(u))*cos(u)
158 (sin u))
159 ((= m 1)
160 ;; jacobi_sn(u,1) = tanh(u). Should we use A&S 16.15.1 if m
161 ;; is close enough to 1?
162 ;;
163 ;; sn(u,m) = tanh(u) + 1/4*(1-m)*(sinh(u)*cosh(u)-u)*sech(u)^2
164 (tanh u))
165 (t
166 ;; Use the ascending Landen transformation to compute sn.
167 (let ((s (elliptic-sn-descending u m)))
168 (if (and (realp u) (realp m))
169 (realpart s)
170 s)))))
171
172 (defun dn (u m)
173 (cond ((zerop m)
174 ;; jacobi_dn(u,0) = 1. Should we use A&S 16.13.3 for small m?
175 ;;
176 ;; dn(u,m) = 1 - 1/2*m*sin(u)^2
177 1)
178 ((= m 1)
179 ;; jacobi_dn(u,1) = sech(u). Should we use A&S 16.15.3 if m
180 ;; is close enough to 1?
181 ;;
182 ;; dn(u,m) = sech(u) + 1/4*(1-m)*(sinh(u)*cosh(u)+u)*tanh(u)*sech(u)
183 (/ (cosh u)))
184 (t
185 ;; Use the Gauss transformation from
186 ;; http://functions.wolfram.com/09.29.16.0013.01:
187 ;;
188 ;;
189 ;; dn((1+sqrt(m))*z, 4*sqrt(m)/(1+sqrt(m))^2)
190 ;; = (1-sqrt(m)*sn(z, m)^2)/(1+sqrt(m)*sn(z,m)^2)
191 ;;
192 ;; So
193 ;;
194 ;; dn(y, mu) = (1-sqrt(m)*sn(z, m)^2)/(1+sqrt(m)*sn(z,m)^2)
195 ;;
196 ;; where z = y/(1+sqrt(m)) and mu=4*sqrt(m)/(1+sqrt(m))^2.
197 ;;
198 ;; Solve for m, and we get
199 ;;
200 ;; sqrt(m) = -(mu+2*sqrt(1-mu)-2)/mu or (-mu+2*sqrt(1-mu)+2)/mu.
201 ;;
202 ;; I don't think it matters which sqrt we use, so I (rtoy)
203 ;; arbitrarily choose the first one above.
204 ;;
205 ;; Note that (1-sqrt(1-mu))/(1+sqrt(1-mu)) is the same as
206 ;; -(mu+2*sqrt(1-mu)-2)/mu. Also, the former is more
207 ;; accurate for small mu.
208 (let* ((root (let ((root-1-m (sqrt (- 1 m))))
209 (/ (- 1 root-1-m)
210 (+ 1 root-1-m))))
211 (z (/ u (+ 1 root)))
212 (s (elliptic-sn-descending z (* root root)))
213 (p (* root s s )))
214 (/ (- 1 p)
215 (+ 1 p))))))
216
217 (defun cn (u m)
218 (cond ((zerop m)
219 ;; jacobi_cn(u,0) = cos(u). Should we use A&S 16.13.2 for
220 ;; small m?
221 ;;
222 ;; cn(u,m) = cos(u) + 1/4*m*(u-sin(u)*cos(u))*sin(u)
223 (cos u))
224 ((= m 1)
225 ;; jacobi_cn(u,1) = sech(u). Should we use A&S 16.15.3 if m
226 ;; is close enough to 1?
227 ;;
228 ;; cn(u,m) = sech(u) - 1/4*(1-m)*(sinh(u)*cosh(u)-u)*tanh(u)*sech(u)
229 (/ (cosh u)))
230 (t
231 ;; Use the ascending Landen transformation, A&S 16.14.3.
232 (multiple-value-bind (v mu root-mu1)
233 (ascending-transform u m)
234 (let ((d (dn v mu)))
235 (* (/ (+ 1 root-mu1) mu)
236 (/ (- (* d d) root-mu1)
237 d)))))))
238
239 (in-package :maxima)
240
241 ;; Tell maxima what the derivatives are.
242 ;;
243 ;; Lawden says the derivative wrt to k but that's not what we want.
244 ;;
245 ;; Here's the derivation we used, based on how Lawden get's his results.
246 ;;
247 ;; Let
248 ;;
249 ;; diff(sn(u,m),m) = s
250 ;; diff(cn(u,m),m) = p
251 ;; diff(dn(u,m),m) = q
252 ;;
253 ;; From the derivatives of sn, cn, dn wrt to u, we have
254 ;;
255 ;; diff(sn(u,m),u) = cn(u)*dn(u)
256 ;; diff(cn(u,m),u) = -cn(u)*dn(u)
257 ;; diff(dn(u,m),u) = -m*sn(u)*cn(u)
258 ;;
259
260 ;; Differentiate these wrt to m:
261 ;;
262 ;; diff(s,u) = p*dn + cn*q
263 ;; diff(p,u) = -p*dn - q*dn
264 ;; diff(q,u) = -sn*cn - m*s*cn - m*sn*q
265 ;;
266 ;; Also recall that
267 ;;
268 ;; sn(u)^2 + cn(u)^2 = 1
269 ;; dn(u)^2 + m*sn(u)^2 = 1
270 ;;
271 ;; Differentiate these wrt to m:
272 ;;
273 ;; sn*s + cn*p = 0
274 ;; 2*dn*q + sn^2 + 2*m*sn*s = 0
275 ;;
276 ;; Thus,
277 ;;
278 ;; p = -s*sn/cn
279 ;; q = -m*s*sn/dn - sn^2/dn/2
280 ;;
281 ;; So
282 ;; diff(s,u) = -s*sn*dn/cn - m*s*sn*cn/dn - sn^2*cn/dn/2
283 ;;
284 ;; or
285 ;;
286 ;; diff(s,u) + s*(sn*dn/cn + m*sn*cn/dn) = -1/2*sn^2*cn/dn
287 ;;
288 ;; diff(s,u) + s*sn/cn/dn*(dn^2 + m*cn^2) = -1/2*sn^2*cn/dn
289 ;;
290 ;; Multiply through by the integrating factor 1/cn/dn:
291 ;;
292 ;; diff(s/cn/dn, u) = -1/2*sn^2/dn^2 = -1/2*sd^2.
293 ;;
294 ;; Integrate this to get
295 ;;
296 ;; s/cn/dn = C + -1/2*int sd^2
297 ;;
298 ;; It can be shown that C is zero.
299 ;;
300 ;; We know that (by differentiating this expression)
301 ;;
302 ;; int dn^2 = (1-m)*u+m*sn*cd + m*(1-m)*int sd^2
303 ;;
304 ;; or
305 ;;
306 ;; int sd^2 = 1/m/(1-m)*int dn^2 - u/m - sn*cd/(1-m)
307 ;;
308 ;; Thus, we get
309 ;;
310 ;; s/cn/dn = u/(2*m) + sn*cd/(2*(1-m)) - 1/2/m/(1-m)*int dn^2
311 ;;
312 ;; or
313 ;;
314 ;; s = 1/(2*m)*u*cn*dn + 1/(2*(1-m))*sn*cn^2 - 1/2/(m*(1-m))*cn*dn*E(u)
315 ;;
316 ;; where E(u) = int dn^2 = elliptic_e(am(u)) = elliptic_e(asin(sn(u)))
317 ;;
318 ;; This is our desired result:
319 ;;
320 ;; s = 1/(2*m)*cn*dn*[u - elliptic_e(asin(sn(u)),m)/(1-m)] + sn*cn^2/2/(1-m)
321 ;;
322 ;;
323 ;; Since diff(cn(u,m),m) = p = -s*sn/cn, we have
324 ;;
325 ;; p = -1/(2*m)*sn*dn[u - elliptic_e(asin(sn(u)),m)/(1-m)] - sn^2*cn/2/(1-m)
326 ;;
327 ;; diff(dn(u,m),m) = q = -m*s*sn/dn - sn^2/dn/2
328 ;;
329 ;; q = -1/2*sn*cn*[u-elliptic_e(asin(sn),m)/(1-m)] - m*sn^2*cn^2/dn/2/(1-m)
330 ;;
331 ;; - sn^2/dn/2
332 ;;
333 ;; = -1/2*sn*cn*[u-elliptic_e(asin(sn),m)/(1-m)] + dn*sn^2/2/(m-1)
334 ;;
335 (defprop %jacobi_sn
336 ((u m)
337 ((mtimes) ((%jacobi_cn) u m) ((%jacobi_dn) u m))
338 ((mplus simp)
339 ((mtimes simp) ((rat simp) 1 2)
340 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m)) -1)
341 ((mexpt simp) ((%jacobi_cn simp) u m) 2) ((%jacobi_sn simp) u m))
342 ((mtimes simp) ((rat simp) 1 2) ((mexpt simp) m -1)
343 ((%jacobi_cn simp) u m) ((%jacobi_dn simp) u m)
344 ((mplus simp) u
345 ((mtimes simp) -1 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m)) -1)
346 ((%elliptic_e simp) ((%asin simp) ((%jacobi_sn simp) u m)) m))))))
347 grad)
348
349 (defprop %jacobi_cn
350 ((u m)
351 ((mtimes simp) -1 ((%jacobi_sn simp) u m) ((%jacobi_dn simp) u m))
352 ((mplus simp)
353 ((mtimes simp) ((rat simp) -1 2)
354 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m)) -1)
355 ((%jacobi_cn simp) u m) ((mexpt simp) ((%jacobi_sn simp) u m) 2))
356 ((mtimes simp) ((rat simp) -1 2) ((mexpt simp) m -1)
357 ((%jacobi_dn simp) u m) ((%jacobi_sn simp) u m)
358 ((mplus simp) u
359 ((mtimes simp) -1 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m)) -1)
360 ((%elliptic_e simp) ((%asin simp) ((%jacobi_sn simp) u m)) m))))))
361 grad)
362
363 (defprop %jacobi_dn
364 ((u m)
365 ((mtimes) -1 m ((%jacobi_sn) u m) ((%jacobi_cn) u m))
366 ((mplus simp)
367 ((mtimes simp) ((rat simp) -1 2)
368 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m)) -1)
369 ((%jacobi_dn simp) u m) ((mexpt simp) ((%jacobi_sn simp) u m) 2))
370 ((mtimes simp) ((rat simp) -1 2) ((%jacobi_cn simp) u m)
371 ((%jacobi_sn simp) u m)
372 ((mplus simp) u
373 ((mtimes simp) -1
374 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m)) -1)
375 ((%elliptic_e simp) ((%asin simp) ((%jacobi_sn simp) u m)) m))))))
376 grad)
377
378 ;; The inverse elliptic functions.
379 ;;
380 ;; F(phi|m) = asn(sin(phi),m)
381 ;;
382 ;; so asn(u,m) = F(asin(u)|m)
383 (defprop %inverse_jacobi_sn
384 ((x m)
385 ;; Lawden 3.1.2:
386 ;; inverse_jacobi_sn(x) = integrate(1/sqrt(1-t^2)/sqrt(1-m*t^2),t,0,x)
387 ;; -> 1/sqrt(1-x^2)/sqrt(1-m*x^2)
388 ((mtimes simp)
389 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 ((mexpt simp) x 2)))
390 ((rat simp) -1 2))
391 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m ((mexpt simp) x 2)))
392 ((rat simp) -1 2)))
393 ;; diff(F(asin(u)|m),m)
394 ((mtimes simp) ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m)) -1)
395 ((mplus simp)
396 ((mtimes simp) -1 x
397 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 ((mexpt simp) x 2)))
398 ((rat simp) 1 2))
399 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m ((mexpt simp) x 2)))
400 ((rat simp) -1 2)))
401 ((mtimes simp) ((mexpt simp) m -1)
402 ((mplus simp) ((%elliptic_e simp) ((%asin simp) x) m)
403 ((mtimes simp) -1 ((mplus simp) 1 ((mtimes simp) -1 m))
404 ((%elliptic_f simp) ((%asin simp) x) m)))))))
405 grad)
406
407 ;; Let u = inverse_jacobi_cn(x). Then jacobi_cn(u) = x or
408 ;; sqrt(1-jacobi_sn(u)^2) = x. Or
409 ;;
410 ;; jacobi_sn(u) = sqrt(1-x^2)
411 ;;
412 ;; So u = inverse_jacobi_sn(sqrt(1-x^2),m) = inverse_jacob_cn(x,m)
413 ;;
414 (defprop %inverse_jacobi_cn
415 ((x m)
416 ;; Whittaker and Watson, 22.121
417 ;; inverse_jacobi_cn(u,m) = integrate(1/sqrt(1-t^2)/sqrt(1-m+m*t^2), t, u, 1)
418 ;; -> -1/sqrt(1-x^2)/sqrt(1-m+m*x^2)
419 ((mtimes simp) -1
420 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 ((mexpt simp) x 2)))
421 ((rat simp) -1 2))
422 ((mexpt simp)
423 ((mplus simp) 1 ((mtimes simp) -1 m)
424 ((mtimes simp) m ((mexpt simp) x 2)))
425 ((rat simp) -1 2)))
426 ((mtimes simp) ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m)) -1)
427 ((mplus simp)
428 ((mtimes simp) -1
429 ((mexpt simp)
430 ((mplus simp) 1
431 ((mtimes simp) -1 m ((mplus simp) 1 ((mtimes simp) -1 ((mexpt simp) x 2)))))
432 ((rat simp) -1 2))
433 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 ((mexpt simp) x 2))) ((rat simp) 1 2))
434 ((mabs simp) x))
435 ((mtimes simp) ((mexpt simp) m -1)
436 ((mplus simp)
437 ((%elliptic_e simp)
438 ((%asin simp)
439 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 ((mexpt simp) x 2))) ((rat simp) 1 2)))
440 m)
441 ((mtimes simp) -1 ((mplus simp) 1 ((mtimes simp) -1 m))
442 ((%elliptic_f simp)
443 ((%asin simp)
444 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 ((mexpt simp) x 2))) ((rat simp) 1 2)))
445 m)))))))
446 grad)
447
448 ;; Let u = inverse_jacobi_dn(x). Then
449 ;;
450 ;; jacobi_dn(u) = x or
451 ;;
452 ;; x^2 = jacobi_dn(u)^2 = 1 - m*jacobi_sn(u)^2
453 ;;
454 ;; so jacobi_sn(u) = sqrt(1-x^2)/sqrt(m)
455 ;;
456 ;; or u = inverse_jacobi_sn(sqrt(1-x^2)/sqrt(m))
457 (defprop %inverse_jacobi_dn
458 ((x m)
459 ;; Whittaker and Watson, 22.121
460 ;; inverse_jacobi_dn(u,m) = integrate(1/sqrt(1-t^2)/sqrt(t^2-(1-m)), t, u, 1)
461 ;; -> -1/sqrt(1-x^2)/sqrt(x^2+m-1)
462 ((mtimes simp)
463 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 ((mexpt simp) x 2)))
464 ((rat simp) -1 2))
465 ((mexpt simp) ((mplus simp) -1 m ((mexpt simp) x 2)) ((rat simp) -1 2)))
466 ((mplus simp)
467 ((mtimes simp) ((rat simp) -1 2) ((mexpt simp) m ((rat simp) -3 2))
468 ((mexpt simp)
469 ((mplus simp) 1
470 ((mtimes simp) -1 ((mexpt simp) m -1)
471 ((mplus simp) 1 ((mtimes simp) -1 ((mexpt simp) x 2)))))
472 ((rat simp) -1 2))
473 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 ((mexpt simp) x 2)))
474 ((rat simp) 1 2))
475 ((mexpt simp) ((mabs simp) x) -1))
476 ((mtimes simp) ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m)) -1)
477 ((mplus simp)
478 ((mtimes simp) -1 ((mexpt simp) m ((rat simp) -1 2))
479 ((mexpt simp)
480 ((mplus simp) 1
481 ((mtimes simp) -1 ((mexpt simp) m -1)
482 ((mplus simp) 1 ((mtimes simp) -1 ((mexpt simp) x 2)))))
483 ((rat simp) 1 2))
484 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 ((mexpt simp) x 2)))
485 ((rat simp) 1 2))
486 ((mexpt simp) ((mabs simp) x) -1))
487 ((mtimes simp) ((mexpt simp) m -1)
488 ((mplus simp)
489 ((%elliptic_e simp)
490 ((%asin simp)
491 ((mtimes simp) ((mexpt simp) m ((rat simp) -1 2))
492 ((mexpt simp) ((mplus simp) 1
493 ((mtimes simp) -1 ((mexpt simp) x 2)))
494 ((rat simp) 1 2))))
495 m)
496 ((mtimes simp) -1 ((mplus simp) 1 ((mtimes simp) -1 m))
497 ((%elliptic_f simp)
498 ((%asin simp)
499 ((mtimes simp) ((mexpt simp) m ((rat simp) -1 2))
500 ((mexpt simp) ((mplus simp) 1
501 ((mtimes simp) -1 ((mexpt simp) x 2)))
502 ((rat simp) 1 2))))
503 m))))))))
504 grad)
505
506
507 ;; Possible forms of a complex number:
508 ;;
509 ;; 2.3
510 ;; $%i
511 ;; ((mplus simp) 2.3 ((mtimes simp) 2.3 $%i))
512 ;; ((mplus simp) 2.3 $%i))
513 ;; ((mtimes simp) 2.3 $%i)
514 ;;
515
516
517 ;; Is argument u a complex number with real and imagpart satisfying predicate ntypep?
518 (defun complex-number-p (u &optional (ntypep 'numberp))
519 (let ((R 0) (I 0))
520 (labels ((a1 (x) (cadr x))
521 (a2 (x) (caddr x))
522 (a3+ (x) (cdddr x))
523 (N (x) (funcall ntypep x)) ; N
524 (i (x) (and (eq x '$%i) (N 1))) ; %i
525 (N+i (x) (and (null (a3+ x)) ; mplus test is precondition
526 (N (setq R (a1 x)))
527 (or (and (i (a2 x)) (setq I 1) t)
528 (and (mtimesp (a2 x)) (N*i (a2 x))))))
529 (N*i (x) (and (null (a3+ x)) ; mtimes test is precondition
530 (N (setq I (a1 x)))
531 (eq (a2 x) '$%i))))
532 (declare (inline a1 a2 a3+ N i N+i N*i))
533 (cond ((N u) (values t u 0)) ;2.3
534 ((atom u) (if (i u) (values t 0 1))) ;%i
535 ((mplusp u) (if (N+i u) (values t R I))) ;N+%i, N+N*%i
536 ((mtimesp u) (if (N*i u) (values t R I))) ;N*%i
537 (t nil)))))
538
539 (defun complexify (x)
540 ;; Convert a Lisp number to a maxima number
541 (cond ((realp x) x)
542 ((complexp x) (add (realpart x) (mul '$%i (imagpart x))))
543 (t (merror (intl:gettext "COMPLEXIFY: argument must be a Lisp real or complex number.~%COMPLEXIFY: found: ~:M") x))))
544
545 (defun kc-arg (exp m)
546 ;; Replace elliptic_kc(m) in the expression with sym. Check to see
547 ;; if the resulting expression is linear in sym and the constant
548 ;; term is zero. If so, return the coefficient of sym, i.e, the
549 ;; coefficient of elliptic_kc(m).
550 (let* ((sym (gensym))
551 (arg (maxima-substitute sym `((%elliptic_kc) ,m) exp)))
552 (if (and (not (equalp arg exp))
553 (linearp arg sym)
554 (zerop1 (coefficient arg sym 0)))
555 (coefficient arg sym 1)
556 nil)))
557
558 (defun kc-arg2 (exp m)
559 ;; Replace elliptic_kc(m) in the expression with sym. Check to see
560 ;; if the resulting expression is linear in sym and the constant
561 ;; term is zero. If so, return the coefficient of sym, i.e, the
562 ;; coefficient of elliptic_kc(m), and the constant term. Otherwise,
563 ;; return NIL.
564 (let* ((sym (gensym))
565 (arg (maxima-substitute sym `((%elliptic_kc) ,m) exp)))
566 (if (and (not (equalp arg exp))
567 (linearp arg sym))
568 (list (coefficient arg sym 1)
569 (coefficient arg sym 0))
570 nil)))
571
572 ;; Tell maxima how to simplify the functions
573
574 (def-simplifier jacobi_sn (u m)
575 (let (coef args)
576 (cond
577 ((float-numerical-eval-p u m)
578 (to (bigfloat::sn (bigfloat:to ($float u)) (bigfloat:to ($float m)))))
579 ((setf args (complex-float-numerical-eval-p u m))
580 (destructuring-bind (u m)
581 args
582 (to (bigfloat::sn (bigfloat:to ($float u)) (bigfloat:to ($float m))))))
583 ((bigfloat-numerical-eval-p u m)
584 (to (bigfloat::sn (bigfloat:to ($bfloat u)) (bigfloat:to ($bfloat m)))))
585 ((setf args (complex-bigfloat-numerical-eval-p u m))
586 (destructuring-bind (u m)
587 args
588 (to (bigfloat::sn (bigfloat:to ($bfloat u)) (bigfloat:to ($bfloat m))))))
589 ((zerop1 u)
590 ;; A&S 16.5.1
591 0)
592 ((zerop1 m)
593 ;; A&S 16.6.1
594 (ftake '%sin u))
595 ((onep1 m)
596 ;; A&S 16.6.1
597 (ftake '%tanh u))
598 ((and $trigsign (mminusp* u))
599 (neg (ftake* '%jacobi_sn (neg u) m)))
600 ((and $triginverses
601 (listp u)
602 (member (caar u) '(%inverse_jacobi_sn
603 %inverse_jacobi_ns
604 %inverse_jacobi_cn
605 %inverse_jacobi_nc
606 %inverse_jacobi_dn
607 %inverse_jacobi_nd
608 %inverse_jacobi_sc
609 %inverse_jacobi_cs
610 %inverse_jacobi_sd
611 %inverse_jacobi_ds
612 %inverse_jacobi_cd
613 %inverse_jacobi_dc))
614 (alike1 (third u) m))
615 (let ((inv-arg (second u)))
616 (ecase (caar u)
617 (%inverse_jacobi_sn
618 ;; jacobi_sn(inverse_jacobi_sn(u,m), m) = u
619 inv-arg)
620 (%inverse_jacobi_ns
621 ;; inverse_jacobi_ns(u,m) = inverse_jacobi_sn(1/u,m)
622 (div 1 inv-arg))
623 (%inverse_jacobi_cn
624 ;; sn(x)^2 + cn(x)^2 = 1 so sn(x) = sqrt(1-cn(x)^2)
625 (power (sub 1 (mul inv-arg inv-arg)) 1//2))
626 (%inverse_jacobi_nc
627 ;; inverse_jacobi_nc(u) = inverse_jacobi_cn(1/u)
628 (ftake '%jacobi_sn (ftake '%inverse_jacobi_cn (div 1 inv-arg) m)
629 m))
630 (%inverse_jacobi_dn
631 ;; dn(x)^2 + m*sn(x)^2 = 1 so
632 ;; sn(x) = 1/sqrt(m)*sqrt(1-dn(x)^2)
633 (mul (div 1 (power m 1//2))
634 (power (sub 1 (mul inv-arg inv-arg)) 1//2)))
635 (%inverse_jacobi_nd
636 ;; inverse_jacobi_nd(u) = inverse_jacobi_dn(1/u)
637 (ftake '%jacobi_sn (ftake '%inverse_jacobi_dn (div 1 inv-arg) m)
638 m))
639 (%inverse_jacobi_sc
640 ;; See below for inverse_jacobi_sc.
641 (div inv-arg (power (add 1 (mul inv-arg inv-arg)) 1//2)))
642 (%inverse_jacobi_cs
643 ;; inverse_jacobi_cs(u) = inverse_jacobi_sc(1/u)
644 (ftake '%jacobi_sn (ftake '%inverse_jacobi_sc (div 1 inv-arg) m)
645 m))
646 (%inverse_jacobi_sd
647 ;; See below for inverse_jacobi_sd
648 (div inv-arg (power (add 1 (mul m (mul inv-arg inv-arg))) 1//2)))
649 (%inverse_jacobi_ds
650 ;; inverse_jacobi_ds(u) = inverse_jacobi_sd(1/u)
651 (ftake '%jacobi_sn (ftake '%inverse_jacobi_sd (div 1 inv-arg) m)
652 m))
653 (%inverse_jacobi_cd
654 ;; See below
655 (div (power (sub 1 (mul inv-arg inv-arg)) 1//2)
656 (power (sub 1 (mul m (mul inv-arg inv-arg))) 1//2)))
657 (%inverse_jacobi_dc
658 (ftake '%jacobi_sn (ftake '%inverse_jacobi_cd (div 1 inv-arg) m) m)))))
659 ;; A&S 16.20.1 (Jacobi's Imaginary transformation)
660 ((and $%iargs (multiplep u '$%i))
661 (mul '$%i
662 (ftake* '%jacobi_sc (coeff u '$%i 1) (add 1 (neg m)))))
663 ((setq coef (kc-arg2 u m))
664 ;; sn(m*K+u)
665 ;;
666 ;; A&S 16.8.1
667 (destructuring-bind (lin const)
668 coef
669 (cond ((integerp lin)
670 (ecase (mod lin 4)
671 (0
672 ;; sn(4*m*K + u) = sn(u), sn(0) = 0
673 (if (zerop1 const)
674 0
675 (ftake '%jacobi_sn const m)))
676 (1
677 ;; sn(4*m*K + K + u) = sn(K+u) = cd(u)
678 ;; sn(K) = 1
679 (if (zerop1 const)
680 1
681 (ftake '%jacobi_cd const m)))
682 (2
683 ;; sn(4*m*K+2*K + u) = sn(2*K+u) = -sn(u)
684 ;; sn(2*K) = 0
685 (if (zerop1 const)
686 0
687 (neg (ftake '%jacobi_sn const m))))
688 (3
689 ;; sn(4*m*K+3*K+u) = sn(2*K + K + u) = -sn(K+u) = -cd(u)
690 ;; sn(3*K) = -1
691 (if (zerop1 const)
692 -1
693 (neg (ftake '%jacobi_cd const m))))))
694 ((and (alike1 lin 1//2)
695 (zerop1 const))
696 ;; A&S 16.5.2
697 ;;
698 ;; sn(1/2*K) = 1/sqrt(1+sqrt(1-m))
699 (div 1
700 (power (add 1 (power (sub 1 m) 1//2))
701 1//2)))
702 ((and (alike1 lin 3//2)
703 (zerop1 const))
704 ;; A&S 16.5.2
705 ;;
706 ;; sn(1/2*K + K) = cd(1/2*K,m)
707 (ftake '%jacobi_cd (mul 1//2
708 (ftake '%elliptic_kc m))
709 m))
710 (t
711 (give-up)))))
712 (t
713 ;; Nothing to do
714 (give-up)))))
715
716 (def-simplifier jacobi_cn (u m)
717 (let (coef args)
718 (cond
719 ((float-numerical-eval-p u m)
720 (to (bigfloat::cn (bigfloat:to ($float u)) (bigfloat:to ($float m)))))
721 ((setf args (complex-float-numerical-eval-p u m))
722 (destructuring-bind (u m)
723 args
724 (to (bigfloat::cn (bigfloat:to ($float u)) (bigfloat:to ($float m))))))
725 ((bigfloat-numerical-eval-p u m)
726 (to (bigfloat::cn (bigfloat:to ($bfloat u)) (bigfloat:to ($bfloat m)))))
727 ((setf args (complex-bigfloat-numerical-eval-p u m))
728 (destructuring-bind (u m)
729 args
730 (to (bigfloat::cn (bigfloat:to ($bfloat u)) (bigfloat:to ($bfloat m))))))
731 ((zerop1 u)
732 ;; A&S 16.5.1
733 1)
734 ((zerop1 m)
735 ;; A&S 16.6.2
736 (ftake '%cos u))
737 ((onep1 m)
738 ;; A&S 16.6.2
739 (ftake '%sech u))
740 ((and $trigsign (mminusp* u))
741 (ftake* '%jacobi_cn (neg u) m))
742 ((and $triginverses
743 (listp u)
744 (member (caar u) '(%inverse_jacobi_sn
745 %inverse_jacobi_ns
746 %inverse_jacobi_cn
747 %inverse_jacobi_nc
748 %inverse_jacobi_dn
749 %inverse_jacobi_nd
750 %inverse_jacobi_sc
751 %inverse_jacobi_cs
752 %inverse_jacobi_sd
753 %inverse_jacobi_ds
754 %inverse_jacobi_cd
755 %inverse_jacobi_dc))
756 (alike1 (third u) m))
757 (cond ((eq (caar u) '%inverse_jacobi_cn)
758 (second u))
759 (t
760 ;; I'm lazy. Use cn(x) = sqrt(1-sn(x)^2). Hope
761 ;; this is right.
762 (power (sub 1 (power (ftake '%jacobi_sn u (third u)) 2))
763 1//2))))
764 ;; A&S 16.20.2 (Jacobi's Imaginary transformation)
765 ((and $%iargs (multiplep u '$%i))
766 (ftake* '%jacobi_nc (coeff u '$%i 1) (add 1 (neg m))))
767 ((setq coef (kc-arg2 u m))
768 ;; cn(m*K+u)
769 ;;
770 ;; A&S 16.8.2
771 (destructuring-bind (lin const)
772 coef
773 (cond ((integerp lin)
774 (ecase (mod lin 4)
775 (0
776 ;; cn(4*m*K + u) = cn(u),
777 ;; cn(0) = 1
778 (if (zerop1 const)
779 1
780 (ftake '%jacobi_cn const m)))
781 (1
782 ;; cn(4*m*K + K + u) = cn(K+u) = -sqrt(m1)*sd(u)
783 ;; cn(K) = 0
784 (if (zerop1 const)
785 0
786 (neg (mul (power (sub 1 m) 1//2)
787 (ftake '%jacobi_sd const m)))))
788 (2
789 ;; cn(4*m*K + 2*K + u) = cn(2*K+u) = -cn(u)
790 ;; cn(2*K) = -1
791 (if (zerop1 const)
792 -1
793 (neg (ftake '%jacobi_cn const m))))
794 (3
795 ;; cn(4*m*K + 3*K + u) = cn(2*K + K + u) =
796 ;; -cn(K+u) = sqrt(m1)*sd(u)
797 ;;
798 ;; cn(3*K) = 0
799 (if (zerop1 const)
800 0
801 (mul (power (sub 1 m) 1//2)
802 (ftake '%jacobi_sd const m))))))
803 ((and (alike1 lin 1//2)
804 (zerop1 const))
805 ;; A&S 16.5.2
806 ;; cn(1/2*K) = (1-m)^(1/4)/sqrt(1+sqrt(1-m))
807 (mul (power (sub 1 m) (div 1 4))
808 (power (add 1
809 (power (sub 1 m)
810 1//2))
811 1//2)))
812 (t
813 (give-up)))))
814 (t
815 (give-up)))))
816
817 (def-simplifier jacobi_dn (u m)
818 (let (coef args)
819 (cond
820 ((float-numerical-eval-p u m)
821 (to (bigfloat::dn (bigfloat:to ($float u)) (bigfloat:to ($float m)))))
822 ((setf args (complex-float-numerical-eval-p u m))
823 (destructuring-bind (u m)
824 args
825 (to (bigfloat::dn (bigfloat:to ($float u)) (bigfloat:to ($float m))))))
826 ((bigfloat-numerical-eval-p u m)
827 (to (bigfloat::dn (bigfloat:to ($bfloat u)) (bigfloat:to ($bfloat m)))))
828 ((setf args (complex-bigfloat-numerical-eval-p u m))
829 (destructuring-bind (u m)
830 args
831 (to (bigfloat::dn (bigfloat:to ($bfloat u)) (bigfloat:to ($bfloat m))))))
832 ((zerop1 u)
833 ;; A&S 16.5.1
834 1)
835 ((zerop1 m)
836 ;; A&S 16.6.3
837 1)
838 ((onep1 m)
839 ;; A&S 16.6.3
840 (ftake '%sech u))
841 ((and $trigsign (mminusp* u))
842 (ftake* '%jacobi_dn (neg u) m))
843 ((and $triginverses
844 (listp u)
845 (member (caar u) '(%inverse_jacobi_sn
846 %inverse_jacobi_ns
847 %inverse_jacobi_cn
848 %inverse_jacobi_nc
849 %inverse_jacobi_dn
850 %inverse_jacobi_nd
851 %inverse_jacobi_sc
852 %inverse_jacobi_cs
853 %inverse_jacobi_sd
854 %inverse_jacobi_ds
855 %inverse_jacobi_cd
856 %inverse_jacobi_dc))
857 (alike1 (third u) m))
858 (cond ((eq (caar u) '%inverse_jacobi_dn)
859 ;; jacobi_dn(inverse_jacobi_dn(u,m), m) = u
860 (second u))
861 (t
862 ;; Express in terms of sn:
863 ;; dn(x) = sqrt(1-m*sn(x)^2)
864 (power (sub 1 (mul m
865 (power (ftake '%jacobi_sn u m) 2)))
866 1//2))))
867 ((zerop1 ($ratsimp (sub u (power (sub 1 m) 1//2))))
868 ;; A&S 16.5.3
869 ;; dn(sqrt(1-m),m) = K(m)
870 (ftake '%elliptic_kc m))
871 ;; A&S 16.20.2 (Jacobi's Imaginary transformation)
872 ((and $%iargs (multiplep u '$%i))
873 (ftake* '%jacobi_dc (coeff u '$%i 1)
874 (add 1 (neg m))))
875 ((setq coef (kc-arg2 u m))
876 ;; A&S 16.8.3
877 ;;
878 ;; dn(m*K+u) has period 2K
879 ;;
880 (destructuring-bind (lin const)
881 coef
882 (cond ((integerp lin)
883 (ecase (mod lin 2)
884 (0
885 ;; dn(2*m*K + u) = dn(u)
886 ;; dn(0) = 1
887 (if (zerop1 const)
888 1
889 ;; dn(4*m*K+2*K + u) = dn(2*K+u) = dn(u)
890 (ftake '%jacobi_dn const m)))
891 (1
892 ;; dn(2*m*K + K + u) = dn(K + u) = sqrt(1-m)*nd(u)
893 ;; dn(K) = sqrt(1-m)
894 (if (zerop1 const)
895 (power (sub 1 m) 1//2)
896 (mul (power (sub 1 m) 1//2)
897 (ftake '%jacobi_nd const m))))))
898 ((and (alike1 lin 1//2)
899 (zerop1 const))
900 ;; A&S 16.5.2
901 ;; dn(1/2*K) = (1-m)^(1/4)
902 (power (sub 1 m)
903 (div 1 4)))
904 (t
905 (give-up)))))
906 (t (give-up)))))
907
908 ;; Should we simplify the inverse elliptic functions into the
909 ;; appropriate incomplete elliptic integral? I think we should leave
910 ;; it, but perhaps allow some way to do that transformation if
911 ;; desired.
912
913 (def-simplifier inverse_jacobi_sn (u m)
914 (let (args)
915 ;; To numerically evaluate inverse_jacobi_sn (asn), use
916 ;;
917 ;; asn(x,m) = F(asin(x),m)
918 ;;
919 ;; But F(phi,m) = sin(phi)*rf(cos(phi)^2, 1-m*sin(phi)^2,1). Thus
920 ;;
921 ;; asn(x,m) = F(asin(x),m)
922 ;; = x*rf(1-x^2,1-m*x^2,1)
923 ;;
924 ;; I (rtoy) am not 100% about the first identity above for all
925 ;; complex values of x and m, but tests seem to indicate that it
926 ;; produces the correct value as verified by verifying
927 ;; jacobi_sn(inverse_jacobi_sn(x,m),m) = x.
928 (cond ((float-numerical-eval-p u m)
929 (let ((uu (bigfloat:to ($float u)))
930 (mm (bigfloat:to ($float m))))
931 (complexify
932 (* uu
933 (bigfloat::bf-rf (bigfloat:to (- 1 (* uu uu)))
934 (bigfloat:to (- 1 (* mm uu uu)))
935 1)))))
936 ((setf args (complex-float-numerical-eval-p u m))
937 (destructuring-bind (u m)
938 args
939 (let ((uu (bigfloat:to ($float u)))
940 (mm (bigfloat:to ($float m))))
941 (complexify (* uu (bigfloat::bf-rf (- 1 (* uu uu))
942 (- 1 (* mm uu uu))
943 1))))))
944 ((bigfloat-numerical-eval-p u m)
945 (let ((uu (bigfloat:to u))
946 (mm (bigfloat:to m)))
947 (to (bigfloat:* uu
948 (bigfloat::bf-rf (bigfloat:- 1 (bigfloat:* uu uu))
949 (bigfloat:- 1 (bigfloat:* mm uu uu))
950 1)))))
951 ((setf args (complex-bigfloat-numerical-eval-p u m))
952 (destructuring-bind (u m)
953 args
954 (let ((uu (bigfloat:to u))
955 (mm (bigfloat:to m)))
956 (to (bigfloat:* uu
957 (bigfloat::bf-rf (bigfloat:- 1 (bigfloat:* uu uu))
958 (bigfloat:- 1 (bigfloat:* mm uu uu))
959 1))))))
960 ((zerop1 u)
961 ;; asn(0,m) = 0
962 0)
963 ((onep1 u)
964 ;; asn(1,m) = elliptic_kc(m)
965 (ftake '%elliptic_kc m))
966 ((and (numberp u) (onep1 (- u)))
967 ;; asn(-1,m) = -elliptic_kc(m)
968 (mul -1 (ftake '%elliptic_kc m)))
969 ((zerop1 m)
970 ;; asn(x,0) = F(asin(x),0) = asin(x)
971 (ftake '%asin u))
972 ((onep1 m)
973 ;; asn(x,1) = F(asin(x),1) = log(tan(pi/4+asin(x)/2))
974 (ftake '%elliptic_f (ftake '%asin u) 1))
975 ((and (eq $triginverses '$all)
976 (listp u)
977 (eq (caar u) '%jacobi_sn)
978 (alike1 (third u) m))
979 ;; inverse_jacobi_sn(sn(u)) = u
980 (second u))
981 (t
982 ;; Nothing to do
983 (give-up)))))
984
985 (def-simplifier inverse_jacobi_cn (u m)
986 (let (args)
987 (cond ((float-numerical-eval-p u m)
988 ;; Numerically evaluate acn
989 ;;
990 ;; acn(x,m) = F(acos(x),m)
991 (to (elliptic-f (cl:acos ($float u)) ($float m))))
992 ((setf args (complex-float-numerical-eval-p u m))
993 (destructuring-bind (u m)
994 args
995 (to (elliptic-f (cl:acos (bigfloat:to ($float u)))
996 (bigfloat:to ($float m))))))
997 ((bigfloat-numerical-eval-p u m)
998 (to (bigfloat::bf-elliptic-f (bigfloat:acos (bigfloat:to u))
999 (bigfloat:to m))))
1000 ((setf args (complex-bigfloat-numerical-eval-p u m))
1001 (destructuring-bind (u m)
1002 args
1003 (to (bigfloat::bf-elliptic-f (bigfloat:acos (bigfloat:to u))
1004 (bigfloat:to m)))))
1005 ((zerop1 m)
1006 ;; asn(x,0) = F(acos(x),0) = acos(x)
1007 (ftake '%elliptic_f (ftake '%acos u) 0))
1008 ((onep1 m)
1009 ;; asn(x,1) = F(asin(x),1) = log(tan(pi/2+asin(x)/2))
1010 (ftake '%elliptic_f (ftake '%acos u) 1))
1011 ((zerop1 u)
1012 (ftake '%elliptic_kc m))
1013 ((onep1 u)
1014 0)
1015 ((and (eq $triginverses '$all)
1016 (listp u)
1017 (eq (caar u) '%jacobi_cn)
1018 (alike1 (third u) m))
1019 ;; inverse_jacobi_cn(cn(u)) = u
1020 (second u))
1021 (t
1022 ;; Nothing to do
1023 (give-up)))))
1024
1025 (def-simplifier inverse_jacobi_dn (u m)
1026 (let (args)
1027 (cond ((float-numerical-eval-p u m)
1028 (to (bigfloat::bf-inverse-jacobi-dn (bigfloat:to (float u))
1029 (bigfloat:to (float m)))))
1030 ((setf args (complex-float-numerical-eval-p u m))
1031 (destructuring-bind (u m)
1032 args
1033 (let ((uu (bigfloat:to ($float u)))
1034 (mm (bigfloat:to ($float m))))
1035 (to (bigfloat::bf-inverse-jacobi-dn uu mm)))))
1036 ((bigfloat-numerical-eval-p u m)
1037 (let ((uu (bigfloat:to u))
1038 (mm (bigfloat:to m)))
1039 (to (bigfloat::bf-inverse-jacobi-dn uu mm))))
1040 ((setf args (complex-bigfloat-numerical-eval-p u m))
1041 (destructuring-bind (u m)
1042 args
1043 (to (bigfloat::bf-inverse-jacobi-dn (bigfloat:to u) (bigfloat:to m)))))
1044 ((onep1 m)
1045 ;; x = dn(u,1) = sech(u). so u = asech(x)
1046 (ftake '%asech u))
1047 ((onep1 u)
1048 ;; jacobi_dn(0,m) = 1
1049 0)
1050 ((zerop1 ($ratsimp (sub u (power (sub 1 m) 1//2))))
1051 ;; jacobi_dn(K(m),m) = sqrt(1-m) so
1052 ;; inverse_jacobi_dn(sqrt(1-m),m) = K(m)
1053 (ftake '%elliptic_kc m))
1054 ((and (eq $triginverses '$all)
1055 (listp u)
1056 (eq (caar u) '%jacobi_dn)
1057 (alike1 (third u) m))
1058 ;; inverse_jacobi_dn(dn(u)) = u
1059 (second u))
1060 (t
1061 ;; Nothing to do
1062 (give-up)))))
1063
1064 ;;;; Elliptic integrals
1065
1066 (let ((errtol (expt (* 4 flonum-epsilon) 1/6))
1067 (c1 (float 1/24))
1068 (c2 (float 3/44))
1069 (c3 (float 1/14)))
1070 (declare (type flonum errtol c1 c2 c3))
1071 (defun crf (x y z)
1072 "Compute Carlson's incomplete or complete elliptic integral of the
1073 first kind:
1074
1075 INF
1076 /
1077 [ 1
1078 RF(x, y, z) = I ----------------------------------- dt
1079 ] SQRT(x + t) SQRT(y + t) SQRT(z + t)
1080 /
1081 0
1082
1083 x, y, and z may be complex.
1084 "
1085 (declare (number x y z))
1086 (let ((x (coerce x '(complex flonum)))
1087 (y (coerce y '(complex flonum)))
1088 (z (coerce z '(complex flonum))))
1089 (declare (type (complex flonum) x y z))
1090 (loop
1091 (let* ((mu (/ (+ x y z) 3))
1092 (x-dev (- 2 (/ (+ mu x) mu)))
1093 (y-dev (- 2 (/ (+ mu y) mu)))
1094 (z-dev (- 2 (/ (+ mu z) mu))))
1095 (when (< (max (abs x-dev) (abs y-dev) (abs z-dev)) errtol)
1096 (let ((e2 (- (* x-dev y-dev) (* z-dev z-dev)))
1097 (e3 (* x-dev y-dev z-dev)))
1098 (return (/ (+ 1
1099 (* e2 (- (* c1 e2)
1100 1/10
1101 (* c2 e3)))
1102 (* c3 e3))
1103 (sqrt mu)))))
1104 (let* ((x-root (sqrt x))
1105 (y-root (sqrt y))
1106 (z-root (sqrt z))
1107 (lam (+ (* x-root (+ y-root z-root)) (* y-root z-root))))
1108 (setf x (* (+ x lam) 1/4))
1109 (setf y (* (+ y lam) 1/4))
1110 (setf z (* (+ z lam) 1/4))))))))
1111
1112 ;; Elliptic integral of the first kind (Legendre's form):
1113 ;;
1114 ;;
1115 ;; phi
1116 ;; /
1117 ;; [ 1
1118 ;; I ------------------- ds
1119 ;; ] 2
1120 ;; / SQRT(1 - m SIN (s))
1121 ;; 0
1122
1123 (defun elliptic-f (phi-arg m-arg)
1124 (flet ((base (phi-arg m-arg)
1125 (cond ((and (realp m-arg) (realp phi-arg))
1126 (let ((phi (float phi-arg))
1127 (m (float m-arg)))
1128 (cond ((> m 1)
1129 ;; A&S 17.4.15
1130 ;;
1131 ;; F(phi|m) = 1/sqrt(m)*F(theta|1/m)
1132 ;;
1133 ;; with sin(theta) = sqrt(m)*sin(phi)
1134 (/ (elliptic-f (cl:asin (* (sqrt m) (sin phi))) (/ m))
1135 (sqrt m)))
1136 ((< m 0)
1137 ;; A&S 17.4.17
1138 (let* ((m (- m))
1139 (m+1 (+ 1 m))
1140 (root (sqrt m+1))
1141 (m/m+1 (/ m m+1)))
1142 (- (/ (elliptic-f (float (/ pi 2)) m/m+1)
1143 root)
1144 (/ (elliptic-f (- (float (/ pi 2)) phi) m/m+1)
1145 root))))
1146 ((= m 0)
1147 ;; A&S 17.4.19
1148 phi)
1149 ((= m 1)
1150 ;; A&S 17.4.21
1151 ;;
1152 1 ;; F(phi,1) = log(sec(phi)+tan(phi))
1153 ;; = log(tan(pi/4+pi/2))
1154 (log (cl:tan (+ (/ phi 2) (float (/ pi 4))))))
1155 ((minusp phi)
1156 (- (elliptic-f (- phi) m)))
1157 ((> phi pi)
1158 ;; A&S 17.4.3
1159 (multiple-value-bind (s phi-rem)
1160 (truncate phi (float pi))
1161 (+ (* 2 s (elliptic-k m))
1162 (elliptic-f phi-rem m))))
1163 ((<= phi (/ pi 2))
1164 (let ((sin-phi (sin phi))
1165 (cos-phi (cos phi))
1166 (k (sqrt m)))
1167 (* sin-phi
1168 (bigfloat::bf-rf (* cos-phi cos-phi)
1169 (* (- 1 (* k sin-phi))
1170 (+ 1 (* k sin-phi)))
1171 1.0))))
1172 ((< phi pi)
1173 (+ (* 2 (elliptic-k m))
1174 (elliptic-f (- phi (float pi)) m)))
1175 (t
1176 (error "Shouldn't happen! Unhandled case in elliptic-f: ~S ~S~%"
1177 phi-arg m-arg)))))
1178 (t
1179 (let ((phi (coerce phi-arg '(complex flonum)))
1180 (m (coerce m-arg '(complex flonum))))
1181 (let ((sin-phi (sin phi))
1182 (cos-phi (cos phi))
1183 (k (sqrt m)))
1184 (* sin-phi
1185 (crf (* cos-phi cos-phi)
1186 (* (- 1 (* k sin-phi))
1187 (+ 1 (* k sin-phi)))
1188 1.0))))))))
1189 ;; Elliptic F is quasi-periodic wrt to z:
1190 ;;
1191 ;; F(z|m) = F(z - pi*round(Re(z)/pi)|m) + 2*round(Re(z)/pi)*K(m)
1192 (let ((period (round (realpart phi-arg) pi)))
1193 (+ (base (- phi-arg (* pi period)) m-arg)
1194 (if (zerop period)
1195 0
1196 (* 2 period
1197 (bigfloat:to (elliptic-k m-arg))))))))
1198
1199 ;; Complete elliptic integral of the first kind
1200 (defun elliptic-k (m)
1201 (cond ((realp m)
1202 (cond ((< m 0)
1203 ;; A&S 17.4.17
1204 (let* ((m (- m))
1205 (m+1 (+ 1 m))
1206 (root (sqrt m+1))
1207 (m/m+1 (/ m m+1)))
1208 (- (/ (elliptic-k m/m+1)
1209 root)
1210 (/ (elliptic-f 0.0 m/m+1)
1211 root))))
1212 ((= m 0)
1213 ;; A&S 17.4.19
1214 (float (/ pi 2)))
1215 ((= m 1)
1216 (maxima::merror
1217 (intl:gettext "elliptic_kc: elliptic_kc(1) is undefined.")))
1218 (t
1219 (bigfloat::bf-rf 0.0 (- 1 m)
1220 1.0))))
1221 (t
1222 (bigfloat::bf-rf 0.0 (- 1 m)
1223 1.0))))
1224
1225 ;; Elliptic integral of the second kind (Legendre's form):
1226 ;;
1227 ;;
1228 ;; phi
1229 ;; /
1230 ;; [ 2
1231 ;; I SQRT(1 - m SIN (s)) ds
1232 ;; ]
1233 ;; /
1234 ;; 0
1235
1236 (defun elliptic-e (phi m)
1237 (declare (type flonum phi m))
1238 (flet ((base (phi m)
1239 (cond ((= m 0)
1240 ;; A&S 17.4.23
1241 phi)
1242 ((= m 1)
1243 ;; A&S 17.4.25
1244 (sin phi))
1245 (t
1246 (let* ((sin-phi (sin phi))
1247 (cos-phi (cos phi))
1248 (k (sqrt m))
1249 (y (* (- 1 (* k sin-phi))
1250 (+ 1 (* k sin-phi)))))
1251 (to (- (* sin-phi
1252 (bigfloat::bf-rf (* cos-phi cos-phi) y 1.0))
1253 (* (/ m 3)
1254 (expt sin-phi 3)
1255 (bigfloat::bf-rd (* cos-phi cos-phi) y 1.0)))))))))
1256 ;; Elliptic E is quasi-periodic wrt to phi:
1257 ;;
1258 ;; E(z|m) = E(z - %pi*round(Re(z)/%pi)|m) + 2*round(Re(z)/%pi)*E(m)
1259 (let ((period (round (realpart phi) pi)))
1260 (+ (base (- phi (* pi period)) m)
1261 (* 2 period (elliptic-ec m))))))
1262
1263 ;; Complete version
1264 (defun elliptic-ec (m)
1265 (declare (type flonum m))
1266 (cond ((= m 0)
1267 ;; A&S 17.4.23
1268 (float (/ pi 2)))
1269 ((= m 1)
1270 ;; A&S 17.4.25
1271 1.0)
1272 (t
1273 (let* ((y (- 1 m)))
1274 (to (- (bigfloat::bf-rf 0.0 y 1.0)
1275 (* (/ m 3)
1276 (bigfloat::bf-rd 0.0 y 1.0))))))))
1277
1278
1279 ;; Define the elliptic integrals for maxima
1280 ;;
1281 ;; We use the definitions given in A&S 17.2.6 and 17.2.8. In particular:
1282 ;;
1283 ;; phi
1284 ;; /
1285 ;; [ 1
1286 ;; F(phi|m) = I ------------------- ds
1287 ;; ] 2
1288 ;; / SQRT(1 - m SIN (s))
1289 ;; 0
1290 ;;
1291 ;; and
1292 ;;
1293 ;; phi
1294 ;; /
1295 ;; [ 2
1296 ;; E(phi|m) = I SQRT(1 - m SIN (s)) ds
1297 ;; ]
1298 ;; /
1299 ;; 0
1300 ;;
1301 ;; That is, we do not use the modular angle, alpha, as the second arg;
1302 ;; the parameter m = sin(alpha)^2 is used.
1303 ;;
1304
1305
1306 ;; The derivative of F(phi|m) wrt to phi is easy. The derivative wrt
1307 ;; to m is harder. Here is a derivation. Hope I got it right.
1308 ;;
1309 ;; diff(integrate(1/sqrt(1-m*sin(x)^2),x,0,phi), m);
1310 ;;
1311 ;; PHI
1312 ;; / 2
1313 ;; [ SIN (x)
1314 ;; I ------------------ dx
1315 ;; ] 2 3/2
1316 ;; / (1 - m SIN (x))
1317 ;; 0
1318 ;; --------------------------
1319 ;; 2
1320 ;;
1321 ;;
1322 ;; Now use the following relationship that is easily verified:
1323 ;;
1324 ;; 2 2
1325 ;; (1 - m) SIN (x) COS (x) COS(x) SIN(x)
1326 ;; ------------------- = ------------------- - DIFF(-------------------, x)
1327 ;; 2 2 2
1328 ;; SQRT(1 - m SIN (x)) SQRT(1 - m SIN (x)) SQRT(1 - m SIN (x))
1329 ;;
1330 ;;
1331 ;; Now integrate this to get:
1332 ;;
1333 ;;
1334 ;; PHI
1335 ;; / 2
1336 ;; [ SIN (x)
1337 ;; (1 - m) I ------------------- dx =
1338 ;; ] 2
1339 ;; / SQRT(1 - m SIN (x))
1340 ;; 0
1341
1342 ;;
1343 ;; PHI
1344 ;; / 2
1345 ;; [ COS (x)
1346 ;; + I ------------------- dx
1347 ;; ] 2
1348 ;; / SQRT(1 - m SIN (x))
1349 ;; 0
1350 ;; COS(PHI) SIN(PHI)
1351 ;; - ---------------------
1352 ;; 2
1353 ;; SQRT(1 - m SIN (PHI))
1354 ;;
1355 ;; Use the fact that cos(x)^2 = 1 - sin(x)^2 to show that this
1356 ;; integral on the RHS is:
1357 ;;
1358 ;;
1359 ;; (1 - m) elliptic_F(PHI, m) + elliptic_E(PHI, m)
1360 ;; -------------------------------------------
1361 ;; m
1362 ;; So, finally, we have
1363 ;;
1364 ;;
1365 ;;
1366 ;; d
1367 ;; 2 -- (elliptic_F(PHI, m)) =
1368 ;; dm
1369 ;;
1370 ;; elliptic_E(PHI, m) - (1 - m) elliptic_F(PHI, m) COS(PHI) SIN(PHI)
1371 ;; ---------------------------------------------- - ---------------------
1372 ;; m 2
1373 ;; SQRT(1 - m SIN (PHI))
1374 ;; ----------------------------------------------------------------------
1375 ;; 1 - m
1376
1377 (defprop %elliptic_f
1378 ((phi m)
1379 ;; diff wrt phi
1380 ;; 1/sqrt(1-m*sin(phi)^2)
1381 ((mexpt simp)
1382 ((mplus simp) 1 ((mtimes simp) -1 m ((mexpt simp) ((%sin simp) phi) 2)))
1383 ((rat simp) -1 2))
1384 ;; diff wrt m
1385 ((mtimes simp) ((rat simp) 1 2)
1386 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m)) -1)
1387 ((mplus simp)
1388 ((mtimes simp) ((mexpt simp) m -1)
1389 ((mplus simp) ((%elliptic_e simp) phi m)
1390 ((mtimes simp) -1 ((mplus simp) 1 ((mtimes simp) -1 m))
1391 ((%elliptic_f simp) phi m))))
1392 ((mtimes simp) -1 ((%cos simp) phi) ((%sin simp) phi)
1393 ((mexpt simp)
1394 ((mplus simp) 1
1395 ((mtimes simp) -1 m ((mexpt simp) ((%sin simp) phi) 2)))
1396 ((rat simp) -1 2))))))
1397 grad)
1398
1399 ;;
1400 ;; The derivative of E(phi|m) wrt to m is much simpler to derive than F(phi|m).
1401 ;;
1402 ;; Take the derivative of the definition to get
1403 ;;
1404 ;; PHI
1405 ;; / 2
1406 ;; [ SIN (x)
1407 ;; I ------------------- dx
1408 ;; ] 2
1409 ;; / SQRT(1 - m SIN (x))
1410 ;; 0
1411 ;; - ---------------------------
1412 ;; 2
1413 ;;
1414 ;; It is easy to see that
1415 ;;
1416 ;; PHI
1417 ;; / 2
1418 ;; [ SIN (x)
1419 ;; elliptic_F(PHI, m) - m I ------------------- dx = elliptic_E(PHI, m)
1420 ;; ] 2
1421 ;; / SQRT(1 - m SIN (x))
1422 ;; 0
1423 ;;
1424 ;; So we finally have
1425 ;;
1426 ;; d elliptic_E(PHI, m) - elliptic_F(PHI, m)
1427 ;; -- (elliptic_E(PHI, m)) = ---------------------------------------
1428 ;; dm 2 m
1429
1430 (defprop %elliptic_e
1431 ((phi m)
1432 ;; sqrt(1-m*sin(phi)^2)
1433 ((mexpt simp)
1434 ((mplus simp) 1 ((mtimes simp) -1 m ((mexpt simp) ((%sin simp) phi) 2)))
1435 ((rat simp) 1 2))
1436 ;; diff wrt m
1437 ((mtimes simp) ((rat simp) 1 2) ((mexpt simp) m -1)
1438 ((mplus simp) ((%elliptic_e simp) phi m)
1439 ((mtimes simp) -1 ((%elliptic_f simp) phi m)))))
1440 grad)
1441
1442 (def-simplifier elliptic_f (phi m)
1443 (let (args)
1444 (cond ((float-numerical-eval-p phi m)
1445 ;; Numerically evaluate it
1446 (to (elliptic-f ($float phi) ($float m))))
1447 ((setf args (complex-float-numerical-eval-p phi m))
1448 (destructuring-bind (phi m)
1449 args
1450 (to (elliptic-f (bigfloat:to ($float phi))
1451 (bigfloat:to ($float m))))))
1452 ((bigfloat-numerical-eval-p phi m)
1453 (to (bigfloat::bf-elliptic-f (bigfloat:to ($bfloat phi))
1454 (bigfloat:to ($bfloat m)))))
1455 ((setf args (complex-bigfloat-numerical-eval-p phi m))
1456 (destructuring-bind (phi m)
1457 args
1458 (to (bigfloat::bf-elliptic-f (bigfloat:to ($bfloat phi))
1459 (bigfloat:to ($bfloat m))))))
1460 ((zerop1 phi)
1461 0)
1462 ((zerop1 m)
1463 ;; A&S 17.4.19
1464 phi)
1465 ((onep1 m)
1466 ;; A&S 17.4.21. Let's pick the log tan form. But this
1467 ;; isn't right if we know that abs(phi) > %pi/2, where
1468 ;; elliptic_f is undefined (or infinity).
1469 (cond ((not (eq '$pos (csign (sub ($abs phi) (div '$%pi 2)))))
1470 (ftake '%log
1471 (ftake '%tan
1472 (add (mul '$%pi (div 1 4))
1473 (mul 1//2 phi)))))
1474 (t
1475 (merror (intl:gettext "elliptic_f: elliptic_f(~:M, ~:M) is undefined.")
1476 phi m))))
1477 ((alike1 phi '((mtimes) ((rat) 1 2) $%pi))
1478 ;; Complete elliptic integral
1479 (ftake '%elliptic_kc m))
1480 (t
1481 ;; Nothing to do
1482 (give-up)))))
1483
1484 (def-simplifier elliptic_e (phi m)
1485 (let (args)
1486 (cond ((float-numerical-eval-p phi m)
1487 ;; Numerically evaluate it
1488 (elliptic-e ($float phi) ($float m)))
1489 ((complex-float-numerical-eval-p phi m)
1490 (complexify (bigfloat::bf-elliptic-e (complex ($float ($realpart phi)) ($float ($imagpart phi)))
1491 (complex ($float ($realpart m)) ($float ($imagpart m))))))
1492 ((bigfloat-numerical-eval-p phi m)
1493 (to (bigfloat::bf-elliptic-e (bigfloat:to ($bfloat phi))
1494 (bigfloat:to ($bfloat m)))))
1495 ((setf args (complex-bigfloat-numerical-eval-p phi m))
1496 (destructuring-bind (phi m)
1497 args
1498 (to (bigfloat::bf-elliptic-e (bigfloat:to ($bfloat phi))
1499 (bigfloat:to ($bfloat m))))))
1500 ((zerop1 phi)
1501 0)
1502 ((zerop1 m)
1503 ;; A&S 17.4.23
1504 phi)
1505 ((onep1 m)
1506 ;; A&S 17.4.25, but handle periodicity:
1507 ;; elliptic_e(x,m) = elliptic_e(x-%pi*round(x/%pi), m)
1508 ;; + 2*round(x/%pi)*elliptic_ec(m)
1509 ;;
1510 ;; Or
1511 ;;
1512 ;; elliptic_e(x,1) = sin(x-%pi*round(x/%pi)) + 2*round(x/%pi)*elliptic_ec(m)
1513 ;;
1514 (let ((mult-pi (ftake '%round (div phi '$%pi))))
1515 (add (ftake '%sin (sub phi
1516 (mul '$%pi
1517 mult-pi)))
1518 (mul 2
1519 (mul mult-pi
1520 (ftake '%elliptic_ec m))))))
1521 ((alike1 phi '((mtimes) ((rat) 1 2) $%pi))
1522 ;; Complete elliptic integral
1523 (ftake '%elliptic_ec m))
1524 ((and ($numberp phi)
1525 (let ((r ($round (div phi '$%pi))))
1526 (and ($numberp r)
1527 (not (zerop1 r)))))
1528 ;; Handle the case where phi is a number where we can apply
1529 ;; the periodicity property without blowing up the
1530 ;; expression.
1531 (add (ftake '%elliptic_e
1532 (add phi
1533 (mul (mul -1 '$%pi)
1534 (ftake '%round (div phi '$%pi))))
1535 m)
1536 (mul 2
1537 (mul (ftake '%round (div phi '$%pi))
1538 (ftake '%elliptic_ec m)))))
1539 (t
1540 ;; Nothing to do
1541 (give-up)))))
1542
1543 ;; Complete elliptic integrals
1544 ;;
1545 ;; elliptic_kc(m) = elliptic_f(%pi/2, m)
1546 ;;
1547 ;; elliptic_ec(m) = elliptic_e(%pi/2, m)
1548 ;;
1549
1550 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
1551
1552 ;;; We support a simplim%function. The function is looked up in simplimit and
1553 ;;; handles specific values of the function.
1554
1555 (defprop %elliptic_kc simplim%elliptic_kc simplim%function)
1556
1557 (defun simplim%elliptic_kc (expr var val)
1558 ;; Look for the limit of the argument
1559 (let ((m (limit (cadr expr) var val 'think)))
1560 (cond ((onep1 m)
1561 ;; For an argument 1 return $infinity.
1562 '$infinity)
1563 (t
1564 ;; All other cases are handled by the simplifier of the function.
1565 (simplify (list '(%elliptic_kc) m))))))
1566
1567 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
1568
1569 (def-simplifier elliptic_kc (m)
1570 (let (args)
1571 (cond ((onep1 m)
1572 ;; elliptic_kc(1) is complex infinity. Maxima can not handle
1573 ;; infinities correctly, throw a Maxima error.
1574 (merror
1575 (intl:gettext "elliptic_kc: elliptic_kc(~:M) is undefined.")
1576 m))
1577 ((float-numerical-eval-p m)
1578 ;; Numerically evaluate it
1579 (to (elliptic-k ($float m))))
1580 ((complex-float-numerical-eval-p m)
1581 (complexify (bigfloat::bf-elliptic-k (complex ($float ($realpart m)) ($float ($imagpart m))))))
1582 ((setf args (complex-bigfloat-numerical-eval-p m))
1583 (destructuring-bind (m)
1584 args
1585 (to (bigfloat::bf-elliptic-k (bigfloat:to ($bfloat m))))))
1586 ((zerop1 m)
1587 '((mtimes) ((rat) 1 2) $%pi))
1588 ((alike1 m 1//2)
1589 ;; http://functions.wolfram.com/EllipticIntegrals/EllipticK/03/01/
1590 ;;
1591 ;; elliptic_kc(1/2) = 8*%pi^(3/2)/gamma(-1/4)^2
1592 (div (mul 8 (power '$%pi (div 3 2)))
1593 (power (gm (div -1 4)) 2)))
1594 ((eql -1 m)
1595 ;; elliptic_kc(-1) = gamma(1/4)^2/(4*sqrt(2*%pi))
1596 (div (power (gm (div 1 4)) 2)
1597 (mul 4 (power (mul 2 '$%pi) 1//2))))
1598 ((alike1 m (add 17 (mul -12 (power 2 1//2))))
1599 ;; elliptic_kc(17-12*sqrt(2)) = 2*(2+sqrt(2))*%pi^(3/2)/gamma(-1/4)^2
1600 (div (mul 2 (mul (add 2 (power 2 1//2))
1601 (power '$%pi (div 3 2))))
1602 (power (gm (div -1 4)) 2)))
1603 (t
1604 ;; Nothing to do
1605 (give-up)))))
1606
1607 (defprop %elliptic_kc
1608 ((m)
1609 ;; diff wrt m
1610 ((mtimes)
1611 ((rat) 1 2)
1612 ((mplus) ((%elliptic_ec) m)
1613 ((mtimes) -1
1614 ((%elliptic_kc) m)
1615 ((mplus) 1 ((mtimes) -1 m))))
1616 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
1617 ((mexpt) m -1)))
1618 grad)
1619
1620 (def-simplifier elliptic_ec (m)
1621 (let (args)
1622 (cond ((float-numerical-eval-p m)
1623 ;; Numerically evaluate it
1624 (elliptic-ec ($float m)))
1625 ((setf args (complex-float-numerical-eval-p m))
1626 (destructuring-bind (m)
1627 args
1628 (complexify (bigfloat::bf-elliptic-ec (bigfloat:to ($float m))))))
1629 ((setf args (complex-bigfloat-numerical-eval-p m))
1630 (destructuring-bind (m)
1631 args
1632 (to (bigfloat::bf-elliptic-ec (bigfloat:to ($bfloat m))))))
1633 ;; Some special cases we know about.
1634 ((zerop1 m)
1635 '((mtimes) ((rat) 1 2) $%pi))
1636 ((onep1 m)
1637 1)
1638 ((alike1 m 1//2)
1639 ;; elliptic_ec(1/2). Use the identity
1640 ;;
1641 ;; elliptic_ec(z)*elliptic_kc(1-z) - elliptic_kc(z)*elliptic_kc(1-z)
1642 ;; + elliptic_ec(1-z)*elliptic_kc(z) = %pi/2;
1643 ;;
1644 ;; Let z = 1/2 to get
1645 ;;
1646 ;; %pi^(3/2)*'elliptic_ec(1/2)/gamma(3/4)^2-%pi^3/(4*gamma(3/4)^4) = %pi/2
1647 ;;
1648 ;; since we know that elliptic_kc(1/2) = %pi^(3/2)/(2*gamma(3/4)^2). Hence
1649 ;;
1650 ;; elliptic_ec(1/2)
1651 ;; = (2*%pi*gamma(3/4)^4+%pi^3)/(4*%pi^(3/2)*gamma(3/4)^2)
1652 ;; = gamma(3/4)^2/(2*sqrt(%pi))+%pi^(3/2)/(4*gamma(3/4)^2)
1653 ;;
1654 (add (div (power (ftake '%gamma (div 3 4)) 2)
1655 (mul 2 (power '$%pi 1//2)))
1656 (div (power '$%pi (div 3 2))
1657 (mul 4 (power (ftake '%gamma (div 3 4)) 2)))))
1658 ((zerop1 (add 1 m))
1659 ;; elliptic_ec(-1). Use the identity
1660 ;; http://functions.wolfram.com/08.01.17.0002.01
1661 ;;
1662 ;;
1663 ;; elliptic_ec(z) = sqrt(1 - z)*elliptic_ec(z/(z-1))
1664 ;;
1665 ;; Let z = -1 to get
1666 ;;
1667 ;; elliptic_ec(-1) = sqrt(2)*elliptic_ec(1/2)
1668 ;;
1669 ;; Should we expand out elliptic_ec(1/2) using the above result?
1670 (mul (power 2 1//2)
1671 (ftake '%elliptic_ec 1//2)))
1672 (t
1673 ;; Nothing to do
1674 (give-up)))))
1675
1676 (defprop %elliptic_ec
1677 ((m)
1678 ((mtimes) ((rat) 1 2)
1679 ((mplus) ((%elliptic_ec) m)
1680 ((mtimes) -1 ((%elliptic_kc)
1681 m)))
1682 ((mexpt) m -1)))
1683 grad)
1684
1685 ;;
1686 ;; Elliptic integral of the third kind:
1687 ;;
1688 ;; (A&S 17.2.14)
1689 ;;
1690 ;; phi
1691 ;; /
1692 ;; [ 1
1693 ;; PI(n;phi|m) = I ----------------------------------- ds
1694 ;; ] 2 2
1695 ;; / SQRT(1 - m SIN (s)) (1 - n SIN (s))
1696 ;; 0
1697 ;;
1698 ;; As with E and F, we do not use the modular angle alpha but the
1699 ;; parameter m = sin(alpha)^2.
1700 ;;
1701 (def-simplifier elliptic_pi (n phi m)
1702 (let (args)
1703 (cond
1704 ((float-numerical-eval-p n phi m)
1705 ;; Numerically evaluate it
1706 (elliptic-pi ($float n) ($float phi) ($float m)))
1707 ((setf args (complex-float-numerical-eval-p n phi m))
1708 (destructuring-bind (n phi m)
1709 args
1710 (elliptic-pi (bigfloat:to ($float n))
1711 (bigfloat:to ($float phi))
1712 (bigfloat:to ($float m)))))
1713 ((bigfloat-numerical-eval-p n phi m)
1714 (to (bigfloat::bf-elliptic-pi (bigfloat:to n)
1715 (bigfloat:to phi)
1716 (bigfloat:to m))))
1717 ((setq args (complex-bigfloat-numerical-eval-p n phi m))
1718 (destructuring-bind (n phi m)
1719 args
1720 (to (bigfloat::bf-elliptic-pi (bigfloat:to n)
1721 (bigfloat:to phi)
1722 (bigfloat:to m)))))
1723 ((zerop1 n)
1724 (ftake '%elliptic_f phi m))
1725 ((zerop1 m)
1726 ;; 3 cases depending on n < 1, n > 1, or n = 1.
1727 (let ((s (asksign (add -1 n))))
1728 (case s
1729 ($positive
1730 (div (ftake '%atanh (mul (power (add n -1) 1//2)
1731 (ftake '%tan phi)))
1732 (power (add n -1) 1//2)))
1733 ($negative
1734 (div (ftake '%atan (mul (power (sub 1 n) 1//2)
1735 (ftake '%tan phi)))
1736 (power (sub 1 n) 1//2)))
1737 ($zero
1738 (ftake '%tan phi)))))
1739 (t
1740 ;; Nothing to do
1741 (give-up)))))
1742
1743 ;; Complete elliptic-pi. That is phi = %pi/2. Then
1744 ;; elliptic_pi(n,m)
1745 ;; = Rf(0, 1-m,1) + Rj(0,1-m,1-n)*n/3;
1746 (defun elliptic-pi-complete (n m)
1747 (to (bigfloat:+ (bigfloat::bf-rf 0 (- 1 m) 1)
1748 (bigfloat:* 1/3 n (bigfloat::bf-rj 0 (- 1 m) 1 (- 1 n))))))
1749
1750 ;; To compute elliptic_pi for all z, we use the property
1751 ;; (http://functions.wolfram.com/08.06.16.0002.01)
1752 ;;
1753 ;; elliptic_pi(n, z + %pi*k, m)
1754 ;; = 2*k*elliptic_pi(n, %pi/2, m) + elliptic_pi(n, z, m)
1755 ;;
1756 ;; So we are left with computing the integral for 0 <= z < %pi. Using
1757 ;; Carlson's formulation produces the wrong values for %pi/2 < z <
1758 ;; %pi. How to do that?
1759 ;;
1760 ;; Let
1761 ;;
1762 ;; I(a,b) = integrate(1/(1-n*sin(x)^2)/sqrt(1 - m*sin(x)^2), x, a, b)
1763 ;;
1764 ;; That is, I(a,b) is the integral for the elliptic_pi function but
1765 ;; with a lower limit of a and an upper limit of b.
1766 ;;
1767 ;; Then, we want to compute I(0, z), with %pi <= z < %pi. Let w = z +
1768 ;; %pi/2, 0 <= w < %pi/2. Then
1769 ;;
1770 ;; I(0, w+%pi/2) = I(0, %pi/2) + I(%pi/2, w+%pi/2)
1771 ;;
1772 ;; To evaluate I(%pi/2, w+%pi/2), use a change of variables:
1773 ;;
1774 ;; changevar('integrate(1/(1-n*sin(x)^2)/sqrt(1 - m*sin(x)^2), x, %pi/2, w + %pi/2),
1775 ;; x-%pi+u,u,x)
1776 ;;
1777 ;; = integrate(-1/(sqrt(1-m*sin(u)^2)*(1-n*sin(u)^2)),u,%pi/2-w,%pi/2)
1778 ;; = I(%pi/2-w,%pi/2)
1779 ;; = I(0,%pi/2) - I(0,%pi/2-w)
1780 ;;
1781 ;; Thus,
1782 ;;
1783 ;; I(0,%pi/2+w) = 2*I(0,%pi/2) - I(0,%pi/2-w)
1784 ;;
1785 ;; This allows us to compute the general result with 0 <= z < %pi
1786 ;;
1787 ;; I(0, k*%pi + z) = 2*k*I(0,%pi/2) + I(0,z);
1788 ;;
1789 ;; If 0 <= z < %pi/2, then the we are done. If %pi/2 <= z < %pi, let
1790 ;; z = w+%pi/2. Then
1791 ;;
1792 ;; I(0,z) = 2*I(0,%pi/2) - I(0,%pi/2-w)
1793 ;;
1794 ;; Or, since w = z-%pi/2:
1795 ;;
1796 ;; I(0,z) = 2*I(0,%pi/2) - I(0,%pi-z)
1797
1798 (defun elliptic-pi (n phi m)
1799 ;; elliptic_pi(n, -phi, m) = -elliptic_pi(n, phi, m). That is, it
1800 ;; is an odd function of phi.
1801 (when (minusp (realpart phi))
1802 (return-from elliptic-pi (- (elliptic-pi n (- phi) m))))
1803
1804 ;; Note: Carlson's DRJ has n defined as the negative of the n given
1805 ;; in A&S.
1806 (flet ((base (n phi m)
1807 ;; elliptic_pi(n,phi,m) =
1808 ;; sin(phi)*Rf(cos(phi)^2, 1-m*sin(phi)^2, 1)
1809 ;; - (-n / 3) * sin(phi)^3
1810 ;; * Rj(cos(phi)^2, 1-m*sin(phi)^2, 1, 1-n*sin(phi)^2)
1811 (let* ((nn (- n))
1812 (sin-phi (sin phi))
1813 (cos-phi (cos phi))
1814 (k (sqrt m))
1815 (k2sin (* (- 1 (* k sin-phi))
1816 (+ 1 (* k sin-phi)))))
1817 (- (* sin-phi (bigfloat::bf-rf (expt cos-phi 2) k2sin 1.0))
1818 (* (/ nn 3) (expt sin-phi 3)
1819 (bigfloat::bf-rj (expt cos-phi 2) k2sin 1.0
1820 (- 1 (* n (expt sin-phi 2)))))))))
1821 ;; FIXME: Reducing the arg by pi has significant round-off.
1822 ;; Consider doing something better.
1823 (let* ((cycles (round (realpart phi) pi))
1824 (rem (- phi (* cycles pi))))
1825 (let ((complete (elliptic-pi-complete n m)))
1826 (to (+ (* 2 cycles complete)
1827 (base n rem m)))))))
1828
1829 ;;; Deriviatives from functions.wolfram.com
1830 ;;; http://functions.wolfram.com/EllipticIntegrals/EllipticPi3/20/
1831 (defprop %elliptic_pi
1832 ((n z m)
1833 ;Derivative wrt first argument
1834 ((mtimes) ((rat) 1 2)
1835 ((mexpt) ((mplus) m ((mtimes) -1 n)) -1)
1836 ((mexpt) ((mplus) -1 n) -1)
1837 ((mplus)
1838 ((mtimes) ((mexpt) n -1)
1839 ((mplus) ((mtimes) -1 m) ((mexpt) n 2))
1840 ((%elliptic_pi) n z m))
1841 ((%elliptic_e) z m)
1842 ((mtimes) ((mplus) m ((mtimes) -1 n)) ((mexpt) n -1)
1843 ((%elliptic_f) z m))
1844 ((mtimes) ((rat) -1 2) n
1845 ((mexpt)
1846 ((mplus) 1 ((mtimes) -1 m ((mexpt) ((%sin) z) 2)))
1847 ((rat) 1 2))
1848 ((mexpt)
1849 ((mplus) 1 ((mtimes) -1 n ((mexpt) ((%sin) z) 2)))
1850 -1)
1851 ((%sin) ((mtimes) 2 z)))))
1852 ;derivative wrt second argument
1853 ((mtimes)
1854 ((mexpt)
1855 ((mplus) 1 ((mtimes) -1 m ((mexpt) ((%sin) z) 2)))
1856 ((rat) -1 2))
1857 ((mexpt)
1858 ((mplus) 1 ((mtimes) -1 n ((mexpt) ((%sin) z) 2))) -1))
1859 ;Derivative wrt third argument
1860 ((mtimes) ((rat) 1 2)
1861 ((mexpt) ((mplus) ((mtimes) -1 m) n) -1)
1862 ((mplus) ((%elliptic_pi) n z m)
1863 ((mtimes) ((mexpt) ((mplus) -1 m) -1)
1864 ((%elliptic_e) z m))
1865 ((mtimes) ((rat) -1 2) ((mexpt) ((mplus) -1 m) -1) m
1866 ((mexpt)
1867 ((mplus) 1 ((mtimes) -1 m ((mexpt) ((%sin) z) 2)))
1868 ((rat) -1 2))
1869 ((%sin) ((mtimes) 2 z))))))
1870 grad)
1871
1872 (in-package #:bigfloat)
1873 ;; Translation of Jim FitzSimons' bigfloat implementation of elliptic
1874 ;; integrals from http://www.getnet.com/~cherry/elliptbf3.mac.
1875 ;;
1876 ;; The algorithms are based on B.C. Carlson's "Numerical Computation
1877 ;; of Real or Complex Elliptic Integrals". These are updated to the
1878 ;; algorithms in Journal of Computational and Applied Mathematics 118
1879 ;; (2000) 71-85 "Reduction Theorems for Elliptic Integrands with the
1880 ;; Square Root of two quadritic factors"
1881 ;;
1882
1883 ;; NOTE: Despite the names indicating these are for bigfloat numbers,
1884 ;; the algorithms and routines are generic and will work with floats
1885 ;; and bigfloats.
1886
1887 (defun bferrtol (&rest args)
1888 ;; Compute error tolerance as sqrt(2^(-fpprec)). Not sure this is
1889 ;; quite right, but it makes the routines more accurate as fpprec
1890 ;; increases.
1891 (sqrt (reduce #'min (mapcar #'(lambda (x)
1892 (if (rationalp (realpart x))
1893 maxima::flonum-epsilon
1894 (epsilon x)))
1895 args))))
1896
1897 ;; rc(x,y) = integrate(1/2*(t+x)^(-1/2)/(t+y), t, 0, inf)
1898 ;;
1899 ;; log(x) = (x-1)*rc(((1+x)/2)^2, x), x > 0
1900 ;; asin(x) = x * rc(1-x^2, 1), |x|<= 1
1901 ;; acos(x) = sqrt(1-x^2)*rc(x^2,1), 0 <= x <=1
1902 ;; atan(x) = x * rc(1,1+x^2)
1903 ;; asinh(x) = x * rc(1+x^2,1)
1904 ;; acosh(x) = sqrt(x^2-1) * rc(x^2,1), x >= 1
1905 ;; atanh(x) = x * rc(1,1-x^2), |x|<=1
1906
1907 (defun bf-rc (x y)
1908 (let ((yn y)
1909 xn z w a an pwr4 n epslon lambda sn s)
1910 (cond ((and (zerop (imagpart yn))
1911 (minusp (realpart yn)))
1912 (setf xn (- x y))
1913 (setf yn (- yn))
1914 (setf z yn)
1915 (setf w (sqrt (/ x xn))))
1916 (t
1917 (setf xn x)
1918 (setf z yn)
1919 (setf w 1)))
1920 (setf a (/ (+ xn yn yn) 3))
1921 (setf epslon (/ (abs (- a xn)) (bferrtol x y)))
1922 (setf an a)
1923 (setf pwr4 1)
1924 (setf n 0)
1925 (loop while (> (* epslon pwr4) (abs an))
1926 do
1927 (setf pwr4 (/ pwr4 4))
1928 (setf lambda (+ (* 2 (sqrt xn) (sqrt yn)) yn))
1929 (setf an (/ (+ an lambda) 4))
1930 (setf xn (/ (+ xn lambda) 4))
1931 (setf yn (/ (+ yn lambda) 4))
1932 (incf n))
1933 ;; c2=3/10,c3=1/7,c4=3/8,c5=9/22,c6=159/208,c7=9/8
1934 (setf sn (/ (* pwr4 (- z a)) an))
1935 (setf s (* sn sn (+ 3/10
1936 (* sn (+ 1/7
1937 (* sn (+ 3/8
1938 (* sn (+ 9/22
1939 (* sn (+ 159/208
1940 (* sn 9/8))))))))))))
1941 (/ (* w (+ 1 s))
1942 (sqrt an))))
1943
1944
1945
1946 ;; See https://dlmf.nist.gov/19.16.E5:
1947 ;;
1948 ;; rd(x,y,z) = integrate(3/2/sqrt(t+x)/sqrt(t+y)/sqrt(t+z)/(t+z), t, 0, inf)
1949 ;;
1950 ;; rd(1,1,1) = 1
1951 ;; E(K) = rf(0, 1-K^2, 1) - (K^2/3)*rd(0,1-K^2,1)
1952 ;;
1953 ;; B = integrate(s^2/sqrt(1-s^4), s, 0 ,1)
1954 ;; = beta(3/4,1/2)/4
1955 ;; = sqrt(%pi)*gamma(3/4)/gamma(1/4)
1956 ;; = 1/3*rd(0,2,1)
1957
1958 (defun bf-rd (x y z)
1959 (let* ((xn x)
1960 (yn y)
1961 (zn z)
1962 (a (/ (+ xn yn (* 3 zn)) 5))
1963 (epslon (/ (max (abs (- a xn))
1964 (abs (- a yn))
1965 (abs (- a zn)))
1966 (bferrtol x y z)))
1967 (an a)
1968 (sigma 0)
1969 (power4 1)
1970 (n 0)
1971 xnroot ynroot znroot lam)
1972 (loop while (> (* power4 epslon) (abs an))
1973 do
1974 (setf xnroot (sqrt xn))
1975 (setf ynroot (sqrt yn))
1976 (setf znroot (sqrt zn))
1977 (setf lam (+ (* xnroot ynroot)
1978 (* xnroot znroot)
1979 (* ynroot znroot)))
1980 (setf sigma (+ sigma (/ power4
1981 (* znroot (+ zn lam)))))
1982 (setf power4 (* power4 1/4))
1983 (setf xn (* (+ xn lam) 1/4))
1984 (setf yn (* (+ yn lam) 1/4))
1985 (setf zn (* (+ zn lam) 1/4))
1986 (setf an (* (+ an lam) 1/4))
1987 (incf n))
1988 ;; c1=-3/14,c2=1/6,c3=9/88,c4=9/22,c5=-3/22,c6=-9/52,c7=3/26
1989 (let* ((xndev (/ (* (- a x) power4) an))
1990 (yndev (/ (* (- a y) power4) an))
1991 (zndev (- (* (+ xndev yndev) 1/3)))
1992 (ee2 (- (* xndev yndev) (* 6 zndev zndev)))
1993 (ee3 (* (- (* 3 xndev yndev)
1994 (* 8 zndev zndev))
1995 zndev))
1996 (ee4 (* 3 (- (* xndev yndev) (* zndev zndev)) zndev zndev))
1997 (ee5 (* xndev yndev zndev zndev zndev))
1998 (s (+ 1
1999 (* -3/14 ee2)
2000 (* 1/6 ee3)
2001 (* 9/88 ee2 ee2)
2002 (* -3/22 ee4)
2003 (* -9/52 ee2 ee3)
2004 (* 3/26 ee5)
2005 (* -1/16 ee2 ee2 ee2)
2006 (* 3/10 ee3 ee3)
2007 (* 3/20 ee2 ee4)
2008 (* 45/272 ee2 ee2 ee3)
2009 (* -9/68 (+ (* ee2 ee5) (* ee3 ee4))))))
2010 (+ (* 3 sigma)
2011 (/ (* power4 s)
2012 (expt an 3/2))))))
2013
2014 ;; See https://dlmf.nist.gov/19.16.E1
2015 ;;
2016 ;; rf(x,y,z) = 1/2*integrate(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+z)), t, 0, inf);
2017 ;;
2018 ;; rf(1,1,1) = 1
2019 (defun bf-rf (x y z)
2020 (let* ((xn x)
2021 (yn y)
2022 (zn z)
2023 (a (/ (+ xn yn zn) 3))
2024 (epslon (/ (max (abs (- a xn))
2025 (abs (- a yn))
2026 (abs (- a zn)))
2027 (bferrtol x y z)))
2028 (an a)
2029 (power4 1)
2030 (n 0)
2031 xnroot ynroot znroot lam)
2032 (loop while (> (* power4 epslon) (abs an))
2033 do
2034 (setf xnroot (sqrt xn))
2035 (setf ynroot (sqrt yn))
2036 (setf znroot (sqrt zn))
2037 (setf lam (+ (* xnroot ynroot)
2038 (* xnroot znroot)
2039 (* ynroot znroot)))
2040 (setf power4 (* power4 1/4))
2041 (setf xn (* (+ xn lam) 1/4))
2042 (setf yn (* (+ yn lam) 1/4))
2043 (setf zn (* (+ zn lam) 1/4))
2044 (setf an (* (+ an lam) 1/4))
2045 (incf n))
2046 ;; c1=-3/14,c2=1/6,c3=9/88,c4=9/22,c5=-3/22,c6=-9/52,c7=3/26
2047 (let* ((xndev (/ (* (- a x) power4) an))
2048 (yndev (/ (* (- a y) power4) an))
2049 (zndev (- (+ xndev yndev)))
2050 (ee2 (- (* xndev yndev) (* 6 zndev zndev)))
2051 (ee3 (* xndev yndev zndev))
2052 (s (+ 1
2053 (* -1/10 ee2)
2054 (* 1/14 ee3)
2055 (* 1/24 ee2 ee2)
2056 (* -3/44 ee2 ee3))))
2057 (/ s (sqrt an)))))
2058
2059 (defun bf-rj1 (x y z p)
2060 (let* ((xn x)
2061 (yn y)
2062 (zn z)
2063 (pn p)
2064 (en (* (- pn xn)
2065 (- pn yn)
2066 (- pn zn)))
2067 (sigma 0)
2068 (power4 1)
2069 (k 0)
2070 (a (/ (+ xn yn zn pn pn) 5))
2071 (epslon (/ (max (abs (- a xn))
2072 (abs (- a yn))
2073 (abs (- a zn))
2074 (abs (- a pn)))
2075 (bferrtol x y z p)))
2076 (an a)
2077 xnroot ynroot znroot pnroot lam dn)
2078 (loop while (> (* power4 epslon) (abs an))
2079 do
2080 (setf xnroot (sqrt xn))
2081 (setf ynroot (sqrt yn))
2082 (setf znroot (sqrt zn))
2083 (setf pnroot (sqrt pn))
2084 (setf lam (+ (* xnroot ynroot)
2085 (* xnroot znroot)
2086 (* ynroot znroot)))
2087 (setf dn (* (+ pnroot xnroot)
2088 (+ pnroot ynroot)
2089 (+ pnroot znroot)))
2090 (setf sigma (+ sigma
2091 (/ (* power4
2092 (bf-rc 1 (+ 1 (/ en (* dn dn)))))
2093 dn)))
2094 (setf power4 (* power4 1/4))
2095 (setf en (/ en 64))
2096 (setf xn (* (+ xn lam) 1/4))
2097 (setf yn (* (+ yn lam) 1/4))
2098 (setf zn (* (+ zn lam) 1/4))
2099 (setf pn (* (+ pn lam) 1/4))
2100 (setf an (* (+ an lam) 1/4))
2101 (incf k))
2102 (let* ((xndev (/ (* (- a x) power4) an))
2103 (yndev (/ (* (- a y) power4) an))
2104 (zndev (/ (* (- a z) power4) an))
2105 (pndev (* -0.5 (+ xndev yndev zndev)))
2106 (ee2 (+ (* xndev yndev)
2107 (* xndev zndev)
2108 (* yndev zndev)
2109 (* -3 pndev pndev)))
2110 (ee3 (+ (* xndev yndev zndev)
2111 (* 2 ee2 pndev)
2112 (* 4 pndev pndev pndev)))
2113 (ee4 (* (+ (* 2 xndev yndev zndev)
2114 (* ee2 pndev)
2115 (* 3 pndev pndev pndev))
2116 pndev))
2117 (ee5 (* xndev yndev zndev pndev pndev))
2118 (s (+ 1
2119 (* -3/14 ee2)
2120 (* 1/6 ee3)
2121 (* 9/88 ee2 ee2)
2122 (* -3/22 ee4)
2123 (* -9/52 ee2 ee3)
2124 (* 3/26 ee5)
2125 (* -1/16 ee2 ee2 ee2)
2126 (* 3/10 ee3 ee3)
2127 (* 3/20 ee2 ee4)
2128 (* 45/272 ee2 ee2 ee3)
2129 (* -9/68 (+ (* ee2 ee5) (* ee3 ee4))))))
2130 (+ (* 6 sigma)
2131 (/ (* power4 s)
2132 (sqrt (* an an an)))))))
2133
2134 (defun bf-rj (x y z p)
2135 (let* ((xn x)
2136 (yn y)
2137 (zn z)
2138 (qn (- p)))
2139 (cond ((and (and (zerop (imagpart xn)) (>= (realpart xn) 0))
2140 (and (zerop (imagpart yn)) (>= (realpart yn) 0))
2141 (and (zerop (imagpart zn)) (>= (realpart zn) 0))
2142 (and (zerop (imagpart qn)) (> (realpart qn) 0)))
2143 (destructuring-bind (xn yn zn)
2144 (sort (list xn yn zn) #'<)
2145 (let* ((pn (+ yn (* (- zn yn) (/ (- yn xn) (+ yn qn)))))
2146 (s (- (* (- pn yn) (bf-rj1 xn yn zn pn))
2147 (* 3 (bf-rf xn yn zn)))))
2148 (setf s (+ s (* 3 (sqrt (/ (* xn yn zn)
2149 (+ (* xn zn) (* pn qn))))
2150 (bf-rc (+ (* xn zn) (* pn qn)) (* pn qn)))))
2151 (/ s (+ yn qn)))))
2152 (t
2153 (bf-rj1 x y z p)))))
2154
2155 (defun bf-rg (x y z)
2156 (* 0.5
2157 (+ (* z (bf-rf x y z))
2158 (* (- z x)
2159 (- y z)
2160 (bf-rd x y z)
2161 1/3)
2162 (sqrt (/ (* x y) z)))))
2163
2164 ;; elliptic_f(phi,m) = sin(phi)*rf(cos(phi)^2, 1-m*sin(phi)^2,1)
2165 (defun bf-elliptic-f (phi m)
2166 (flet ((base (phi m)
2167 (cond ((= m 1)
2168 ;; F(z|1) = log(tan(z/2+%pi/4))
2169 (log (tan (+ (/ phi 2) (/ (%pi phi) 4)))))
2170 (t
2171 (let ((s (sin phi))
2172 (c (cos phi)))
2173 (* s (bf-rf (* c c) (- 1 (* m s s)) 1)))))))
2174 ;; Handle periodicity (see elliptic-f)
2175 (let* ((bfpi (%pi phi))
2176 (period (round (realpart phi) bfpi)))
2177 (+ (base (- phi (* bfpi period)) m)
2178 (if (zerop period)
2179 0
2180 (* 2 period (bf-elliptic-k m)))))))
2181
2182 ;; elliptic_kc(k) = rf(0, 1-k^2,1)
2183 ;;
2184 ;; or
2185 ;; elliptic_kc(m) = rf(0, 1-m,1)
2186
2187 (defun bf-elliptic-k (m)
2188 (cond ((= m 0)
2189 (if (maxima::$bfloatp m)
2190 (maxima::$bfloat (maxima::div 'maxima::$%pi 2))
2191 (float (/ pi 2) 1e0)))
2192 ((= m 1)
2193 (maxima::merror
2194 (intl:gettext "elliptic_kc: elliptic_kc(1) is undefined.")))
2195 (t
2196 (bf-rf 0 (- 1 m) 1))))
2197
2198 ;; elliptic_e(phi, k) = sin(phi)*rf(cos(phi)^2,1-k^2*sin(phi)^2,1)
2199 ;; - (k^2/3)*sin(phi)^3*rd(cos(phi)^2, 1-k^2*sin(phi)^2,1)
2200 ;;
2201 ;;
2202 ;; or
2203 ;; elliptic_e(phi, m) = sin(phi)*rf(cos(phi)^2,1-m*sin(phi)^2,1)
2204 ;; - (m/3)*sin(phi)^3*rd(cos(phi)^2, 1-m*sin(phi)^2,1)
2205 ;;
2206 (defun bf-elliptic-e (phi m)
2207 (flet ((base (phi m)
2208 (let* ((s (sin phi))
2209 (c (cos phi))
2210 (c2 (* c c))
2211 (s2 (- 1 (* m s s))))
2212 (- (* s (bf-rf c2 s2 1))
2213 (* (/ m 3) (* s s s) (bf-rd c2 s2 1))))))
2214 ;; Elliptic E is quasi-periodic wrt to phi:
2215 ;;
2216 ;; E(z|m) = E(z - %pi*round(Re(z)/%pi)|m) + 2*round(Re(z)/%pi)*E(m)
2217 (let* ((bfpi (%pi phi))
2218 (period (round (realpart phi) bfpi)))
2219 (+ (base (- phi (* bfpi period)) m)
2220 (* 2 period (bf-elliptic-ec m))))))
2221
2222
2223 ;; elliptic_ec(k) = rf(0,1-k^2,1) - (k^2/3)*rd(0,1-k^2,1);
2224 ;;
2225 ;; or
2226 ;; elliptic_ec(m) = rf(0,1-m,1) - (m/3)*rd(0,1-m,1);
2227
2228 (defun bf-elliptic-ec (m)
2229 (cond ((= m 0)
2230 (if (typep m 'bigfloat)
2231 (bigfloat (maxima::$bfloat (maxima::div 'maxima::$%pi 2)))
2232 (float (/ pi 2) 1e0)))
2233 ((= m 1)
2234 (if (typep m 'bigfloat)
2235 (bigfloat 1)
2236 1e0))
2237 (t
2238 (let ((m1 (- 1 m)))
2239 (- (bf-rf 0 m1 1)
2240 (* m 1/3 (bf-rd 0 m1 1)))))))
2241
2242 (defun bf-elliptic-pi-complete (n m)
2243 (+ (bf-rf 0 (- 1 m) 1)
2244 (* 1/3 n (bf-rj 0 (- 1 m) 1 (- 1 n)))))
2245
2246 (defun bf-elliptic-pi (n phi m)
2247 ;; Note: Carlson's DRJ has n defined as the negative of the n given
2248 ;; in A&S.
2249 (flet ((base (n phi m)
2250 (let* ((nn (- n))
2251 (sin-phi (sin phi))
2252 (cos-phi (cos phi))
2253 (k (sqrt m))
2254 (k2sin (* (- 1 (* k sin-phi))
2255 (+ 1 (* k sin-phi)))))
2256 (- (* sin-phi (bf-rf (expt cos-phi 2) k2sin 1.0))
2257 (* (/ nn 3) (expt sin-phi 3)
2258 (bf-rj (expt cos-phi 2) k2sin 1.0
2259 (- 1 (* n (expt sin-phi 2)))))))))
2260 ;; FIXME: Reducing the arg by pi has significant round-off.
2261 ;; Consider doing something better.
2262 (let* ((bf-pi (%pi (realpart phi)))
2263 (cycles (round (realpart phi) bf-pi))
2264 (rem (- phi (* cycles bf-pi))))
2265 (let ((complete (bf-elliptic-pi-complete n m)))
2266 (+ (* 2 cycles complete)
2267 (base n rem m))))))
2268
2269 ;; Compute inverse_jacobi_sn, for float or bigfloat args.
2270 (defun bf-inverse-jacobi-sn (u m)
2271 (* u (bf-rf (- 1 (* u u))
2272 (- 1 (* m u u))
2273 1)))
2274
2275 ;; Compute inverse_jacobi_dn. We use the following identity
2276 ;; from Gradshteyn & Ryzhik, 8.153.6
2277 ;;
2278 ;; w = dn(z|m) = cn(sqrt(m)*z, 1/m)
2279 ;;
2280 ;; Solve for z to get
2281 ;;
2282 ;; z = inverse_jacobi_dn(w,m)
2283 ;; = 1/sqrt(m) * inverse_jacobi_cn(w, 1/m)
2284 (defun bf-inverse-jacobi-dn (w m)
2285 (cond ((= w 1)
2286 (float 0 w))
2287 ((= m 1)
2288 ;; jacobi_dn(x,1) = sech(x) so the inverse is asech(x)
2289 (maxima::take '(maxima::%asech) (maxima::to w)))
2290 (t
2291 ;; We should do something better to make sure that things
2292 ;; that should be real are real.
2293 (/ (to (maxima::take '(maxima::%inverse_jacobi_cn)
2294 (maxima::to w)
2295 (maxima::to (/ m))))
2296 (sqrt m)))))
2297
2298 (in-package :maxima)
2299
2300 ;; Define Carlson's elliptic integrals.
2301
2302 (def-simplifier carlson_rc (x y)
2303 (let (args)
2304 (flet ((calc (x y)
2305 (flet ((floatify (z)
2306 ;; If z is a complex rational, convert to a
2307 ;; complex double-float. Otherwise, leave it as
2308 ;; is. If we don't do this, %i is handled as
2309 ;; #c(0 1), which makes bf-rc use single-float
2310 ;; arithmetic instead of the desired
2311 ;; double-float.
2312 (if (and (complexp z) (rationalp (realpart z)))
2313 (complex (float (realpart z))
2314 (float (imagpart z)))
2315 z)))
2316 (to (bigfloat::bf-rc (floatify (bigfloat:to x))
2317 (floatify (bigfloat:to y)))))))
2318 ;; See comments from bf-rc
2319 (cond ((float-numerical-eval-p x y)
2320 (calc ($float x) ($float y)))
2321 ((bigfloat-numerical-eval-p x y)
2322 (calc ($bfloat x) ($bfloat y)))
2323 ((setf args (complex-float-numerical-eval-p x y))
2324 (destructuring-bind (x y)
2325 args
2326 (calc ($float x) ($float y))))
2327 ((setf args (complex-bigfloat-numerical-eval-p x y))
2328 (destructuring-bind (x y)
2329 args
2330 (calc ($bfloat x) ($bfloat y))))
2331 ((and (zerop1 x)
2332 (onep1 y))
2333 ;; rc(0, 1) = %pi/2
2334 (div '$%pi 2))
2335 ((and (zerop1 x)
2336 (alike1 y (div 1 4)))
2337 ;; rc(0,1/4) = %pi
2338 '$%pi)
2339 ((and (eql x 2)
2340 (onep1 y))
2341 ;; rc(2,1) = 1/2*integrate(1/sqrt(t+2)/(t+1), t, 0, inf)
2342 ;; = (log(sqrt(2)+1)-log(sqrt(2)-1))/2
2343 ;; ratsimp(logcontract(%)),algebraic:
2344 ;; = -log(3-2^(3/2))/2
2345 ;; = -log(sqrt(3-2^(3/2)))
2346 ;; = -log(sqrt(2)-1)
2347 ;; = log(1/(sqrt(2)-1))
2348 ;; ratsimp(%),algebraic;
2349 ;; = log(sqrt(2)+1)
2350 (ftake '%log (add 1 (power 2 1//2))))
2351 ((and (alike x '$%i)
2352 (alike y (add 1 '$%i)))
2353 ;; rc(%i, %i+1) = 1/2*integrate(1/sqrt(t+%i)/(t+%i+1), t, 0, inf)
2354 ;; = %pi/2-atan((-1)^(1/4))
2355 ;; ratsimp(logcontract(ratsimp(rectform(%o42)))),algebraic;
2356 ;; = (%i*log(3-2^(3/2))+%pi)/4
2357 ;; = (%i*log(3-2^(3/2)))/4+%pi/4
2358 ;; = %i*log(sqrt(3-2^(3/2)))/2+%pi/4
2359 ;; sqrtdenest(%);
2360 ;; = %pi/4 + %i*log(sqrt(2)-1)/2
2361 (add (div '$%pi 4)
2362 (mul '$%i
2363 1//2
2364 (ftake '%log (sub (power 2 1//2) 1)))))
2365 ((and (zerop1 x)
2366 (alike1 y '$%i))
2367 ;; rc(0,%i) = 1/2*integrate(1/(sqrt(t)*(t+%i)), t, 0, inf)
2368 ;; = -((sqrt(2)*%i-sqrt(2))*%pi)/4
2369 ;; = ((1-%i)*%pi)/2^(3/2)
2370 (div (mul (sub 1 '$%i)
2371 '$%pi)
2372 (power 2 3//2)))
2373 ((and (alike1 x y)
2374 (eq ($sign ($realpart x)) '$pos))
2375 ;; carlson_rc(x,x) = 1/2*integrate(1/sqrt(t+x)/(t+x), t, 0, inf)
2376 ;; = 1/sqrt(x)
2377 (power x -1//2))
2378 ((and (alike1 x (power (div (add 1 y) 2) 2))
2379 (eq ($sign ($realpart y)) '$pos))
2380 ;; Rc(((1+x)/2)^2,x) = log(x)/(x-1) for x > 0.
2381 ;;
2382 ;; This is done by looking at Rc(x,y) and seeing if
2383 ;; ((1+y)/2)^2 is the same as x.
2384 (div (ftake '%log y)
2385 (sub y 1)))
2386 (t
2387 (give-up))))))
2388
2389 (def-simplifier carlson_rd (x y z)
2390 (let (args)
2391 (flet ((calc (x y z)
2392 (to (bigfloat::bf-rd (bigfloat:to x)
2393 (bigfloat:to y)
2394 (bigfloat:to z)))))
2395 ;; See https://dlmf.nist.gov/19.20.E18
2396 (cond ((and (eql x 1)
2397 (eql x y)
2398 (eql y z))
2399 ;; Rd(1,1,1) = 1
2400 1)
2401 ((and (alike1 x y)
2402 (alike1 y z))
2403 ;; Rd(x,x,x) = x^(-3/2)
2404 (power x (div -3 2)))
2405 ((and (zerop1 x)
2406 (alike1 y z))
2407 ;; Rd(0,y,y) = 3/4*%pi*y^(-3/2)
2408 (mul (div 3 4)
2409 '$%pi
2410 (power y (div -3 2))))
2411 ((alike1 y z)
2412 ;; Rd(x,y,y) = 3/(2*(y-x))*(Rc(x, y) - sqrt(x)/y)
2413 (mul (div 3 (mul 2 (sub y x)))
2414 (sub (ftake '%carlson_rc x y)
2415 (div (power x 1//2)
2416 y))))
2417 ((alike1 x y)
2418 ;; Rd(x,x,z) = 3/(z-x)*(Rc(z,x) - 1/sqrt(z))
2419 (mul (div 3 (sub z x))
2420 (sub (ftake '%carlson_rc z x)
2421 (div 1 (power z 1//2)))))
2422 ((and (eql z 1)
2423 (or (and (eql x 0)
2424 (eql y 2))
2425 (and (eql x 2)
2426 (eql y 0))))
2427 ;; Rd(0,2,1) = 3*(gamma(3/4)^2)/sqrt(2*%pi)
2428 ;; See https://dlmf.nist.gov/19.20.E22.
2429 ;;
2430 ;; But that's the same as
2431 ;; 3*sqrt(%pi)*gamma(3/4)/gamma(1/4). We can see this by
2432 ;; taking the ratio to get
2433 ;; gamma(1/4)*gamma(3/4)/sqrt(2)*%pi. But
2434 ;; gamma(1/4)*gamma(3/4) = beta(1/4,3/4) = sqrt(2)*%pi.
2435 ;; Hence, the ratio is 1.
2436 ;;
2437 ;; Note also that Rd(x,y,z) = Rd(y,x,z)
2438 (mul 3
2439 (power '$%pi 1//2)
2440 (div (ftake '%gamma (div 3 4))
2441 (ftake '%gamma (div 1 4)))))
2442 ((and (or (eql x 0) (eql y 0))
2443 (eql z 1))
2444 ;; 1/3*m*Rd(0,1-m,1) = K(m) - E(m).
2445 ;; See https://dlmf.nist.gov/19.25.E1
2446 ;;
2447 ;; Thus, Rd(0,y,1) = 3/(1-y)*(K(1-y) - E(1-y))
2448 ;;
2449 ;; Note that Rd(x,y,z) = Rd(y,x,z).
2450 (let ((m (sub 1 y)))
2451 (mul (div 3 m)
2452 (sub (ftake '%elliptic_kc m)
2453 (ftake '%elliptic_ec m)))))
2454 ((or (and (eql x 0)
2455 (eql y 1))
2456 (and (eql x 1)
2457 (eql y 0)))
2458 ;; 1/3*m*(1-m)*Rd(0,1,1-m) = E(m) - (1-m)*K(m)
2459 ;; See https://dlmf.nist.gov/19.25.E1
2460 ;;
2461 ;; Thus
2462 ;; Rd(0,1,z) = 3/(z*(1-z))*(E(1-z) - z*K(1-z))
2463 ;; Recall that Rd(x,y,z) = Rd(y,x,z).
2464 (mul (div 3 (mul z (sub 1 z)))
2465 (sub (ftake '%elliptic_ec (sub 1 z))
2466 (mul z
2467 (ftake '%elliptic_kc (sub 1 z))))))
2468 ((float-numerical-eval-p x y z)
2469 (calc ($float x) ($float y) ($float z)))
2470 ((bigfloat-numerical-eval-p x y z)
2471 (calc ($bfloat x) ($bfloat y) ($bfloat z)))
2472 ((setf args (complex-float-numerical-eval-p x y z))
2473 (destructuring-bind (x y z)
2474 args
2475 (calc ($float x) ($float y) ($float z))))
2476 ((setf args (complex-bigfloat-numerical-eval-p x y z))
2477 (destructuring-bind (x y z)
2478 args
2479 (calc ($bfloat x) ($bfloat y) ($bfloat z))))
2480 (t
2481 (give-up))))))
2482
2483 (def-simplifier carlson_rf (x y z)
2484 (let (args)
2485 (flet ((calc (x y z)
2486 (to (bigfloat::bf-rf (bigfloat:to x)
2487 (bigfloat:to y)
2488 (bigfloat:to z)))))
2489 ;; See https://dlmf.nist.gov/19.20.i
2490 (cond ((and (alike1 x y)
2491 (alike1 y z))
2492 ;; Rf(x,x,x) = x^(-1/2)
2493 (power x -1//2))
2494 ((and (zerop1 x)
2495 (alike1 y z))
2496 ;; Rf(0,y,y) = 1/2*%pi*y^(-1/2)
2497 (mul 1//2 '$%pi
2498 (power y -1//2)))
2499 ((alike1 y z)
2500 (ftake '%carlson_rc x y))
2501 ((some #'(lambda (args)
2502 (destructuring-bind (x y z)
2503 args
2504 (and (zerop1 x)
2505 (eql y 1)
2506 (eql z 2))))
2507 (list (list x y z)
2508 (list x z y)
2509 (list y x z)
2510 (list y z x)
2511 (list z x y)
2512 (list z y x)))
2513 ;; Rf(0,1,2) = (gamma(1/4))^2/(4*sqrt(2*%pi))
2514 ;;
2515 ;; And Rf is symmetric in all the args, so check every
2516 ;; permutation too. This could probably be simplified
2517 ;; without consing all the lists, but I'm lazy.
2518 (div (power (ftake '%gamma (div 1 4)) 2)
2519 (mul 4 (power (mul 2 '$%pi) 1//2))))
2520 ((some #'(lambda (args)
2521 (destructuring-bind (x y z)
2522 args
2523 (and (alike1 x '$%i)
2524 (alike1 y (mul -1 '$%i))
2525 (eql z 0))))
2526 (list (list x y z)
2527 (list x z y)
2528 (list y x z)
2529 (list y z x)
2530 (list z x y)
2531 (list z y x)))
2532 ;; rf(%i, -%i, 0)
2533 ;; = 1/2*integrate(1/sqrt(t^2+1)/sqrt(t),t,0,inf)
2534 ;; = beta(1/4,1/4)/4;
2535 ;; makegamma(%)
2536 ;; = gamma(1/4)^2/(4*sqrt(%pi))
2537 ;;
2538 ;; Rf is symmetric, so check all the permutations too.
2539 (div (power (ftake '%gamma (div 1 4)) 2)
2540 (mul 4 (power '$%pi 1//2))))
2541 ((setf args
2542 (some #'(lambda (args)
2543 (destructuring-bind (x y z)
2544 args
2545 ;; Check that x = 0 and z = 1, and
2546 ;; return y.
2547 (and (zerop1 x)
2548 (eql z 1)
2549 y)))
2550 (list (list x y z)
2551 (list x z y)
2552 (list y x z)
2553 (list y z x)
2554 (list z x y)
2555 (list z y x))))
2556 ;; Rf(0,1-m,1) = elliptic_kc(m).
2557 ;; See https://dlmf.nist.gov/19.25.E1
2558 (ftake '%elliptic_kc (sub 1 args)))
2559 ((some #'(lambda (args)
2560 (destructuring-bind (x y z)
2561 args
2562 (and (alike1 x '$%i)
2563 (alike1 y (mul -1 '$%i))
2564 (eql z 0))))
2565 (list (list x y z)
2566 (list x z y)
2567 (list y x z)
2568 (list y z x)
2569 (list z x y)
2570 (list z y x)))
2571 ;; rf(%i, -%i, 0)
2572 ;; = 1/2*integrate(1/sqrt(t^2+1)/sqrt(t),t,0,inf)
2573 ;; = beta(1/4,1/4)/4;
2574 ;; makegamma(%)
2575 ;; = gamma(1/4)^2/(4*sqrt(%pi))
2576 ;;
2577 ;; Rf is symmetric, so check all the permutations too.
2578 (div (power (ftake '%gamma (div 1 4)) 2)
2579 (mul 4 (power '$%pi 1//2))))
2580 ((float-numerical-eval-p x y z)
2581 (calc ($float x) ($float y) ($float z)))
2582 ((bigfloat-numerical-eval-p x y z)
2583 (calc ($bfloat x) ($bfloat y) ($bfloat z)))
2584 ((setf args (complex-float-numerical-eval-p x y z))
2585 (destructuring-bind (x y z)
2586 args
2587 (calc ($float x) ($float y) ($float z))))
2588 ((setf args (complex-bigfloat-numerical-eval-p x y z))
2589 (destructuring-bind (x y z)
2590 args
2591 (calc ($bfloat x) ($bfloat y) ($bfloat z))))
2592 (t
2593 (give-up))))))
2594
2595 (def-simplifier carlson_rj (x y z p)
2596 (let (args)
2597 (flet ((calc (x y z p)
2598 (to (bigfloat::bf-rj (bigfloat:to x)
2599 (bigfloat:to y)
2600 (bigfloat:to z)
2601 (bigfloat:to p)))))
2602 ;; See https://dlmf.nist.gov/19.20.iii
2603 (cond ((and (alike1 x y)
2604 (alike1 y z)
2605 (alike1 z p))
2606 ;; Rj(x,x,x,x) = x^(-3/2)
2607 (power x (div -3 2)))
2608 ((alike1 z p)
2609 ;; Rj(x,y,z,z) = Rd(x,y,z)
2610 (ftake '%carlson_rd x y z))
2611 ((and (zerop1 x)
2612 (alike1 y z))
2613 ;; Rj(0,y,y,p) = 3*%pi/(2*(y*sqrt(p)+p*sqrt(y)))
2614 (div (mul 3 '$%pi)
2615 (mul 2
2616 (add (mul y (power p 1//2))
2617 (mul p (power y 1//2))))))
2618 ((alike1 y z)
2619 ;; Rj(x,y,y,p) = 3/(p-y)*(Rc(x,y) - Rc(x,p))
2620 (mul (div 3 (sub p y))
2621 (sub (ftake '%carlson_rc x y)
2622 (ftake '%carlson_rc x p))))
2623 ((and (alike1 y z)
2624 (alike1 y p))
2625 ;; Rj(x,y,y,y) = Rd(x,y,y)
2626 (ftake '%carlson_rd x y y))
2627 ((float-numerical-eval-p x y z p)
2628 (calc ($float x) ($float y) ($float z) ($float p)))
2629 ((bigfloat-numerical-eval-p x y z p)
2630 (calc ($bfloat x) ($bfloat y) ($bfloat z) ($bfloat p)))
2631 ((setf args (complex-float-numerical-eval-p x y z p))
2632 (destructuring-bind (x y z p)
2633 args
2634 (calc ($float x) ($float y) ($float z) ($float p))))
2635 ((setf args (complex-bigfloat-numerical-eval-p x y z p))
2636 (destructuring-bind (x y z p)
2637 args
2638 (calc ($bfloat x) ($bfloat y) ($bfloat z) ($bfloat p))))
2639 (t
2640 (give-up))))))
2641
2642 ;;; Other Jacobian elliptic functions
2643
2644 ;; jacobi_ns(u,m) = 1/jacobi_sn(u,m)
2645 (defprop %jacobi_ns
2646 ((u m)
2647 ;; diff wrt u
2648 ((mtimes) -1 ((%jacobi_cn) u m) ((%jacobi_dn) u m)
2649 ((mexpt) ((%jacobi_sn) u m) -2))
2650 ;; diff wrt m
2651 ((mtimes) -1 ((mexpt) ((%jacobi_sn) u m) -2)
2652 ((mplus)
2653 ((mtimes) ((rat) 1 2)
2654 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
2655 ((mexpt) ((%jacobi_cn) u m) 2)
2656 ((%jacobi_sn) u m))
2657 ((mtimes) ((rat) 1 2) ((mexpt) m -1)
2658 ((%jacobi_cn) u m) ((%jacobi_dn) u m)
2659 ((mplus) u
2660 ((mtimes) -1
2661 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
2662 ((%elliptic_e) ((%asin) ((%jacobi_sn) u m))
2663 m)))))))
2664 grad)
2665
2666 (def-simplifier jacobi_ns (u m)
2667 (let (coef args)
2668 (cond
2669 ((float-numerical-eval-p u m)
2670 (to (bigfloat:/ (bigfloat::sn (bigfloat:to ($float u))
2671 (bigfloat:to ($float m))))))
2672 ((setf args (complex-float-numerical-eval-p u m))
2673 (destructuring-bind (u m)
2674 args
2675 (to (bigfloat:/ (bigfloat::sn (bigfloat:to ($float u))
2676 (bigfloat:to ($float m)))))))
2677 ((bigfloat-numerical-eval-p u m)
2678 (let ((uu (bigfloat:to ($bfloat u)))
2679 (mm (bigfloat:to ($bfloat m))))
2680 (to (bigfloat:/ (bigfloat::sn uu mm)))))
2681 ((setf args (complex-bigfloat-numerical-eval-p u m))
2682 (destructuring-bind (u m)
2683 args
2684 (let ((uu (bigfloat:to ($bfloat u)))
2685 (mm (bigfloat:to ($bfloat m))))
2686 (to (bigfloat:/ (bigfloat::sn uu mm))))))
2687 ((zerop1 m)
2688 ;; A&S 16.6.10
2689 (ftake '%csc u))
2690 ((onep1 m)
2691 ;; A&S 16.6.10
2692 (ftake '%coth u))
2693 ((zerop1 u)
2694 (dbz-err1 'jacobi_ns))
2695 ((and $trigsign (mminusp* u))
2696 ;; ns is odd
2697 (neg (ftake* '%jacobi_ns (neg u) m)))
2698 ((and $triginverses
2699 (listp u)
2700 (member (caar u) '(%inverse_jacobi_sn
2701 %inverse_jacobi_ns
2702 %inverse_jacobi_cn
2703 %inverse_jacobi_nc
2704 %inverse_jacobi_dn
2705 %inverse_jacobi_nd
2706 %inverse_jacobi_sc
2707 %inverse_jacobi_cs
2708 %inverse_jacobi_sd
2709 %inverse_jacobi_ds
2710 %inverse_jacobi_cd
2711 %inverse_jacobi_dc))
2712 (alike1 (third u) m))
2713 (cond ((eq (caar u) '%inverse_jacobi_ns)
2714 (second u))
2715 (t
2716 ;; Express in terms of sn:
2717 ;; ns(x) = 1/sn(x)
2718 (div 1 (ftake '%jacobi_sn u m)))))
2719 ;; A&S 16.20 (Jacobi's Imaginary transformation)
2720 ((and $%iargs (multiplep u '$%i))
2721 ;; ns(i*u) = 1/sn(i*u) = -i/sc(u,m1) = -i*cs(u,m1)
2722 (neg (mul '$%i
2723 (ftake* '%jacobi_cs (coeff u '$%i 1) (add 1 (neg m))))))
2724 ((setq coef (kc-arg2 u m))
2725 ;; A&S 16.8.10
2726 ;;
2727 ;; ns(m*K+u) = 1/sn(m*K+u)
2728 ;;
2729 (destructuring-bind (lin const)
2730 coef
2731 (cond ((integerp lin)
2732 (ecase (mod lin 4)
2733 (0
2734 ;; ns(4*m*K+u) = ns(u)
2735 ;; ns(0) = infinity
2736 (if (zerop1 const)
2737 (dbz-err1 'jacobi_ns)
2738 (ftake '%jacobi_ns const m)))
2739 (1
2740 ;; ns(4*m*K + K + u) = ns(K+u) = dc(u)
2741 ;; ns(K) = 1
2742 (if (zerop1 const)
2743 1
2744 (ftake '%jacobi_dc const m)))
2745 (2
2746 ;; ns(4*m*K+2*K + u) = ns(2*K+u) = -ns(u)
2747 ;; ns(2*K) = infinity
2748 (if (zerop1 const)
2749 (dbz-err1 'jacobi_ns)
2750 (neg (ftake '%jacobi_ns const m))))
2751 (3
2752 ;; ns(4*m*K+3*K+u) = ns(2*K + K + u) = -ns(K+u) = -dc(u)
2753 ;; ns(3*K) = -1
2754 (if (zerop1 const)
2755 -1
2756 (neg (ftake '%jacobi_dc const m))))))
2757 ((and (alike1 lin 1//2)
2758 (zerop1 const))
2759 (div 1 (ftake '%jacobi_sn u m)))
2760 (t
2761 (give-up)))))
2762 (t
2763 ;; Nothing to do
2764 (give-up)))))
2765
2766 ;; jacobi_nc(u,m) = 1/jacobi_cn(u,m)
2767 (defprop %jacobi_nc
2768 ((u m)
2769 ;; wrt u
2770 ((mtimes) ((mexpt) ((%jacobi_cn) u m) -2)
2771 ((%jacobi_dn) u m) ((%jacobi_sn) u m))
2772 ;; wrt m
2773 ((mtimes) -1 ((mexpt) ((%jacobi_cn) u m) -2)
2774 ((mplus)
2775 ((mtimes) ((rat) -1 2)
2776 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
2777 ((%jacobi_cn) u m) ((mexpt) ((%jacobi_sn) u m) 2))
2778 ((mtimes) ((rat) -1 2) ((mexpt) m -1)
2779 ((%jacobi_dn) u m) ((%jacobi_sn) u m)
2780 ((mplus) u
2781 ((mtimes) -1 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
2782 ((%elliptic_e) ((%asin) ((%jacobi_sn) u m)) m)))))))
2783 grad)
2784
2785 (def-simplifier jacobi_nc (u m)
2786 (let (coef args)
2787 (cond
2788 ((float-numerical-eval-p u m)
2789 (to (bigfloat:/ (bigfloat::cn (bigfloat:to ($float u))
2790 (bigfloat:to ($float m))))))
2791 ((setf args (complex-float-numerical-eval-p u m))
2792 (destructuring-bind (u m)
2793 args
2794 (to (bigfloat:/ (bigfloat::cn (bigfloat:to ($float u))
2795 (bigfloat:to ($float m)))))))
2796 ((bigfloat-numerical-eval-p u m)
2797 (let ((uu (bigfloat:to ($bfloat u)))
2798 (mm (bigfloat:to ($bfloat m))))
2799 (to (bigfloat:/ (bigfloat::cn uu mm)))))
2800 ((setf args (complex-bigfloat-numerical-eval-p u m))
2801 (destructuring-bind (u m)
2802 args
2803 (let ((uu (bigfloat:to ($bfloat u)))
2804 (mm (bigfloat:to ($bfloat m))))
2805 (to (bigfloat:/ (bigfloat::cn uu mm))))))
2806 ((zerop1 u)
2807 1)
2808 ((zerop1 m)
2809 ;; A&S 16.6.8
2810 (ftake '%sec u))
2811 ((onep1 m)
2812 ;; A&S 16.6.8
2813 (ftake '%cosh u))
2814 ((and $trigsign (mminusp* u))
2815 ;; nc is even
2816 (ftake* '%jacobi_nc (neg u) m))
2817 ((and $triginverses
2818 (listp u)
2819 (member (caar u) '(%inverse_jacobi_sn
2820 %inverse_jacobi_ns
2821 %inverse_jacobi_cn
2822 %inverse_jacobi_nc
2823 %inverse_jacobi_dn
2824 %inverse_jacobi_nd
2825 %inverse_jacobi_sc
2826 %inverse_jacobi_cs
2827 %inverse_jacobi_sd
2828 %inverse_jacobi_ds
2829 %inverse_jacobi_cd
2830 %inverse_jacobi_dc))
2831 (alike1 (third u) m))
2832 (cond ((eq (caar u) '%inverse_jacobi_nc)
2833 (second u))
2834 (t
2835 ;; Express in terms of cn:
2836 ;; nc(x) = 1/cn(x)
2837 (div 1 (ftake '%jacobi_cn u m)))))
2838 ;; A&S 16.20 (Jacobi's Imaginary transformation)
2839 ((and $%iargs (multiplep u '$%i))
2840 ;; nc(i*u) = 1/cn(i*u) = 1/nc(u,1-m) = cn(u,1-m)
2841 (ftake* '%jacobi_cn (coeff u '$%i 1) (add 1 (neg m))))
2842 ((setq coef (kc-arg2 u m))
2843 ;; A&S 16.8.8
2844 ;;
2845 ;; nc(u) = 1/cn(u)
2846 ;;
2847 (destructuring-bind (lin const)
2848 coef
2849 (cond ((integerp lin)
2850 (ecase (mod lin 4)
2851 (0
2852 ;; nc(4*m*K+u) = nc(u)
2853 ;; nc(0) = 1
2854 (if (zerop1 const)
2855 1
2856 (ftake '%jacobi_nc const m)))
2857 (1
2858 ;; nc(4*m*K+K+u) = nc(K+u) = -ds(u)/sqrt(1-m)
2859 ;; nc(K) = infinity
2860 (if (zerop1 const)
2861 (dbz-err1 'jacobi_nc)
2862 (neg (div (ftake '%jacobi_ds const m)
2863 (power (sub 1 m) 1//2)))))
2864 (2
2865 ;; nc(4*m*K+2*K+u) = nc(2*K+u) = -nc(u)
2866 ;; nc(2*K) = -1
2867 (if (zerop1 const)
2868 -1
2869 (neg (ftake '%jacobi_nc const m))))
2870 (3
2871 ;; nc(4*m*K+3*K+u) = nc(3*K+u) = nc(2*K+K+u) =
2872 ;; -nc(K+u) = ds(u)/sqrt(1-m)
2873 ;;
2874 ;; nc(3*K) = infinity
2875 (if (zerop1 const)
2876 (dbz-err1 'jacobi_nc)
2877 (div (ftake '%jacobi_ds const m)
2878 (power (sub 1 m) 1//2))))))
2879 ((and (alike1 1//2 lin)
2880 (zerop1 const))
2881 (div 1 (ftake '%jacobi_cn u m)))
2882 (t
2883 (give-up)))))
2884 (t
2885 ;; Nothing to do
2886 (give-up)))))
2887
2888 ;; jacobi_nd(u,m) = 1/jacobi_dn(u,m)
2889 (defprop %jacobi_nd
2890 ((u m)
2891 ;; wrt u
2892 ((mtimes) m ((%jacobi_cn) u m)
2893 ((mexpt) ((%jacobi_dn) u m) -2) ((%jacobi_sn) u m))
2894 ;; wrt m
2895 ((mtimes) -1 ((mexpt) ((%jacobi_dn) u m) -2)
2896 ((mplus)
2897 ((mtimes) ((rat) -1 2)
2898 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
2899 ((%jacobi_dn) u m)
2900 ((mexpt) ((%jacobi_sn) u m) 2))
2901 ((mtimes) ((rat) -1 2) ((%jacobi_cn) u m)
2902 ((%jacobi_sn) u m)
2903 ((mplus) u
2904 ((mtimes) -1
2905 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
2906 ((%elliptic_e) ((%asin) ((%jacobi_sn) u m))
2907 m)))))))
2908 grad)
2909
2910 (def-simplifier jacobi_nd (u m)
2911 (let (coef args)
2912 (cond
2913 ((float-numerical-eval-p u m)
2914 (to (bigfloat:/ (bigfloat::dn (bigfloat:to ($float u))
2915 (bigfloat:to ($float m))))))
2916 ((setf args (complex-float-numerical-eval-p u m))
2917 (destructuring-bind (u m)
2918 args
2919 (to (bigfloat:/ (bigfloat::dn (bigfloat:to ($float u))
2920 (bigfloat:to ($float m)))))))
2921 ((bigfloat-numerical-eval-p u m)
2922 (let ((uu (bigfloat:to ($bfloat u)))
2923 (mm (bigfloat:to ($bfloat m))))
2924 (to (bigfloat:/ (bigfloat::dn uu mm)))))
2925 ((setf args (complex-bigfloat-numerical-eval-p u m))
2926 (destructuring-bind (u m)
2927 args
2928 (let ((uu (bigfloat:to ($bfloat u)))
2929 (mm (bigfloat:to ($bfloat m))))
2930 (to (bigfloat:/ (bigfloat::dn uu mm))))))
2931 ((zerop1 u)
2932 1)
2933 ((zerop1 m)
2934 ;; A&S 16.6.6
2935 1)
2936 ((onep1 m)
2937 ;; A&S 16.6.6
2938 (ftake '%cosh u))
2939 ((and $trigsign (mminusp* u))
2940 ;; nd is even
2941 (ftake* '%jacobi_nd (neg u) m))
2942 ((and $triginverses
2943 (listp u)
2944 (member (caar u) '(%inverse_jacobi_sn
2945 %inverse_jacobi_ns
2946 %inverse_jacobi_cn
2947 %inverse_jacobi_nc
2948 %inverse_jacobi_dn
2949 %inverse_jacobi_nd
2950 %inverse_jacobi_sc
2951 %inverse_jacobi_cs
2952 %inverse_jacobi_sd
2953 %inverse_jacobi_ds
2954 %inverse_jacobi_cd
2955 %inverse_jacobi_dc))
2956 (alike1 (third u) m))
2957 (cond ((eq (caar u) '%inverse_jacobi_nd)
2958 (second u))
2959 (t
2960 ;; Express in terms of dn:
2961 ;; nd(x) = 1/dn(x)
2962 (div 1 (ftake '%jacobi_dn u m)))))
2963 ;; A&S 16.20 (Jacobi's Imaginary transformation)
2964 ((and $%iargs (multiplep u '$%i))
2965 ;; nd(i*u) = 1/dn(i*u) = 1/dc(u,1-m) = cd(u,1-m)
2966 (ftake* '%jacobi_cd (coeff u '$%i 1) (add 1 (neg m))))
2967 ((setq coef (kc-arg2 u m))
2968 ;; A&S 16.8.6
2969 ;;
2970 (destructuring-bind (lin const)
2971 coef
2972 (cond ((integerp lin)
2973 ;; nd has period 2K
2974 (ecase (mod lin 2)
2975 (0
2976 ;; nd(2*m*K+u) = nd(u)
2977 ;; nd(0) = 1
2978 (if (zerop1 const)
2979 1
2980 (ftake '%jacobi_nd const m)))
2981 (1
2982 ;; nd(2*m*K+K+u) = nd(K+u) = dn(u)/sqrt(1-m)
2983 ;; nd(K) = 1/sqrt(1-m)
2984 (if (zerop1 const)
2985 (power (sub 1 m) -1//2)
2986 (div (ftake '%jacobi_nd const m)
2987 (power (sub 1 m) 1//2))))))
2988 (t
2989 (give-up)))))
2990 (t
2991 ;; Nothing to do
2992 (give-up)))))
2993
2994 ;; jacobi_sc(u,m) = jacobi_sn/jacobi_cn
2995 (defprop %jacobi_sc
2996 ((u m)
2997 ;; wrt u
2998 ((mtimes) ((mexpt) ((%jacobi_cn) u m) -2)
2999 ((%jacobi_dn) u m))
3000 ;; wrt m
3001 ((mplus)
3002 ((mtimes) ((mexpt) ((%jacobi_cn) u m) -1)
3003 ((mplus)
3004 ((mtimes) ((rat) 1 2)
3005 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3006 ((mexpt) ((%jacobi_cn) u m) 2)
3007 ((%jacobi_sn) u m))
3008 ((mtimes) ((rat) 1 2) ((mexpt) m -1)
3009 ((%jacobi_cn) u m) ((%jacobi_dn) u m)
3010 ((mplus) u
3011 ((mtimes) -1
3012 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3013 ((%elliptic_e) ((%asin) ((%jacobi_sn) u m))
3014 m))))))
3015 ((mtimes) -1 ((mexpt) ((%jacobi_cn) u m) -2)
3016 ((%jacobi_sn) u m)
3017 ((mplus)
3018 ((mtimes) ((rat) -1 2)
3019 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3020 ((%jacobi_cn) u m)
3021 ((mexpt) ((%jacobi_sn) u m) 2))
3022 ((mtimes) ((rat) -1 2) ((mexpt) m -1)
3023 ((%jacobi_dn) u m) ((%jacobi_sn) u m)
3024 ((mplus) u
3025 ((mtimes) -1
3026 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3027 ((%elliptic_e) ((%asin) ((%jacobi_sn) u m))
3028 m))))))))
3029 grad)
3030
3031 (def-simplifier jacobi_sc (u m)
3032 (let (coef args)
3033 (cond
3034 ((float-numerical-eval-p u m)
3035 (let ((fu (bigfloat:to ($float u)))
3036 (fm (bigfloat:to ($float m))))
3037 (to (bigfloat:/ (bigfloat::sn fu fm) (bigfloat::cn fu fm)))))
3038 ((setf args (complex-float-numerical-eval-p u m))
3039 (destructuring-bind (u m)
3040 args
3041 (let ((fu (bigfloat:to ($float u)))
3042 (fm (bigfloat:to ($float m))))
3043 (to (bigfloat:/ (bigfloat::sn fu fm) (bigfloat::cn fu fm))))))
3044 ((bigfloat-numerical-eval-p u m)
3045 (let ((uu (bigfloat:to ($bfloat u)))
3046 (mm (bigfloat:to ($bfloat m))))
3047 (to (bigfloat:/ (bigfloat::sn uu mm)
3048 (bigfloat::cn uu mm)))))
3049 ((setf args (complex-bigfloat-numerical-eval-p u m))
3050 (destructuring-bind (u m)
3051 args
3052 (let ((uu (bigfloat:to ($bfloat u)))
3053 (mm (bigfloat:to ($bfloat m))))
3054 (to (bigfloat:/ (bigfloat::sn uu mm)
3055 (bigfloat::cn uu mm))))))
3056 ((zerop1 u)
3057 0)
3058 ((zerop1 m)
3059 ;; A&S 16.6.9
3060 (ftake '%tan u))
3061 ((onep1 m)
3062 ;; A&S 16.6.9
3063 (ftake '%sinh u))
3064 ((and $trigsign (mminusp* u))
3065 ;; sc is odd
3066 (neg (ftake* '%jacobi_sc (neg u) m)))
3067 ((and $triginverses
3068 (listp u)
3069 (member (caar u) '(%inverse_jacobi_sn
3070 %inverse_jacobi_ns
3071 %inverse_jacobi_cn
3072 %inverse_jacobi_nc
3073 %inverse_jacobi_dn
3074 %inverse_jacobi_nd
3075 %inverse_jacobi_sc
3076 %inverse_jacobi_cs
3077 %inverse_jacobi_sd
3078 %inverse_jacobi_ds
3079 %inverse_jacobi_cd
3080 %inverse_jacobi_dc))
3081 (alike1 (third u) m))
3082 (cond ((eq (caar u) '%inverse_jacobi_sc)
3083 (second u))
3084 (t
3085 ;; Express in terms of sn and cn
3086 ;; sc(x) = sn(x)/cn(x)
3087 (div (ftake '%jacobi_sn u m)
3088 (ftake '%jacobi_cn u m)))))
3089 ;; A&S 16.20 (Jacobi's Imaginary transformation)
3090 ((and $%iargs (multiplep u '$%i))
3091 ;; sc(i*u) = sn(i*u)/cn(i*u) = i*sc(u,m1)/nc(u,m1) = i*sn(u,m1)
3092 (mul '$%i
3093 (ftake* '%jacobi_sn (coeff u '$%i 1) (add 1 (neg m)))))
3094 ((setq coef (kc-arg2 u m))
3095 ;; A&S 16.8.9
3096 ;; sc(2*m*K+u) = sc(u)
3097 (destructuring-bind (lin const)
3098 coef
3099 (cond ((integerp lin)
3100 (ecase (mod lin 2)
3101 (0
3102 ;; sc(2*m*K+ u) = sc(u)
3103 ;; sc(0) = 0
3104 (if (zerop1 const)
3105 1
3106 (ftake '%jacobi_sc const m)))
3107 (1
3108 ;; sc(2*m*K + K + u) = sc(K+u)= - cs(u)/sqrt(1-m)
3109 ;; sc(K) = infinity
3110 (if (zerop1 const)
3111 (dbz-err1 'jacobi_sc)
3112 (mul -1
3113 (div (ftake* '%jacobi_cs const m)
3114 (power (sub 1 m) 1//2)))))))
3115 ((and (alike1 lin 1//2)
3116 (zerop1 const))
3117 ;; From A&S 16.3.3 and 16.5.2:
3118 ;; sc(1/2*K) = 1/(1-m)^(1/4)
3119 (power (sub 1 m) (div -1 4)))
3120 (t
3121 (give-up)))))
3122 (t
3123 ;; Nothing to do
3124 (give-up)))))
3125
3126 ;; jacobi_sd(u,m) = jacobi_sn/jacobi_dn
3127 (defprop %jacobi_sd
3128 ((u m)
3129 ;; wrt u
3130 ((mtimes) ((%jacobi_cn) u m)
3131 ((mexpt) ((%jacobi_dn) u m) -2))
3132 ;; wrt m
3133 ((mplus)
3134 ((mtimes) ((mexpt) ((%jacobi_dn) u m) -1)
3135 ((mplus)
3136 ((mtimes) ((rat) 1 2)
3137 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3138 ((mexpt) ((%jacobi_cn) u m) 2)
3139 ((%jacobi_sn) u m))
3140 ((mtimes) ((rat) 1 2) ((mexpt) m -1)
3141 ((%jacobi_cn) u m) ((%jacobi_dn) u m)
3142 ((mplus) u
3143 ((mtimes) -1
3144 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3145 ((%elliptic_e) ((%asin) ((%jacobi_sn) u m))
3146 m))))))
3147 ((mtimes) -1 ((mexpt) ((%jacobi_dn) u m) -2)
3148 ((%jacobi_sn) u m)
3149 ((mplus)
3150 ((mtimes) ((rat) -1 2)
3151 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3152 ((%jacobi_dn) u m)
3153 ((mexpt) ((%jacobi_sn) u m) 2))
3154 ((mtimes) ((rat) -1 2) ((%jacobi_cn) u m)
3155 ((%jacobi_sn) u m)
3156 ((mplus) u
3157 ((mtimes) -1
3158 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3159 ((%elliptic_e) ((%asin) ((%jacobi_sn) u m))
3160 m))))))))
3161 grad)
3162
3163 (def-simplifier jacobi_sd (u m)
3164 (let (coef args)
3165 (cond
3166 ((float-numerical-eval-p u m)
3167 (let ((fu (bigfloat:to ($float u)))
3168 (fm (bigfloat:to ($float m))))
3169 (to (bigfloat:/ (bigfloat::sn fu fm) (bigfloat::dn fu fm)))))
3170 ((setf args (complex-float-numerical-eval-p u m))
3171 (destructuring-bind (u m)
3172 args
3173 (let ((fu (bigfloat:to ($float u)))
3174 (fm (bigfloat:to ($float m))))
3175 (to (bigfloat:/ (bigfloat::sn fu fm) (bigfloat::dn fu fm))))))
3176 ((bigfloat-numerical-eval-p u m)
3177 (let ((uu (bigfloat:to ($bfloat u)))
3178 (mm (bigfloat:to ($bfloat m))))
3179 (to (bigfloat:/ (bigfloat::sn uu mm)
3180 (bigfloat::dn uu mm)))))
3181 ((setf args (complex-bigfloat-numerical-eval-p u m))
3182 (destructuring-bind (u m)
3183 args
3184 (let ((uu (bigfloat:to ($bfloat u)))
3185 (mm (bigfloat:to ($bfloat m))))
3186 (to (bigfloat:/ (bigfloat::sn uu mm)
3187 (bigfloat::dn uu mm))))))
3188 ((zerop1 u)
3189 0)
3190 ((zerop1 m)
3191 ;; A&S 16.6.5
3192 (ftake '%sin u))
3193 ((onep1 m)
3194 ;; A&S 16.6.5
3195 (ftake '%sinh u))
3196 ((and $trigsign (mminusp* u))
3197 ;; sd is odd
3198 (neg (ftake* '%jacobi_sd (neg u) m)))
3199 ((and $triginverses
3200 (listp u)
3201 (member (caar u) '(%inverse_jacobi_sn
3202 %inverse_jacobi_ns
3203 %inverse_jacobi_cn
3204 %inverse_jacobi_nc
3205 %inverse_jacobi_dn
3206 %inverse_jacobi_nd
3207 %inverse_jacobi_sc
3208 %inverse_jacobi_cs
3209 %inverse_jacobi_sd
3210 %inverse_jacobi_ds
3211 %inverse_jacobi_cd
3212 %inverse_jacobi_dc))
3213 (alike1 (third u) m))
3214 (cond ((eq (caar u) '%inverse_jacobi_sd)
3215 (second u))
3216 (t
3217 ;; Express in terms of sn and dn
3218 (div (ftake '%jacobi_sn u m)
3219 (ftake '%jacobi_dn u m)))))
3220 ;; A&S 16.20 (Jacobi's Imaginary transformation)
3221 ((and $%iargs (multiplep u '$%i))
3222 ;; sd(i*u) = sn(i*u)/dn(i*u) = i*sc(u,m1)/dc(u,m1) = i*sd(u,m1)
3223 (mul '$%i
3224 (ftake* '%jacobi_sd (coeff u '$%i 1) (add 1 (neg m)))))
3225 ((setq coef (kc-arg2 u m))
3226 ;; A&S 16.8.5
3227 ;; sd(4*m*K+u) = sd(u)
3228 (destructuring-bind (lin const)
3229 coef
3230 (cond ((integerp lin)
3231 (ecase (mod lin 4)
3232 (0
3233 ;; sd(4*m*K+u) = sd(u)
3234 ;; sd(0) = 0
3235 (if (zerop1 const)
3236 0
3237 (ftake '%jacobi_sd const m)))
3238 (1
3239 ;; sd(4*m*K+K+u) = sd(K+u) = cn(u)/sqrt(1-m)
3240 ;; sd(K) = 1/sqrt(m1)
3241 (if (zerop1 const)
3242 (power (sub 1 m) 1//2)
3243 (div (ftake '%jacobi_cn const m)
3244 (power (sub 1 m) 1//2))))
3245 (2
3246 ;; sd(4*m*K+2*K+u) = sd(2*K+u) = -sd(u)
3247 ;; sd(2*K) = 0
3248 (if (zerop1 const)
3249 0
3250 (neg (ftake '%jacobi_sd const m))))
3251 (3
3252 ;; sd(4*m*K+3*K+u) = sd(3*K+u) = sd(2*K+K+u) =
3253 ;; -sd(K+u) = -cn(u)/sqrt(1-m)
3254 ;; sd(3*K) = -1/sqrt(m1)
3255 (if (zerop1 const)
3256 (neg (power (sub 1 m) -1//2))
3257 (neg (div (ftake '%jacobi_cn const m)
3258 (power (sub 1 m) 1//2)))))))
3259 ((and (alike1 lin 1//2)
3260 (zerop1 const))
3261 ;; jacobi_sn/jacobi_dn
3262 (div (ftake '%jacobi_sn
3263 (mul 1//2
3264 (ftake '%elliptic_kc m))
3265 m)
3266 (ftake '%jacobi_dn
3267 (mul 1//2
3268 (ftake '%elliptic_kc m))
3269 m)))
3270 (t
3271 (give-up)))))
3272 (t
3273 ;; Nothing to do
3274 (give-up)))))
3275
3276 ;; jacobi_cs(u,m) = jacobi_cn/jacobi_sn
3277 (defprop %jacobi_cs
3278 ((u m)
3279 ;; wrt u
3280 ((mtimes) -1 ((%jacobi_dn) u m)
3281 ((mexpt) ((%jacobi_sn) u m) -2))
3282 ;; wrt m
3283 ((mplus)
3284 ((mtimes) -1 ((%jacobi_cn) u m)
3285 ((mexpt) ((%jacobi_sn) u m) -2)
3286 ((mplus)
3287 ((mtimes) ((rat) 1 2)
3288 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3289 ((mexpt) ((%jacobi_cn) u m) 2)
3290 ((%jacobi_sn) u m))
3291 ((mtimes) ((rat) 1 2) ((mexpt) m -1)
3292 ((%jacobi_cn) u m) ((%jacobi_dn) u m)
3293 ((mplus) u
3294 ((mtimes) -1
3295 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3296 ((%elliptic_e) ((%asin) ((%jacobi_sn) u m))
3297 m))))))
3298 ((mtimes) ((mexpt) ((%jacobi_sn) u m) -1)
3299 ((mplus)
3300 ((mtimes) ((rat) -1 2)
3301 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3302 ((%jacobi_cn) u m)
3303 ((mexpt) ((%jacobi_sn) u m) 2))
3304 ((mtimes) ((rat) -1 2) ((mexpt) m -1)
3305 ((%jacobi_dn) u m) ((%jacobi_sn) u m)
3306 ((mplus) u
3307 ((mtimes) -1
3308 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3309 ((%elliptic_e) ((%asin) ((%jacobi_sn) u m))
3310 m))))))))
3311 grad)
3312
3313 (def-simplifier jacobi_cs (u m)
3314 (let (coef args)
3315 (cond
3316 ((float-numerical-eval-p u m)
3317 (let ((fu (bigfloat:to ($float u)))
3318 (fm (bigfloat:to ($float m))))
3319 (to (bigfloat:/ (bigfloat::cn fu fm) (bigfloat::sn fu fm)))))
3320 ((setf args (complex-float-numerical-eval-p u m))
3321 (destructuring-bind (u m)
3322 args
3323 (let ((fu (bigfloat:to ($float u)))
3324 (fm (bigfloat:to ($float m))))
3325 (to (bigfloat:/ (bigfloat::cn fu fm) (bigfloat::sn fu fm))))))
3326 ((bigfloat-numerical-eval-p u m)
3327 (let ((uu (bigfloat:to ($bfloat u)))
3328 (mm (bigfloat:to ($bfloat m))))
3329 (to (bigfloat:/ (bigfloat::cn uu mm)
3330 (bigfloat::sn uu mm)))))
3331 ((setf args (complex-bigfloat-numerical-eval-p u m))
3332 (destructuring-bind (u m)
3333 args
3334 (let ((uu (bigfloat:to ($bfloat u)))
3335 (mm (bigfloat:to ($bfloat m))))
3336 (to (bigfloat:/ (bigfloat::cn uu mm)
3337 (bigfloat::sn uu mm))))))
3338 ((zerop1 m)
3339 ;; A&S 16.6.12
3340 (ftake '%cot u))
3341 ((onep1 m)
3342 ;; A&S 16.6.12
3343 (ftake '%csch u))
3344 ((zerop1 u)
3345 (dbz-err1 'jacobi_cs))
3346 ((and $trigsign (mminusp* u))
3347 ;; cs is odd
3348 (neg (ftake* '%jacobi_cs (neg u) m)))
3349 ((and $triginverses
3350 (listp u)
3351 (member (caar u) '(%inverse_jacobi_sn
3352 %inverse_jacobi_ns
3353 %inverse_jacobi_cn
3354 %inverse_jacobi_nc
3355 %inverse_jacobi_dn
3356 %inverse_jacobi_nd
3357 %inverse_jacobi_sc
3358 %inverse_jacobi_cs
3359 %inverse_jacobi_sd
3360 %inverse_jacobi_ds
3361 %inverse_jacobi_cd
3362 %inverse_jacobi_dc))
3363 (alike1 (third u) m))
3364 (cond ((eq (caar u) '%inverse_jacobi_cs)
3365 (second u))
3366 (t
3367 ;; Express in terms of cn an sn
3368 (div (ftake '%jacobi_cn u m)
3369 (ftake '%jacobi_sn u m)))))
3370 ;; A&S 16.20 (Jacobi's Imaginary transformation)
3371 ((and $%iargs (multiplep u '$%i))
3372 ;; cs(i*u) = cn(i*u)/sn(i*u) = -i*nc(u,m1)/sc(u,m1) = -i*ns(u,m1)
3373 (neg (mul '$%i
3374 (ftake* '%jacobi_ns (coeff u '$%i 1) (add 1 (neg m))))))
3375 ((setq coef (kc-arg2 u m))
3376 ;; A&S 16.8.12
3377 ;;
3378 ;; cs(2*m*K + u) = cs(u)
3379 (destructuring-bind (lin const)
3380 coef
3381 (cond ((integerp lin)
3382 (ecase (mod lin 2)
3383 (0
3384 ;; cs(2*m*K + u) = cs(u)
3385 ;; cs(0) = infinity
3386 (if (zerop1 const)
3387 (dbz-err1 'jacobi_cs)
3388 (ftake '%jacobi_cs const m)))
3389 (1
3390 ;; cs(K+u) = -sqrt(1-m)*sc(u)
3391 ;; cs(K) = 0
3392 (if (zerop1 const)
3393 0
3394 (neg (mul (power (sub 1 m) 1//2)
3395 (ftake '%jacobi_sc const m)))))))
3396 ((and (alike1 lin 1//2)
3397 (zerop1 const))
3398 ;; 1/jacobi_sc
3399 (div 1
3400 (ftake '%jacobi_sc (mul 1//2
3401 (ftake '%elliptic_kc m))
3402 m)))
3403 (t
3404 (give-up)))))
3405 (t
3406 ;; Nothing to do
3407 (give-up)))))
3408
3409 ;; jacobi_cd(u,m) = jacobi_cn/jacobi_dn
3410 (defprop %jacobi_cd
3411 ((u m)
3412 ;; wrt u
3413 ((mtimes) ((mplus) -1 m)
3414 ((mexpt) ((%jacobi_dn) u m) -2)
3415 ((%jacobi_sn) u m))
3416 ;; wrt m
3417 ((mplus)
3418 ((mtimes) -1 ((%jacobi_cn) u m)
3419 ((mexpt) ((%jacobi_dn) u m) -2)
3420 ((mplus)
3421 ((mtimes) ((rat) -1 2)
3422 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3423 ((%jacobi_dn) u m)
3424 ((mexpt) ((%jacobi_sn) u m) 2))
3425 ((mtimes) ((rat) -1 2) ((%jacobi_cn) u m)
3426 ((%jacobi_sn) u m)
3427 ((mplus) u
3428 ((mtimes) -1
3429 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3430 ((%elliptic_e) ((%asin) ((%jacobi_sn) u m))
3431 m))))))
3432 ((mtimes) ((mexpt) ((%jacobi_dn) u m) -1)
3433 ((mplus)
3434 ((mtimes) ((rat) -1 2)
3435 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3436 ((%jacobi_cn) u m)
3437 ((mexpt) ((%jacobi_sn) u m) 2))
3438 ((mtimes) ((rat) -1 2) ((mexpt) m -1)
3439 ((%jacobi_dn) u m) ((%jacobi_sn) u m)
3440 ((mplus) u
3441 ((mtimes) -1
3442 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3443 ((%elliptic_e) ((%asin) ((%jacobi_sn) u m))
3444 m))))))))
3445 grad)
3446
3447 (def-simplifier jacobi_cd (u m)
3448 (let (coef args)
3449 (cond
3450 ((float-numerical-eval-p u m)
3451 (let ((fu (bigfloat:to ($float u)))
3452 (fm (bigfloat:to ($float m))))
3453 (to (bigfloat:/ (bigfloat::cn fu fm) (bigfloat::dn fu fm)))))
3454 ((setf args (complex-float-numerical-eval-p u m))
3455 (destructuring-bind (u m)
3456 args
3457 (let ((fu (bigfloat:to ($float u)))
3458 (fm (bigfloat:to ($float m))))
3459 (to (bigfloat:/ (bigfloat::cn fu fm) (bigfloat::dn fu fm))))))
3460 ((bigfloat-numerical-eval-p u m)
3461 (let ((uu (bigfloat:to ($bfloat u)))
3462 (mm (bigfloat:to ($bfloat m))))
3463 (to (bigfloat:/ (bigfloat::cn uu mm) (bigfloat::dn uu mm)))))
3464 ((setf args (complex-bigfloat-numerical-eval-p u m))
3465 (destructuring-bind (u m)
3466 args
3467 (let ((uu (bigfloat:to ($bfloat u)))
3468 (mm (bigfloat:to ($bfloat m))))
3469 (to (bigfloat:/ (bigfloat::cn uu mm) (bigfloat::dn uu mm))))))
3470 ((zerop1 u)
3471 1)
3472 ((zerop1 m)
3473 ;; A&S 16.6.4
3474 (ftake '%cos u))
3475 ((onep1 m)
3476 ;; A&S 16.6.4
3477 1)
3478 ((and $trigsign (mminusp* u))
3479 ;; cd is even
3480 (ftake* '%jacobi_cd (neg u) m))
3481 ((and $triginverses
3482 (listp u)
3483 (member (caar u) '(%inverse_jacobi_sn
3484 %inverse_jacobi_ns
3485 %inverse_jacobi_cn
3486 %inverse_jacobi_nc
3487 %inverse_jacobi_dn
3488 %inverse_jacobi_nd
3489 %inverse_jacobi_sc
3490 %inverse_jacobi_cs
3491 %inverse_jacobi_sd
3492 %inverse_jacobi_ds
3493 %inverse_jacobi_cd
3494 %inverse_jacobi_dc))
3495 (alike1 (third u) m))
3496 (cond ((eq (caar u) '%inverse_jacobi_cd)
3497 (second u))
3498 (t
3499 ;; Express in terms of cn and dn
3500 (div (ftake '%jacobi_cn u m)
3501 (ftake '%jacobi_dn u m)))))
3502 ;; A&S 16.20 (Jacobi's Imaginary transformation)
3503 ((and $%iargs (multiplep u '$%i))
3504 ;; cd(i*u) = cn(i*u)/dn(i*u) = nc(u,m1)/dc(u,m1) = nd(u,m1)
3505 (ftake* '%jacobi_nd (coeff u '$%i 1) (add 1 (neg m))))
3506 ((setf coef (kc-arg2 u m))
3507 ;; A&S 16.8.4
3508 ;;
3509 (destructuring-bind (lin const)
3510 coef
3511 (cond ((integerp lin)
3512 (ecase (mod lin 4)
3513 (0
3514 ;; cd(4*m*K + u) = cd(u)
3515 ;; cd(0) = 1
3516 (if (zerop1 const)
3517 1
3518 (ftake '%jacobi_cd const m)))
3519 (1
3520 ;; cd(4*m*K + K + u) = cd(K+u) = -sn(u)
3521 ;; cd(K) = 0
3522 (if (zerop1 const)
3523 0
3524 (neg (ftake '%jacobi_sn const m))))
3525 (2
3526 ;; cd(4*m*K + 2*K + u) = cd(2*K+u) = -cd(u)
3527 ;; cd(2*K) = -1
3528 (if (zerop1 const)
3529 -1
3530 (neg (ftake '%jacobi_cd const m))))
3531 (3
3532 ;; cd(4*m*K + 3*K + u) = cd(2*K + K + u) =
3533 ;; -cd(K+u) = sn(u)
3534 ;; cd(3*K) = 0
3535 (if (zerop1 const)
3536 0
3537 (ftake '%jacobi_sn const m)))))
3538 ((and (alike1 lin 1//2)
3539 (zerop1 const))
3540 ;; jacobi_cn/jacobi_dn
3541 (div (ftake '%jacobi_cn
3542 (mul 1//2
3543 (ftake '%elliptic_kc m))
3544 m)
3545 (ftake '%jacobi_dn
3546 (mul 1//2
3547 (ftake '%elliptic_kc m))
3548 m)))
3549 (t
3550 ;; Nothing to do
3551 (give-up)))))
3552 (t
3553 ;; Nothing to do
3554 (give-up)))))
3555
3556 ;; jacobi_ds(u,m) = jacobi_dn/jacobi_sn
3557 (defprop %jacobi_ds
3558 ((u m)
3559 ;; wrt u
3560 ((mtimes) -1 ((%jacobi_cn) u m)
3561 ((mexpt) ((%jacobi_sn) u m) -2))
3562 ;; wrt m
3563 ((mplus)
3564 ((mtimes) -1 ((%jacobi_dn) u m)
3565 ((mexpt) ((%jacobi_sn) u m) -2)
3566 ((mplus)
3567 ((mtimes) ((rat) 1 2)
3568 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3569 ((mexpt) ((%jacobi_cn) u m) 2)
3570 ((%jacobi_sn) u m))
3571 ((mtimes) ((rat) 1 2) ((mexpt) m -1)
3572 ((%jacobi_cn) u m) ((%jacobi_dn) u m)
3573 ((mplus) u
3574 ((mtimes) -1
3575 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3576 ((%elliptic_e) ((%asin) ((%jacobi_sn) u m))
3577 m))))))
3578 ((mtimes) ((mexpt) ((%jacobi_sn) u m) -1)
3579 ((mplus)
3580 ((mtimes) ((rat) -1 2)
3581 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3582 ((%jacobi_dn) u m)
3583 ((mexpt) ((%jacobi_sn) u m) 2))
3584 ((mtimes) ((rat) -1 2) ((%jacobi_cn) u m)
3585 ((%jacobi_sn) u m)
3586 ((mplus) u
3587 ((mtimes) -1
3588 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3589 ((%elliptic_e) ((%asin) ((%jacobi_sn) u m))
3590 m))))))))
3591 grad)
3592
3593 (def-simplifier jacobi_ds (u m)
3594 (let (coef args)
3595 (cond
3596 ((float-numerical-eval-p u m)
3597 (let ((fu (bigfloat:to ($float u)))
3598 (fm (bigfloat:to ($float m))))
3599 (to (bigfloat:/ (bigfloat::dn fu fm) (bigfloat::sn fu fm)))))
3600 ((setf args (complex-float-numerical-eval-p u m))
3601 (destructuring-bind (u m)
3602 args
3603 (let ((fu (bigfloat:to ($float u)))
3604 (fm (bigfloat:to ($float m))))
3605 (to (bigfloat:/ (bigfloat::dn fu fm) (bigfloat::sn fu fm))))))
3606 ((bigfloat-numerical-eval-p u m)
3607 (let ((uu (bigfloat:to ($bfloat u)))
3608 (mm (bigfloat:to ($bfloat m))))
3609 (to (bigfloat:/ (bigfloat::dn uu mm)
3610 (bigfloat::sn uu mm)))))
3611 ((setf args (complex-bigfloat-numerical-eval-p u m))
3612 (destructuring-bind (u m)
3613 args
3614 (let ((uu (bigfloat:to ($bfloat u)))
3615 (mm (bigfloat:to ($bfloat m))))
3616 (to (bigfloat:/ (bigfloat::dn uu mm)
3617 (bigfloat::sn uu mm))))))
3618 ((zerop1 m)
3619 ;; A&S 16.6.11
3620 (ftake '%csc u))
3621 ((onep1 m)
3622 ;; A&S 16.6.11
3623 (ftake '%csch u))
3624 ((zerop1 u)
3625 (dbz-err1 'jacobi_ds))
3626 ((and $trigsign (mminusp* u))
3627 (neg (ftake* '%jacobi_ds (neg u) m)))
3628 ((and $triginverses
3629 (listp u)
3630 (member (caar u) '(%inverse_jacobi_sn
3631 %inverse_jacobi_ns
3632 %inverse_jacobi_cn
3633 %inverse_jacobi_nc
3634 %inverse_jacobi_dn
3635 %inverse_jacobi_nd
3636 %inverse_jacobi_sc
3637 %inverse_jacobi_cs
3638 %inverse_jacobi_sd
3639 %inverse_jacobi_ds
3640 %inverse_jacobi_cd
3641 %inverse_jacobi_dc))
3642 (alike1 (third u) m))
3643 (cond ((eq (caar u) '%inverse_jacobi_ds)
3644 (second u))
3645 (t
3646 ;; Express in terms of dn and sn
3647 (div (ftake '%jacobi_dn u m)
3648 (ftake '%jacobi_sn u m)))))
3649 ;; A&S 16.20 (Jacobi's Imaginary transformation)
3650 ((and $%iargs (multiplep u '$%i))
3651 ;; ds(i*u) = dn(i*u)/sn(i*u) = -i*dc(u,m1)/sc(u,m1) = -i*ds(u,m1)
3652 (neg (mul '$%i
3653 (ftake* '%jacobi_ds (coeff u '$%i 1) (add 1 (neg m))))))
3654 ((setf coef (kc-arg2 u m))
3655 ;; A&S 16.8.11
3656 (destructuring-bind (lin const)
3657 coef
3658 (cond ((integerp lin)
3659 (ecase (mod lin 4)
3660 (0
3661 ;; ds(4*m*K + u) = ds(u)
3662 ;; ds(0) = infinity
3663 (if (zerop1 const)
3664 (dbz-err1 'jacobi_ds)
3665 (ftake '%jacobi_ds const m)))
3666 (1
3667 ;; ds(4*m*K + K + u) = ds(K+u) = sqrt(1-m)*nc(u)
3668 ;; ds(K) = sqrt(1-m)
3669 (if (zerop1 const)
3670 (power (sub 1 m) 1//2)
3671 (mul (power (sub 1 m) 1//2)
3672 (ftake '%jacobi_nc const m))))
3673 (2
3674 ;; ds(4*m*K + 2*K + u) = ds(2*K+u) = -ds(u)
3675 ;; ds(0) = pole
3676 (if (zerop1 const)
3677 (dbz-err1 'jacobi_ds)
3678 (neg (ftake '%jacobi_ds const m))))
3679 (3
3680 ;; ds(4*m*K + 3*K + u) = ds(2*K + K + u) =
3681 ;; -ds(K+u) = -sqrt(1-m)*nc(u)
3682 ;; ds(3*K) = -sqrt(1-m)
3683 (if (zerop1 const)
3684 (neg (power (sub 1 m) 1//2))
3685 (neg (mul (power (sub 1 m) 1//2)
3686 (ftake '%jacobi_nc u m)))))))
3687 ((and (alike1 lin 1//2)
3688 (zerop1 const))
3689 ;; jacobi_dn/jacobi_sn
3690 (div
3691 (ftake '%jacobi_dn
3692 (mul 1//2 (ftake '%elliptic_kc m))
3693 m)
3694 (ftake '%jacobi_sn
3695 (mul 1//2 (ftake '%elliptic_kc m))
3696 m)))
3697 (t
3698 ;; Nothing to do
3699 (give-up)))))
3700 (t
3701 ;; Nothing to do
3702 (give-up)))))
3703
3704 ;; jacobi_dc(u,m) = jacobi_dn/jacobi_cn
3705 (defprop %jacobi_dc
3706 ((u m)
3707 ;; wrt u
3708 ((mtimes) ((mplus) 1 ((mtimes) -1 m))
3709 ((mexpt) ((%jacobi_cn) u m) -2)
3710 ((%jacobi_sn) u m))
3711 ;; wrt m
3712 ((mplus)
3713 ((mtimes) ((mexpt) ((%jacobi_cn) u m) -1)
3714 ((mplus)
3715 ((mtimes) ((rat) -1 2)
3716 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3717 ((%jacobi_dn) u m)
3718 ((mexpt) ((%jacobi_sn) u m) 2))
3719 ((mtimes) ((rat) -1 2) ((%jacobi_cn) u m)
3720 ((%jacobi_sn) u m)
3721 ((mplus) u
3722 ((mtimes) -1
3723 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3724 ((%elliptic_e) ((%asin) ((%jacobi_sn) u m))
3725 m))))))
3726 ((mtimes) -1 ((mexpt) ((%jacobi_cn) u m) -2)
3727 ((%jacobi_dn) u m)
3728 ((mplus)
3729 ((mtimes) ((rat) -1 2)
3730 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3731 ((%jacobi_cn) u m)
3732 ((mexpt) ((%jacobi_sn) u m) 2))
3733 ((mtimes) ((rat) -1 2) ((mexpt) m -1)
3734 ((%jacobi_dn) u m) ((%jacobi_sn) u m)
3735 ((mplus) u
3736 ((mtimes) -1
3737 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3738 ((%elliptic_e) ((%asin) ((%jacobi_sn) u m))
3739 m))))))))
3740 grad)
3741
3742 (def-simplifier jacobi_dc (u m)
3743 (let (coef args)
3744 (cond
3745 ((float-numerical-eval-p u m)
3746 (let ((fu (bigfloat:to ($float u)))
3747 (fm (bigfloat:to ($float m))))
3748 (to (bigfloat:/ (bigfloat::dn fu fm) (bigfloat::cn fu fm)))))
3749 ((setf args (complex-float-numerical-eval-p u m))
3750 (destructuring-bind (u m)
3751 args
3752 (let ((fu (bigfloat:to ($float u)))
3753 (fm (bigfloat:to ($float m))))
3754 (to (bigfloat:/ (bigfloat::dn fu fm) (bigfloat::cn fu fm))))))
3755 ((bigfloat-numerical-eval-p u m)
3756 (let ((uu (bigfloat:to ($bfloat u)))
3757 (mm (bigfloat:to ($bfloat m))))
3758 (to (bigfloat:/ (bigfloat::dn uu mm)
3759 (bigfloat::cn uu mm)))))
3760 ((setf args (complex-bigfloat-numerical-eval-p u m))
3761 (destructuring-bind (u m)
3762 args
3763 (let ((uu (bigfloat:to ($bfloat u)))
3764 (mm (bigfloat:to ($bfloat m))))
3765 (to (bigfloat:/ (bigfloat::dn uu mm)
3766 (bigfloat::cn uu mm))))))
3767 ((zerop1 u)
3768 1)
3769 ((zerop1 m)
3770 ;; A&S 16.6.7
3771 (ftake '%sec u))
3772 ((onep1 m)
3773 ;; A&S 16.6.7
3774 1)
3775 ((and $trigsign (mminusp* u))
3776 (ftake* '%jacobi_dc (neg u) m))
3777 ((and $triginverses
3778 (listp u)
3779 (member (caar u) '(%inverse_jacobi_sn
3780 %inverse_jacobi_ns
3781 %inverse_jacobi_cn
3782 %inverse_jacobi_nc
3783 %inverse_jacobi_dn
3784 %inverse_jacobi_nd
3785 %inverse_jacobi_sc
3786 %inverse_jacobi_cs
3787 %inverse_jacobi_sd
3788 %inverse_jacobi_ds
3789 %inverse_jacobi_cd
3790 %inverse_jacobi_dc))
3791 (alike1 (third u) m))
3792 (cond ((eq (caar u) '%inverse_jacobi_dc)
3793 (second u))
3794 (t
3795 ;; Express in terms of dn and cn
3796 (div (ftake '%jacobi_dn u m)
3797 (ftake '%jacobi_cn u m)))))
3798 ;; A&S 16.20 (Jacobi's Imaginary transformation)
3799 ((and $%iargs (multiplep u '$%i))
3800 ;; dc(i*u) = dn(i*u)/cn(i*u) = dc(u,m1)/nc(u,m1) = dn(u,m1)
3801 (ftake* '%jacobi_dn (coeff u '$%i 1) (add 1 (neg m))))
3802 ((setf coef (kc-arg2 u m))
3803 ;; See A&S 16.8.7
3804 (destructuring-bind (lin const)
3805 coef
3806 (cond ((integerp lin)
3807 (ecase (mod lin 4)
3808 (0
3809 ;; dc(4*m*K + u) = dc(u)
3810 ;; dc(0) = 1
3811 (if (zerop1 const)
3812 1
3813 (ftake '%jacobi_dc const m)))
3814 (1
3815 ;; dc(4*m*K + K + u) = dc(K+u) = -ns(u)
3816 ;; dc(K) = pole
3817 (if (zerop1 const)
3818 (dbz-err1 'jacobi_dc)
3819 (neg (ftake '%jacobi_ns const m))))
3820 (2
3821 ;; dc(4*m*K + 2*K + u) = dc(2*K+u) = -dc(u)
3822 ;; dc(2K) = -1
3823 (if (zerop1 const)
3824 -1
3825 (neg (ftake '%jacobi_dc const m))))
3826 (3
3827 ;; dc(4*m*K + 3*K + u) = dc(2*K + K + u) =
3828 ;; -dc(K+u) = ns(u)
3829 ;; dc(3*K) = ns(0) = inf
3830 (if (zerop1 const)
3831 (dbz-err1 'jacobi_dc)
3832 (ftake '%jacobi_dc const m)))))
3833 ((and (alike1 lin 1//2)
3834 (zerop1 const))
3835 ;; jacobi_dn/jacobi_cn
3836 (div
3837 (ftake '%jacobi_dn
3838 (mul 1//2 (ftake '%elliptic_kc m))
3839 m)
3840 (ftake '%jacobi_cn
3841 (mul 1//2 (ftake '%elliptic_kc m))
3842 m)))
3843 (t
3844 ;; Nothing to do
3845 (give-up)))))
3846 (t
3847 ;; Nothing to do
3848 (give-up)))))
3849
3850 ;;; Other inverse Jacobian functions
3851
3852 ;; inverse_jacobi_ns(x)
3853 ;;
3854 ;; Let u = inverse_jacobi_ns(x). Then jacobi_ns(u) = x or
3855 ;; 1/jacobi_sn(u) = x or
3856 ;;
3857 ;; jacobi_sn(u) = 1/x
3858 ;;
3859 ;; so u = inverse_jacobi_sn(1/x)
3860 (defprop %inverse_jacobi_ns
3861 ((x m)
3862 ;; Whittaker and Watson, example in 22.122
3863 ;; inverse_jacobi_ns(u,m) = integrate(1/sqrt(t^2-1)/sqrt(t^2-m), t, u, inf)
3864 ;; -> -1/sqrt(x^2-1)/sqrt(x^2-m)
3865 ((mtimes) -1
3866 ((mexpt) ((mplus) -1 ((mexpt) x 2)) ((rat) -1 2))
3867 ((mexpt)
3868 ((mplus) ((mtimes simp ratsimp) -1 m) ((mexpt) x 2))
3869 ((rat) -1 2)))
3870 ;; wrt m
3871 ; ((%derivative) ((%inverse_jacobi_ns) x m) m 1)
3872 nil)
3873 grad)
3874
3875 (def-simplifier inverse_jacobi_ns (u m)
3876 (let (args)
3877 (cond
3878 ((float-numerical-eval-p u m)
3879 ;; Numerically evaluate asn
3880 ;;
3881 ;; ans(x,m) = asn(1/x,m) = F(asin(1/x),m)
3882 (to (elliptic-f (cl:asin (/ ($float u))) ($float m))))
3883 ((complex-float-numerical-eval-p u m)
3884 (to (elliptic-f (cl:asin (/ (complex ($realpart ($float u)) ($imagpart ($float u)))))
3885 (complex ($realpart ($float m)) ($imagpart ($float m))))))
3886 ((bigfloat-numerical-eval-p u m)
3887 (to (bigfloat::bf-elliptic-f (bigfloat:asin (bigfloat:/ (bigfloat:to ($bfloat u))))
3888 (bigfloat:to ($bfloat m)))))
3889 ((setf args (complex-bigfloat-numerical-eval-p u m))
3890 (destructuring-bind (u m)
3891 args
3892 (to (bigfloat::bf-elliptic-f (bigfloat:asin (bigfloat:/ (bigfloat:to ($bfloat u))))
3893 (bigfloat:to ($bfloat m))))))
3894 ((zerop1 m)
3895 ;; ans(x,0) = F(asin(1/x),0) = asin(1/x)
3896 (ftake '%elliptic_f (ftake '%asin (div 1 u)) 0))
3897 ((onep1 m)
3898 ;; ans(x,1) = F(asin(1/x),1) = log(tan(pi/2+asin(1/x)/2))
3899 (ftake '%elliptic_f (ftake '%asin (div 1 u)) 1))
3900 ((onep1 u)
3901 (ftake '%elliptic_kc m))
3902 ((alike1 u -1)
3903 (neg (ftake '%elliptic_kc m)))
3904 ((and (eq $triginverses '$all)
3905 (listp u)
3906 (eq (caar u) '%jacobi_ns)
3907 (alike1 (third u) m))
3908 ;; inverse_jacobi_ns(ns(u)) = u
3909 (second u))
3910 (t
3911 ;; Nothing to do
3912 (give-up)))))
3913
3914 ;; inverse_jacobi_nc(x)
3915 ;;
3916 ;; Let u = inverse_jacobi_nc(x). Then jacobi_nc(u) = x or
3917 ;; 1/jacobi_cn(u) = x or
3918 ;;
3919 ;; jacobi_cn(u) = 1/x
3920 ;;
3921 ;; so u = inverse_jacobi_cn(1/x)
3922 (defprop %inverse_jacobi_nc
3923 ((x m)
3924 ;; Whittaker and Watson, example in 22.122
3925 ;; inverse_jacobi_nc(u,m) = integrate(1/sqrt(t^2-1)/sqrt((1-m)*t^2+m), t, 1, u)
3926 ;; -> 1/sqrt(x^2-1)/sqrt((1-m)*x^2+m)
3927 ((mtimes)
3928 ((mexpt) ((mplus) -1 ((mexpt) x 2)) ((rat) -1 2))
3929 ((mexpt)
3930 ((mplus) m
3931 ((mtimes) -1 ((mplus) -1 m) ((mexpt) x 2)))
3932 ((rat) -1 2)))
3933 ;; wrt m
3934 ; ((%derivative) ((%inverse_jacobi_nc) x m) m 1)
3935 nil)
3936 grad)
3937
3938 (def-simplifier inverse_jacobi_nc (u m)
3939 (cond ((or (float-numerical-eval-p u m)
3940 (complex-float-numerical-eval-p u m)
3941 (bigfloat-numerical-eval-p u m)
3942 (complex-bigfloat-numerical-eval-p u m))
3943 ;;
3944 (ftake '%inverse_jacobi_cn ($rectform (div 1 u)) m))
3945 ((onep1 u)
3946 0)
3947 ((alike1 u -1)
3948 (mul 2 (ftake '%elliptic_kc m)))
3949 ((and (eq $triginverses '$all)
3950 (listp u)
3951 (eq (caar u) '%jacobi_nc)
3952 (alike1 (third u) m))
3953 ;; inverse_jacobi_nc(nc(u)) = u
3954 (second u))
3955 (t
3956 ;; Nothing to do
3957 (give-up))))
3958
3959 ;; inverse_jacobi_nd(x)
3960 ;;
3961 ;; Let u = inverse_jacobi_nd(x). Then jacobi_nd(u) = x or
3962 ;; 1/jacobi_dn(u) = x or
3963 ;;
3964 ;; jacobi_dn(u) = 1/x
3965 ;;
3966 ;; so u = inverse_jacobi_dn(1/x)
3967 (defprop %inverse_jacobi_nd
3968 ((x m)
3969 ;; Whittaker and Watson, example in 22.122
3970 ;; inverse_jacobi_nd(u,m) = integrate(1/sqrt(t^2-1)/sqrt(1-(1-m)*t^2), t, 1, u)
3971 ;; -> 1/sqrt(u^2-1)/sqrt(1-(1-m)*t^2)
3972 ((mtimes)
3973 ((mexpt) ((mplus) -1 ((mexpt simp ratsimp) x 2))
3974 ((rat) -1 2))
3975 ((mexpt)
3976 ((mplus) 1
3977 ((mtimes) ((mplus) -1 m) ((mexpt simp ratsimp) x 2)))
3978 ((rat) -1 2)))
3979 ;; wrt m
3980 ; ((%derivative) ((%inverse_jacobi_nd) x m) m 1)
3981 nil)
3982 grad)
3983
3984 (def-simplifier inverse_jacobi_nd (u m)
3985 (cond ((or (float-numerical-eval-p u m)
3986 (complex-float-numerical-eval-p u m)
3987 (bigfloat-numerical-eval-p u m)
3988 (complex-bigfloat-numerical-eval-p u m))
3989 (ftake '%inverse_jacobi_dn ($rectform (div 1 u)) m))
3990 ((onep1 u)
3991 0)
3992 ((onep1 ($ratsimp (mul (power (sub 1 m) 1//2) u)))
3993 ;; jacobi_nd(1/sqrt(1-m),m) = K(m). This follows from
3994 ;; jacobi_dn(sqrt(1-m),m) = K(m).
3995 (ftake '%elliptic_kc m))
3996 ((and (eq $triginverses '$all)
3997 (listp u)
3998 (eq (caar u) '%jacobi_nd)
3999 (alike1 (third u) m))
4000 ;; inverse_jacobi_nd(nd(u)) = u
4001 (second u))
4002 (t
4003 ;; Nothing to do
4004 (give-up))))
4005
4006 ;; inverse_jacobi_sc(x)
4007 ;;
4008 ;; Let u = inverse_jacobi_sc(x). Then jacobi_sc(u) = x or
4009 ;; x = jacobi_sn(u)/jacobi_cn(u)
4010 ;;
4011 ;; x^2 = sn^2/cn^2
4012 ;; = sn^2/(1-sn^2)
4013 ;;
4014 ;; so
4015 ;;
4016 ;; sn^2 = x^2/(1+x^2)
4017 ;;
4018 ;; sn(u) = x/sqrt(1+x^2)
4019 ;;
4020 ;; u = inverse_sn(x/sqrt(1+x^2))
4021 ;;
4022 (defprop %inverse_jacobi_sc
4023 ((x m)
4024 ;; Whittaker and Watson, example in 22.122
4025 ;; inverse_jacobi_sc(u,m) = integrate(1/sqrt(1+t^2)/sqrt(1+(1-m)*t^2), t, 0, u)
4026 ;; -> 1/sqrt(1+x^2)/sqrt(1+(1-m)*x^2)
4027 ((mtimes)
4028 ((mexpt) ((mplus) 1 ((mexpt) x 2))
4029 ((rat) -1 2))
4030 ((mexpt)
4031 ((mplus) 1
4032 ((mtimes) -1 ((mplus) -1 m) ((mexpt) x 2)))
4033 ((rat) -1 2)))
4034 ;; wrt m
4035 ; ((%derivative) ((%inverse_jacobi_sc) x m) m 1)
4036 nil)
4037 grad)
4038
4039 (def-simplifier inverse_jacobi_sc (u m)
4040 (cond ((or (float-numerical-eval-p u m)
4041 (complex-float-numerical-eval-p u m)
4042 (bigfloat-numerical-eval-p u m)
4043 (complex-bigfloat-numerical-eval-p u m))
4044 (ftake '%inverse_jacobi_sn
4045 ($rectform (div u (power (add 1 (mul u u)) 1//2)))
4046 m))
4047 ((zerop1 u)
4048 ;; jacobi_sc(0,m) = 0
4049 0)
4050 ((and (eq $triginverses '$all)
4051 (listp u)
4052 (eq (caar u) '%jacobi_sc)
4053 (alike1 (third u) m))
4054 ;; inverse_jacobi_sc(sc(u)) = u
4055 (second u))
4056 (t
4057 ;; Nothing to do
4058 (give-up))))
4059
4060 ;; inverse_jacobi_sd(x)
4061 ;;
4062 ;; Let u = inverse_jacobi_sd(x). Then jacobi_sd(u) = x or
4063 ;; x = jacobi_sn(u)/jacobi_dn(u)
4064 ;;
4065 ;; x^2 = sn^2/dn^2
4066 ;; = sn^2/(1-m*sn^2)
4067 ;;
4068 ;; so
4069 ;;
4070 ;; sn^2 = x^2/(1+m*x^2)
4071 ;;
4072 ;; sn(u) = x/sqrt(1+m*x^2)
4073 ;;
4074 ;; u = inverse_sn(x/sqrt(1+m*x^2))
4075 ;;
4076 (defprop %inverse_jacobi_sd
4077 ((x m)
4078 ;; Whittaker and Watson, example in 22.122
4079 ;; inverse_jacobi_sd(u,m) = integrate(1/sqrt(1-(1-m)*t^2)/sqrt(1+m*t^2), t, 0, u)
4080 ;; -> 1/sqrt(1-(1-m)*x^2)/sqrt(1+m*x^2)
4081 ((mtimes)
4082 ((mexpt)
4083 ((mplus) 1 ((mtimes) ((mplus) -1 m) ((mexpt) x 2)))
4084 ((rat) -1 2))
4085 ((mexpt) ((mplus) 1 ((mtimes) m ((mexpt) x 2)))
4086 ((rat) -1 2)))
4087 ;; wrt m
4088 ; ((%derivative) ((%inverse_jacobi_sd) x m) m 1)
4089 nil)
4090 grad)
4091
4092 (def-simplifier inverse_jacobi_sd (u m)
4093 (cond ((or (float-numerical-eval-p u m)
4094 (complex-float-numerical-eval-p u m)
4095 (bigfloat-numerical-eval-p u m)
4096 (complex-bigfloat-numerical-eval-p u m))
4097 (ftake '%inverse_jacobi_sn
4098 ($rectform (div u (power (add 1 (mul m (mul u u))) 1//2)))
4099 m))
4100 ((zerop1 u)
4101 0)
4102 ((eql 0 ($ratsimp (sub u (div 1 (power (sub 1 m) 1//2)))))
4103 ;; inverse_jacobi_sd(1/sqrt(1-m), m) = elliptic_kc(m)
4104 ;;
4105 ;; We can see this from inverse_jacobi_sd(x,m) =
4106 ;; inverse_jacobi_sn(x/sqrt(1+m*x^2), m). So
4107 ;; inverse_jacobi_sd(1/sqrt(1-m),m) = inverse_jacobi_sn(1,m)
4108 (ftake '%elliptic_kc m))
4109 ((and (eq $triginverses '$all)
4110 (listp u)
4111 (eq (caar u) '%jacobi_sd)
4112 (alike1 (third u) m))
4113 ;; inverse_jacobi_sd(sd(u)) = u
4114 (second u))
4115 (t
4116 ;; Nothing to do
4117 (give-up))))
4118
4119 ;; inverse_jacobi_cs(x)
4120 ;;
4121 ;; Let u = inverse_jacobi_cs(x). Then jacobi_cs(u) = x or
4122 ;; 1/x = 1/jacobi_cs(u) = jacobi_sc(u)
4123 ;;
4124 ;; u = inverse_sc(1/x)
4125 ;;
4126 (defprop %inverse_jacobi_cs
4127 ((x m)
4128 ;; Whittaker and Watson, example in 22.122
4129 ;; inverse_jacobi_cs(u,m) = integrate(1/sqrt(t^2+1)/sqrt(t^2+(1-m)), t, u, inf)
4130 ;; -> -1/sqrt(x^2+1)/sqrt(x^2+(1-m))
4131 ((mtimes) -1
4132 ((mexpt) ((mplus) 1 ((mexpt simp ratsimp) x 2))
4133 ((rat) -1 2))
4134 ((mexpt) ((mplus) 1
4135 ((mtimes simp ratsimp) -1 m)
4136 ((mexpt simp ratsimp) x 2))
4137 ((rat) -1 2)))
4138 ;; wrt m
4139 ; ((%derivative) ((%inverse_jacobi_cs) x m) m 1)
4140 nil)
4141 grad)
4142
4143 (def-simplifier inverse_jacobi_cs (u m)
4144 (cond ((or (float-numerical-eval-p u m)
4145 (complex-float-numerical-eval-p u m)
4146 (bigfloat-numerical-eval-p u m)
4147 (complex-bigfloat-numerical-eval-p u m))
4148 (ftake '%inverse_jacobi_sc ($rectform (div 1 u)) m))
4149 ((zerop1 u)
4150 (ftake '%elliptic_kc m))
4151 (t
4152 ;; Nothing to do
4153 (give-up))))
4154
4155 ;; inverse_jacobi_cd(x)
4156 ;;
4157 ;; Let u = inverse_jacobi_cd(x). Then jacobi_cd(u) = x or
4158 ;; x = jacobi_cn(u)/jacobi_dn(u)
4159 ;;
4160 ;; x^2 = cn^2/dn^2
4161 ;; = (1-sn^2)/(1-m*sn^2)
4162 ;;
4163 ;; or
4164 ;;
4165 ;; sn^2 = (1-x^2)/(1-m*x^2)
4166 ;;
4167 ;; sn(u) = sqrt(1-x^2)/sqrt(1-m*x^2)
4168 ;;
4169 ;; u = inverse_sn(sqrt(1-x^2)/sqrt(1-m*x^2))
4170 ;;
4171 (defprop %inverse_jacobi_cd
4172 ((x m)
4173 ;; Whittaker and Watson, example in 22.122
4174 ;; inverse_jacobi_cd(u,m) = integrate(1/sqrt(1-t^2)/sqrt(1-m*t^2), t, u, 1)
4175 ;; -> -1/sqrt(1-x^2)/sqrt(1-m*x^2)
4176 ((mtimes) -1
4177 ((mexpt)
4178 ((mplus) 1 ((mtimes) -1 ((mexpt) x 2)))
4179 ((rat) -1 2))
4180 ((mexpt)
4181 ((mplus) 1 ((mtimes) -1 m ((mexpt) x 2)))
4182 ((rat) -1 2)))
4183 ;; wrt m
4184 ; ((%derivative) ((%inverse_jacobi_cd) x m) m 1)
4185 nil)
4186 grad)
4187
4188 (def-simplifier inverse_jacobi_cd (u m)
4189 (cond ((or (complex-float-numerical-eval-p u m)
4190 (complex-bigfloat-numerical-eval-p u m))
4191 (let (($numer t))
4192 (ftake '%inverse_jacobi_sn
4193 ($rectform (div (power (mul (sub 1 u) (add 1 u)) 1//2)
4194 (power (sub 1 (mul m (mul u u))) 1//2)))
4195 m)))
4196 ((onep1 u)
4197 0)
4198 ((zerop1 u)
4199 (ftake '%elliptic_kc m))
4200 ((and (eq $triginverses '$all)
4201 (listp u)
4202 (eq (caar u) '%jacobi_cd)
4203 (alike1 (third u) m))
4204 ;; inverse_jacobi_cd(cd(u)) = u
4205 (second u))
4206 (t
4207 ;; Nothing to do
4208 (give-up))))
4209
4210 ;; inverse_jacobi_ds(x)
4211 ;;
4212 ;; Let u = inverse_jacobi_ds(x). Then jacobi_ds(u) = x or
4213 ;; 1/x = 1/jacobi_ds(u) = jacobi_sd(u)
4214 ;;
4215 ;; u = inverse_sd(1/x)
4216 ;;
4217 (defprop %inverse_jacobi_ds
4218 ((x m)
4219 ;; Whittaker and Watson, example in 22.122
4220 ;; inverse_jacobi_ds(u,m) = integrate(1/sqrt(t^2-(1-m))/sqrt(t^2+m), t, u, inf)
4221 ;; -> -1/sqrt(x^2-(1-m))/sqrt(x^2+m)
4222 ((mtimes) -1
4223 ((mexpt)
4224 ((mplus) -1 m ((mexpt simp ratsimp) x 2))
4225 ((rat) -1 2))
4226 ((mexpt)
4227 ((mplus) m ((mexpt simp ratsimp) x 2))
4228 ((rat) -1 2)))
4229 ;; wrt m
4230 ; ((%derivative) ((%inverse_jacobi_ds) x m) m 1)
4231 nil)
4232 grad)
4233
4234 (def-simplifier inverse_jacobi_ds (u m)
4235 (cond ((or (float-numerical-eval-p u m)
4236 (complex-float-numerical-eval-p u m)
4237 (bigfloat-numerical-eval-p u m)
4238 (complex-bigfloat-numerical-eval-p u m))
4239 (ftake '%inverse_jacobi_sd ($rectform (div 1 u)) m))
4240 ((and $trigsign (mminusp* u))
4241 (neg (ftake* '%inverse_jacobi_ds (neg u) m)))
4242 ((eql 0 ($ratsimp (sub u (power (sub 1 m) 1//2))))
4243 ;; inverse_jacobi_ds(sqrt(1-m),m) = elliptic_kc(m)
4244 ;;
4245 ;; Since inverse_jacobi_ds(sqrt(1-m), m) =
4246 ;; inverse_jacobi_sd(1/sqrt(1-m),m). And we know from
4247 ;; above that this is elliptic_kc(m)
4248 (ftake '%elliptic_kc m))
4249 ((and (eq $triginverses '$all)
4250 (listp u)
4251 (eq (caar u) '%jacobi_ds)
4252 (alike1 (third u) m))
4253 ;; inverse_jacobi_ds(ds(u)) = u
4254 (second u))
4255 (t
4256 ;; Nothing to do
4257 (give-up))))
4258
4259
4260 ;; inverse_jacobi_dc(x)
4261 ;;
4262 ;; Let u = inverse_jacobi_dc(x). Then jacobi_dc(u) = x or
4263 ;; 1/x = 1/jacobi_dc(u) = jacobi_cd(u)
4264 ;;
4265 ;; u = inverse_cd(1/x)
4266 ;;
4267 (defprop %inverse_jacobi_dc
4268 ((x m)
4269 ;; Note: Whittaker and Watson, example in 22.122 says
4270 ;; inverse_jacobi_dc(u,m) = integrate(1/sqrt(t^2-1)/sqrt(t^2-m),
4271 ;; t, u, 1) but that seems wrong. A&S 17.4.47 says
4272 ;; integrate(1/sqrt(t^2-1)/sqrt(t^2-m), t, a, u) =
4273 ;; inverse_jacobi_cd(x,m). Lawden 3.2.8 says the same.
4274 ;; functions.wolfram.com says the derivative is
4275 ;; 1/sqrt(t^2-1)/sqrt(t^2-m).
4276 ((mtimes)
4277 ((mexpt)
4278 ((mplus) -1 ((mexpt simp ratsimp) x 2))
4279 ((rat) -1 2))
4280 ((mexpt)
4281 ((mplus)
4282 ((mtimes simp ratsimp) -1 m)
4283 ((mexpt simp ratsimp) x 2))
4284 ((rat) -1 2)))
4285 ;; wrt m
4286 ; ((%derivative) ((%inverse_jacobi_dc) x m) m 1)
4287 nil)
4288 grad)
4289
4290 (def-simplifier inverse_jacobi_dc (u m)
4291 (cond ((or (complex-float-numerical-eval-p u m)
4292 (complex-bigfloat-numerical-eval-p u m))
4293 (ftake '%inverse_jacobi_cd ($rectform (div 1 u)) m))
4294 ((onep1 u)
4295 0)
4296 ((and (eq $triginverses '$all)
4297 (listp u)
4298 (eq (caar u) '%jacobi_dc)
4299 (alike1 (third u) m))
4300 ;; inverse_jacobi_dc(dc(u)) = u
4301 (second u))
4302 (t
4303 ;; Nothing to do
4304 (give-up))))
4305
4306 ;; Convert an inverse Jacobian function into the equivalent elliptic
4307 ;; integral F.
4308 ;;
4309 ;; See A&S 17.4.41-17.4.52.
4310 (defun make-elliptic-f (e)
4311 (cond ((atom e)
4312 e)
4313 ((member (caar e) '(%inverse_jacobi_sc %inverse_jacobi_cs
4314 %inverse_jacobi_nd %inverse_jacobi_dn
4315 %inverse_jacobi_sn %inverse_jacobi_cd
4316 %inverse_jacobi_dc %inverse_jacobi_ns
4317 %inverse_jacobi_nc %inverse_jacobi_ds
4318 %inverse_jacobi_sd %inverse_jacobi_cn))
4319 ;; We have some inverse Jacobi function. Convert it to the F form.
4320 (destructuring-bind ((fn &rest ops) u m)
4321 e
4322 (declare (ignore ops))
4323 (ecase fn
4324 (%inverse_jacobi_sc
4325 ;; A&S 17.4.41
4326 (ftake '%elliptic_f (ftake '%atan u) m))
4327 (%inverse_jacobi_cs
4328 ;; A&S 17.4.42
4329 (ftake '%elliptic_f (ftake '%atan (div 1 u)) m))
4330 (%inverse_jacobi_nd
4331 ;; A&S 17.4.43
4332 (ftake '%elliptic_f
4333 (ftake '%asin
4334 (mul (power m -1//2)
4335 (div 1 u)
4336 (power (add -1 (mul u u))
4337 1//2)))
4338 m))
4339 (%inverse_jacobi_dn
4340 ;; A&S 17.4.44
4341 (ftake '%elliptic_f
4342 (ftake '%asin (mul
4343 (power m -1//2)
4344 (power (sub 1 (power u 2)) 1//2)))
4345 m))
4346 (%inverse_jacobi_sn
4347 ;; A&S 17.4.45
4348 (ftake '%elliptic_f (ftake '%asin u) m))
4349 (%inverse_jacobi_cd
4350 ;; A&S 17.4.46
4351 (ftake '%elliptic_f
4352 (ftake '%asin
4353 (power (mul (sub 1 (mul u u))
4354 (sub 1 (mul m u u)))
4355 1//2))
4356 m))
4357 (%inverse_jacobi_dc
4358 ;; A&S 17.4.47
4359 (ftake '%elliptic_f
4360 (ftake '%asin
4361 (power (mul (sub (mul u u) 1)
4362 (sub (mul u u) m))
4363 1//2))
4364 m))
4365 (%inverse_jacobi_ns
4366 ;; A&S 17.4.48
4367 (ftake '%elliptic_f (ftake '%asin (div 1 u)) m))
4368 (%inverse_jacobi_nc
4369 ;; A&S 17.4.49
4370 (ftake '%elliptic_f (ftake '%acos (div 1 u)) m))
4371 (%inverse_jacobi_ds
4372 ;; A&S 17.4.50
4373 (ftake '%elliptic_f
4374 (ftake '%asin
4375 (power (add m (mul u u))
4376 1//2)
4377 m)))
4378 (%inverse_jacobi_sd
4379 ;; A&S 17.4.51
4380 (ftake '%elliptic_f
4381 (ftake '%asin
4382 (div u
4383 (power (add 1 (mul m u u))
4384 1//2)))
4385 m))
4386 (%inverse_jacobi_cn
4387 ;; A&S 17.4.52
4388 (ftake '%elliptic_f (ftake '%acos u) m)))))
4389 (t
4390 (recur-apply #'make-elliptic-f e))))
4391
4392 (defmfun $make_elliptic_f (e)
4393 (if (atom e)
4394 e
4395 (simplify (make-elliptic-f e))))
4396
4397 (defun make-elliptic-e (e)
4398 (cond ((atom e) e)
4399 ((eq (caar e) '$elliptic_eu)
4400 (destructuring-bind ((ffun &rest ops) u m) e
4401 (declare (ignore ffun ops))
4402 (ftake '%elliptic_e (ftake '%asin (ftake '%jacobi_sn u m)) m)))
4403 (t
4404 (recur-apply #'make-elliptic-e e))))
4405
4406 (defmfun $make_elliptic_e (e)
4407 (if (atom e)
4408 e
4409 (simplify (make-elliptic-e e))))
4410
4411
4412 ;; Eu(u,m) = integrate(jacobi_dn(v,m)^2,v,0,u)
4413 ;; = integrate(sqrt((1-m*t^2)/(1-t^2)),t,0,jacobi_sn(u,m))
4414 ;;
4415 ;; Eu(u,m) = E(am(u),m)
4416 ;;
4417 ;; where E(u,m) is elliptic-e above.
4418 ;;
4419 ;; Checks.
4420 ;; Lawden gives the following relationships
4421 ;;
4422 ;; E(u+v) = E(u) + E(v) - m*sn(u)*sn(v)*sn(u+v)
4423 ;; E(u,0) = u, E(u,1) = tanh u
4424 ;;
4425 ;; E(i*u,k) = i*sc(u,k')*dn(u,k') - i*E(u,k') + i*u
4426 ;;
4427 ;; E(2*i*K') = 2*i*(K'-E')
4428 ;;
4429 ;; E(u + 2*i*K') = E(u) + 2*i*(K' - E')
4430 ;;
4431 ;; E(u+K) = E(u) + E - k^2*sn(u)*cd(u)
4432 (defun elliptic-eu (u m)
4433 (cond ((realp u)
4434 ;; E(u + 2*n*K) = E(u) + 2*n*E
4435 (let ((ell-k (to (elliptic-k m)))
4436 (ell-e (elliptic-ec m)))
4437 (multiple-value-bind (n u-rem)
4438 (floor u (* 2 ell-k))
4439 ;; 0 <= u-rem < 2*K
4440 (+ (* 2 n ell-e)
4441 (cond ((>= u-rem ell-k)
4442 ;; 0 <= u-rem < K so
4443 ;; E(u + K) = E(u) + E - m*sn(u)*cd(u)
4444 (let ((u-k (- u ell-k)))
4445 (- (+ (elliptic-e (cl:asin (bigfloat::sn u-k m)) m)
4446 ell-e)
4447 (/ (* m (bigfloat::sn u-k m) (bigfloat::cn u-k m))
4448 (bigfloat::dn u-k m)))))
4449 (t
4450 (elliptic-e (cl:asin (bigfloat::sn u m)) m)))))))
4451 ((complexp u)
4452 ;; From Lawden:
4453 ;;
4454 ;; E(u+i*v, m) = E(u,m) -i*E(v,m') + i*v + i*sc(v,m')*dn(v,m')
4455 ;; -i*m*sn(u,m)*sc(v,m')*sn(u+i*v,m)
4456 ;;
4457 (let ((u-r (realpart u))
4458 (u-i (imagpart u))
4459 (m1 (- 1 m)))
4460 (+ (elliptic-eu u-r m)
4461 (* #c(0 1)
4462 (- (+ u-i
4463 (/ (* (bigfloat::sn u-i m1) (bigfloat::dn u-i m1))
4464 (bigfloat::cn u-i m1)))
4465 (+ (elliptic-eu u-i m1)
4466 (/ (* m (bigfloat::sn u-r m) (bigfloat::sn u-i m1) (bigfloat::sn u m))
4467 (bigfloat::cn u-i m1))))))))))
4468
4469 (defprop $elliptic_eu
4470 ((u m)
4471 ((mexpt) ((%jacobi_dn) u m) 2)
4472 ;; wrt m
4473 )
4474 grad)
4475
4476 (def-simplifier elliptic_eu (u m)
4477 (cond
4478 ;; as it stands, ELLIPTIC-EU can't handle bigfloats or complex bigfloats,
4479 ;; so handle only floats and complex floats here.
4480 ((float-numerical-eval-p u m)
4481 (elliptic-eu ($float u) ($float m)))
4482 ((complex-float-numerical-eval-p u m)
4483 (let ((u-r ($realpart u))
4484 (u-i ($imagpart u))
4485 (m ($float m)))
4486 (complexify (elliptic-eu (complex u-r u-i) m))))
4487 (t
4488 (give-up))))
4489
4490 (def-simplifier jacobi_am (u m)
4491 (cond
4492 ;; as it stands, BIGFLOAT::SN can't handle bigfloats or complex bigfloats,
4493 ;; so handle only floats and complex floats here.
4494 ((float-numerical-eval-p u m)
4495 (cl:asin (bigfloat::sn ($float u) ($float m))))
4496 ((complex-float-numerical-eval-p u m)
4497 (let ((u-r ($realpart ($float u)))
4498 (u-i ($imagpart ($float u)))
4499 (m ($float m)))
4500 (complexify (cl:asin (bigfloat::sn (complex u-r u-i) m)))))
4501 (t
4502 ;; Nothing to do
4503 (give-up))))
4504
4505 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
4506 ;; Integrals. At present with respect to first argument only.
4507 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
4508
4509 ;; A&S 16.24.1: integrate(jacobi_sn(u,m),u)
4510 ;; = log(jacobi_dn(u,m)-sqrt(m)*jacobi_cn(u,m))/sqrt(m)
4511 (defprop %jacobi_sn
4512 ((u m)
4513 ((mtimes simp) ((mexpt simp) m ((rat simp) -1 2))
4514 ((%log simp)
4515 ((mplus simp)
4516 ((mtimes simp) -1 ((mexpt simp) m ((rat simp) 1 2))
4517 ((%jacobi_cn simp) u m))
4518 ((%jacobi_dn simp) u m))))
4519 nil)
4520 integral)
4521
4522 ;; A&S 16.24.2: integrate(jacobi_cn(u,m),u) = acos(jacobi_dn(u,m))/sqrt(m)
4523 (defprop %jacobi_cn
4524 ((u m)
4525 ((mtimes simp) ((mexpt simp) m ((rat simp) -1 2))
4526 ((%acos simp) ((%jacobi_dn simp) u m)))
4527 nil)
4528 integral)
4529
4530 ;; A&S 16.24.3: integrate(jacobi_dn(u,m),u) = asin(jacobi_sn(u,m))
4531 (defprop %jacobi_dn
4532 ((u m)
4533 ((%asin simp) ((%jacobi_sn simp) u m))
4534 nil)
4535 integral)
4536
4537 ;; A&S 16.24.4: integrate(jacobi_cd(u,m),u)
4538 ;; = log(jacobi_nd(u,m)+sqrt(m)*jacobi_sd(u,m))/sqrt(m)
4539 (defprop %jacobi_cd
4540 ((u m)
4541 ((mtimes simp) ((mexpt simp) m ((rat simp) -1 2))
4542 ((%log simp)
4543 ((mplus simp) ((%jacobi_nd simp) u m)
4544 ((mtimes simp) ((mexpt simp) m ((rat simp) 1 2))
4545 ((%jacobi_sd simp) u m)))))
4546 nil)
4547 integral)
4548
4549 ;; integrate(jacobi_sd(u,m),u)
4550 ;;
4551 ;; A&S 16.24.5 gives
4552 ;; asin(-sqrt(m)*jacobi_cd(u,m))/sqrt(m*m_1), where m + m_1 = 1
4553 ;; but this does not pass some simple tests.
4554 ;;
4555 ;; functions.wolfram.com 09.35.21.001.01 gives
4556 ;; -asin(sqrt(m)*jacobi_cd(u,m))*sqrt(1-m*jacobi_cd(u,m)^2)*jacobi_dn(u,m)/((1-m)*sqrt(m))
4557 ;; and this does pass.
4558 (defprop %jacobi_sd
4559 ((u m)
4560 ((mtimes simp) -1
4561 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m)) -1)
4562 ((mexpt simp) m ((rat simp) -1 2))
4563 ((mexpt simp)
4564 ((mplus simp) 1
4565 ((mtimes simp) -1 $m ((mexpt simp) ((%jacobi_cd simp) u m) 2)))
4566 ((rat simp) 1 2))
4567 ((%jacobi_dn simp) u m)
4568 ((%asin simp)
4569 ((mtimes simp) ((mexpt simp) m ((rat simp) 1 2))
4570 ((%jacobi_cd simp) u m))))
4571 nil)
4572 integral)
4573
4574 ;; integrate(jacobi_nd(u,m),u)
4575 ;;
4576 ;; A&S 16.24.6 gives
4577 ;; acos(jacobi_cd(u,m))/sqrt(m_1), where m + m_1 = 1
4578 ;; but this does not pass some simple tests.
4579 ;;
4580 ;; functions.wolfram.com 09.32.21.0001.01 gives
4581 ;; sqrt(1-jacobi_cd(u,m)^2)*acos(jacobi_cd(u,m))/((1-m)*jacobi_sd(u,m))
4582 ;; and this does pass.
4583 (defprop %jacobi_nd
4584 ((u m)
4585 ((mtimes simp) ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m)) -1)
4586 ((mexpt simp)
4587 ((mplus simp) 1
4588 ((mtimes simp) -1 ((mexpt simp) ((%jacobi_cd simp) u m) 2)))
4589 ((rat simp) 1 2))
4590 ((mexpt simp) ((%jacobi_sd simp) u m) -1)
4591 ((%acos simp) ((%jacobi_cd simp) u m)))
4592 nil)
4593 integral)
4594
4595 ;; A&S 16.24.7: integrate(jacobi_dc(u,m),u) = log(jacobi_nc(u,m)+jacobi_sc(u,m))
4596 (defprop %jacobi_dc
4597 ((u m)
4598 ((%log simp) ((mplus simp) ((%jacobi_nc simp) u m) ((%jacobi_sc simp) u m)))
4599 nil)
4600 integral)
4601
4602 ;; A&S 16.24.8: integrate(jacobi_nc(u,m),u)
4603 ;; = log(jacobi_dc(u,m)+sqrt(m_1)*jacobi_sc(u,m))/sqrt(m_1), where m + m_1 = 1
4604 (defprop %jacobi_nc
4605 ((u m)
4606 ((mtimes simp)
4607 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m))
4608 ((rat simp) -1 2))
4609 ((%log simp)
4610 ((mplus simp) ((%jacobi_dc simp) u m)
4611 ((mtimes simp)
4612 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m))
4613 ((rat simp) 1 2))
4614 ((%jacobi_sc simp) u m)))))
4615 nil)
4616 integral)
4617
4618 ;; A&S 16.24.9: integrate(jacobi_sc(u,m),u)
4619 ;; = log(jacobi_dc(u,m)+sqrt(m_1)*jacobi_nc(u,m))/sqrt(m_1), where m + m_1 = 1
4620 (defprop %jacobi_sc
4621 ((u m)
4622 ((mtimes simp)
4623 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m))
4624 ((rat simp) -1 2))
4625 ((%log simp)
4626 ((mplus simp) ((%jacobi_dc simp) u m)
4627 ((mtimes simp)
4628 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m))
4629 ((rat simp) 1 2))
4630 ((%jacobi_nc simp) u m)))))
4631 nil)
4632 integral)
4633
4634 ;; A&S 16.24.10: integrate(jacobi_ns(u,m),u)
4635 ;; = log(jacobi_ds(u,m)-jacobi_cs(u,m))
4636 (defprop %jacobi_ns
4637 ((u m)
4638 ((%log simp)
4639 ((mplus simp) ((mtimes simp) -1 ((%jacobi_cs simp) u m))
4640 ((%jacobi_ds simp) u m)))
4641 nil)
4642 integral)
4643
4644 ;; integrate(jacobi_ds(u,m),u)
4645 ;;
4646 ;; A&S 16.24.11 gives
4647 ;; log(jacobi_ds(u,m)-jacobi_cs(u,m))
4648 ;; but this does not pass some simple tests.
4649 ;;
4650 ;; functions.wolfram.com 09.30.21.0001.01 gives
4651 ;; log((1-jacobi_cn(u,m))/jacobi_sn(u,m))
4652 ;;
4653 (defprop %jacobi_ds
4654 ((u m)
4655 ((%log simp)
4656 ((mtimes simp)
4657 ((mplus simp) 1 ((mtimes simp) -1 ((%jacobi_cn simp) u m)))
4658 ((mexpt simp) ((%jacobi_sn simp) u m) -1)))
4659 nil)
4660 integral)
4661
4662 ;; A&S 16.24.12: integrate(jacobi_cs(u,m),u) = log(jacobi_ns(u,m)-jacobi_ds(u,m))
4663 (defprop %jacobi_cs
4664 ((u m)
4665 ((%log simp)
4666 ((mplus simp) ((mtimes simp) -1 ((%jacobi_ds simp) u m))
4667 ((%jacobi_ns simp) u m)))
4668 nil)
4669 integral)
4670
4671 ;; functions.wolfram.com 09.48.21.0001.01
4672 ;; integrate(inverse_jacobi_sn(u,m),u) =
4673 ;; inverse_jacobi_sn(u,m)*u
4674 ;; - log( jacobi_dn(inverse_jacobi_sn(u,m),m)
4675 ;; -sqrt(m)*jacobi_cn(inverse_jacobi_sn(u,m),m)) / sqrt(m)
4676 (defprop %inverse_jacobi_sn
4677 ((u m)
4678 ((mplus simp) ((mtimes simp) u ((%inverse_jacobi_sn simp) u m))
4679 ((mtimes simp) -1 ((mexpt simp) m ((rat simp) -1 2))
4680 ((%log simp)
4681 ((mplus simp)
4682 ((mtimes simp) -1 ((mexpt simp) m ((rat simp) 1 2))
4683 ((%jacobi_cn simp) ((%inverse_jacobi_sn simp) u m) m))
4684 ((%jacobi_dn simp) ((%inverse_jacobi_sn simp) u m) m)))))
4685 nil)
4686 integral)
4687
4688 ;; functions.wolfram.com 09.38.21.0001.01
4689 ;; integrate(inverse_jacobi_cn(u,m),u) =
4690 ;; u*inverse_jacobi_cn(u,m)
4691 ;; -%i*log(%i*jacobi_dn(inverse_jacobi_cn(u,m),m)/sqrt(m)
4692 ;; -jacobi_sn(inverse_jacobi_cn(u,m),m))
4693 ;; /sqrt(m)
4694 (defprop %inverse_jacobi_cn
4695 ((u m)
4696 ((mplus simp) ((mtimes simp) u ((%inverse_jacobi_cn simp) u m))
4697 ((mtimes simp) -1 $%i ((mexpt simp) m ((rat simp) -1 2))
4698 ((%log simp)
4699 ((mplus simp)
4700 ((mtimes simp) $%i ((mexpt simp) m ((rat simp) -1 2))
4701 ((%jacobi_dn simp) ((%inverse_jacobi_cn simp) u m) m))
4702 ((mtimes simp) -1
4703 ((%jacobi_sn simp) ((%inverse_jacobi_cn simp) u m) m))))))
4704 nil)
4705 integral)
4706
4707 ;; functions.wolfram.com 09.41.21.0001.01
4708 ;; integrate(inverse_jacobi_dn(u,m),u) =
4709 ;; u*inverse_jacobi_dn(u,m)
4710 ;; - %i*log(%i*jacobi_cn(inverse_jacobi_dn(u,m),m)
4711 ;; +jacobi_sn(inverse_jacobi_dn(u,m),m))
4712 (defprop %inverse_jacobi_dn
4713 ((u m)
4714 ((mplus simp) ((mtimes simp) u ((%inverse_jacobi_dn simp) u m))
4715 ((mtimes simp) -1 $%i
4716 ((%log simp)
4717 ((mplus simp)
4718 ((mtimes simp) $%i
4719 ((%jacobi_cn simp) ((%inverse_jacobi_dn simp) u m) m))
4720 ((%jacobi_sn simp) ((%inverse_jacobi_dn simp) u m) m)))))
4721 nil)
4722 integral)
4723
4724 \f
4725 ;; Real and imaginary part for Jacobi elliptic functions.
4726 (defprop %jacobi_sn risplit-sn-cn-dn risplit-function)
4727 (defprop %jacobi_cn risplit-sn-cn-dn risplit-function)
4728 (defprop %jacobi_dn risplit-sn-cn-dn risplit-function)
4729
4730 (defun risplit-sn-cn-dn (expr)
4731 (let* ((arg (second expr))
4732 (param (third expr)))
4733 ;; We only split on the argument, not the order
4734 (destructuring-bind (arg-r . arg-i)
4735 (risplit arg)
4736 (cond ((=0 arg-i)
4737 ;; Pure real
4738 (cons (take (first expr) arg-r param)
4739 0))
4740 (t
4741 (let* ((s (ftake '%jacobi_sn arg-r param))
4742 (c (ftake '%jacobi_cn arg-r param))
4743 (d (ftake '%jacobi_dn arg-r param))
4744 (s1 (ftake '%jacobi_sn arg-i (sub 1 param)))
4745 (c1 (ftake '%jacobi_cn arg-i (sub 1 param)))
4746 (d1 (ftake '%jacobi_dn arg-i (sub 1 param)))
4747 (den (add (mul c1 c1)
4748 (mul param
4749 (mul (mul s s)
4750 (mul s1 s1))))))
4751 ;; Let s = jacobi_sn(x,m)
4752 ;; c = jacobi_cn(x,m)
4753 ;; d = jacobi_dn(x,m)
4754 ;; s1 = jacobi_sn(y,1-m)
4755 ;; c1 = jacobi_cn(y,1-m)
4756 ;; d1 = jacobi_dn(y,1-m)
4757 (case (caar expr)
4758 (%jacobi_sn
4759 ;; A&S 16.21.1
4760 ;; jacobi_sn(x+%i*y,m) =
4761 ;;
4762 ;; s*d1 + %i*c*d*s1*c1
4763 ;; -------------------
4764 ;; c1^2+m*s^2*s1^2
4765 ;;
4766 (cons (div (mul s d1) den)
4767 (div (mul c (mul d (mul s1 c1)))
4768 den)))
4769 (%jacobi_cn
4770 ;; A&S 16.21.2
4771 ;;
4772 ;; cn(x+%i_y, m) =
4773 ;;
4774 ;; c*c1 - %i*s*d*s1*d1
4775 ;; -------------------
4776 ;; c1^2+m*s^2*s1^2
4777 (cons (div (mul c c1) den)
4778 (div (mul -1
4779 (mul s (mul d (mul s1 d1))))
4780 den)))
4781 (%jacobi_dn
4782 ;; A&S 16.21.3
4783 ;;
4784 ;; dn(x+%i_y, m) =
4785 ;;
4786 ;; d*c1*d1 - %i*m*s*c*s1
4787 ;; ---------------------
4788 ;; c1^2+m*s^2*s1^2
4789 (cons (div (mul d (mul c1 d1))
4790 den)
4791 (div (mul -1 (mul param (mul s (mul c s1))))
4792 den))))))))))
4793
Maxima fork w/ changes for Cygwin