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1 ;;; -*- Mode: Lisp; Package: Maxima; Syntax: Common-Lisp; Base: 10-*- ;;;;
2 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
3 ;;; The data in this file contains enhancments. ;;;;;
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 (defvar 3//2 '((rat simp) 3 2))
18 (defvar 1//2 '((rat simp) 1 2))
19 (defvar -1//2 '((rat simp) -1 2))
20
21 ;;;
22 ;;; Jacobian elliptic functions and elliptic integrals.
23 ;;;
24 ;;; References:
25 ;;;
26 ;;; [1] Abramowitz and Stegun
27 ;;; [2] Lawden, Elliptic Functions and Applications, Springer-Verlag, 1989
28 ;;; [3] Whittaker & Watson, A Course of Modern Analysis
29 ;;;
30 ;;; We use the definitions from Abramowitz and Stegun where our
31 ;;; sn(u,m) is sn(u|m). That is, the second arg is the parameter,
32 ;;; instead of the modulus k or modular angle alpha.
33 ;;;
34 ;;; Note that m = k^2 and k = sin(alpha).
35 ;;;
36
37 ;;
38 ;; Routines for computing the basic elliptic functions sn, cn, and dn.
39 ;;
40 ;;
41 ;; A&S gives several methods for computing elliptic functions
42 ;; including the AGM method (16.4) and ascending and descending Landen
43 ;; transformations (16.12 and 16.14). The latter are actually quite
44 ;; fast, only requiring simple arithmetic and square roots for the
45 ;; transformation until the last step. The AGM requires evaluation of
46 ;; several trigonometric functions at each stage.
47 ;;
48 ;; However, the Landen transformations appear to have some round-off
49 ;; issues. For example, using the ascending transform to compute cn,
50 ;; cn(100,.7) > 1e10. This is clearly not right since |cn| <= 1.
51 ;;
52
53 (in-package #-gcl #:bigfloat #+gcl "BIGFLOAT")
54
55 (declaim (inline descending-transform ascending-transform))
56
57 (defun ascending-transform (u m)
58 ;; A&S 16.14.1
59 ;;
60 ;; Take care in computing this transform. For the case where
61 ;; m is complex, we should compute sqrt(mu1) first as
62 ;; (1-sqrt(m))/(1+sqrt(m)), and then square this to get mu1.
63 ;; If not, we may choose the wrong branch when computing
64 ;; sqrt(mu1).
65 (let* ((root-m (sqrt m))
66 (mu (/ (* 4 root-m)
67 (expt (1+ root-m) 2)))
68 (root-mu1 (/ (- 1 root-m) (+ 1 root-m)))
69 (v (/ u (1+ root-mu1))))
70 (values v mu root-mu1)))
71
72 (defun descending-transform (u m)
73 ;; Note: Don't calculate mu first, as given in 16.12.1. We
74 ;; should calculate sqrt(mu) = (1-sqrt(m1)/(1+sqrt(m1)), and
75 ;; then compute mu = sqrt(mu)^2. If we calculate mu first,
76 ;; sqrt(mu) loses information when m or m1 is complex.
77 (let* ((root-m1 (sqrt (- 1 m)))
78 (root-mu (/ (- 1 root-m1) (+ 1 root-m1)))
79 (mu (* root-mu root-mu))
80 (v (/ u (1+ root-mu))))
81 (values v mu root-mu)))
82
83 ;;
84 ;; This appears to work quite well for both real and complex values
85 ;; of u.
86 (defun elliptic-sn-descending (u m)
87 (cond ((= m 1)
88 ;; A&S 16.6.1
89 (tanh u))
90 ((< (abs m) (epsilon u))
91 ;; A&S 16.6.1
92 (sin u))
93 (t
94 (multiple-value-bind (v mu root-mu)
95 (descending-transform u m)
96 (let* ((new-sn (elliptic-sn-descending v mu)))
97 (/ (* (1+ root-mu) new-sn)
98 (1+ (* root-mu new-sn new-sn))))))))
99
100 ;; AGM scale. See A&S 17.6
101 ;;
102 ;; The AGM scale is
103 ;;
104 ;; 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.
105 ;;
106 ;; We stop when abs(c[n]) <= 10*eps
107 ;;
108 ;; A list of (n a[n] b[n] c[n]) is returned.
109 (defun agm-scale (a b c)
110 (loop for n from 0
111 while (> (abs c) (* 10 (epsilon c)))
112 collect (list n a b c)
113 do (psetf a (/ (+ a b) 2)
114 b (sqrt (* a b))
115 c (/ (- a b) 2))))
116
117 ;; WARNING: This seems to have accuracy problems when u is complex. I
118 ;; (rtoy) do not know why. For example (jacobi-agm #c(1e0 1e0) .7e0)
119 ;; returns
120 ;;
121 ;; #C(1.134045970915582 0.3522523454566013)
122 ;; #C(0.57149659007575 -0.6989899153338323)
123 ;; #C(0.6229715431044184 -0.4488635962149656)
124 ;;
125 ;; But the actual value of sn(1+%i, .7) is .3522523469224946 %i +
126 ;; 1.134045971912365. We've lost about 7 digits of accuracy!
127 (defun jacobi-agm (u m)
128 ;; A&S 16.4.
129 ;;
130 ;; Compute the AGM scale with a = 1, b = sqrt(1-m), c = sqrt(m).
131 ;;
132 ;; Then phi[N] = 2^N*a[N]*u and compute phi[n] from
133 ;;
134 ;; sin(2*phi[n-1] - phi[n]) = c[n]/a[n]*sin(phi[n])
135 ;;
136 ;; Finally,
137 ;;
138 ;; sn(u|m) = sin(phi[0]), cn(u|m) = cos(phi[0])
139 ;; dn(u|m) = cos(phi[0])/cos(phi[1]-phi[0])
140 ;;
141 ;; Returns the three values sn, cn, dn.
142 (let* ((agm-data (nreverse (rest (agm-scale 1 (sqrt (- 1 m)) (sqrt m)))))
143 (phi (destructuring-bind (n a b c)
144 (first agm-data)
145 (declare (ignore b c))
146 (* a u (ash 1 n))))
147 (phi1 0e0))
148 (dolist (agm agm-data)
149 (destructuring-bind (n a b c)
150 agm
151 (declare (ignore n b))
152 (setf phi1 phi
153 phi (/ (+ phi (asin (* (/ c a) (sin phi)))) 2))))
154 (values (sin phi) (cos phi) (/ (cos phi) (cos (- phi1 phi))))))
155
156 (defun sn (u m)
157 (cond ((zerop m)
158 ;; jacobi_sn(u,0) = sin(u). Should we use A&S 16.13.1 if m
159 ;; is small enough?
160 ;;
161 ;; sn(u,m) = sin(u) - 1/4*m(u-sin(u)*cos(u))*cos(u)
162 (sin u))
163 ((= m 1)
164 ;; jacobi_sn(u,1) = tanh(u). Should we use A&S 16.15.1 if m
165 ;; is close enough to 1?
166 ;;
167 ;; sn(u,m) = tanh(u) + 1/4*(1-m)*(sinh(u)*cosh(u)-u)*sech(u)^2
168 (tanh u))
169 (t
170 ;; Use the ascending Landen transformation to compute sn.
171 (let ((s (elliptic-sn-descending u m)))
172 (if (and (realp u) (realp m))
173 (realpart s)
174 s)))))
175
176 (defun dn (u m)
177 (cond ((zerop m)
178 ;; jacobi_dn(u,0) = 1. Should we use A&S 16.13.3 for small m?
179 ;;
180 ;; dn(u,m) = 1 - 1/2*m*sin(u)^2
181 1)
182 ((= m 1)
183 ;; jacobi_dn(u,1) = sech(u). Should we use A&S 16.15.3 if m
184 ;; is close enough to 1?
185 ;;
186 ;; dn(u,m) = sech(u) + 1/4*(1-m)*(sinh(u)*cosh(u)+u)*tanh(u)*sech(u)
187 (/ (cosh u)))
188 (t
189 ;; Use the Gauss transformation from
190 ;; http://functions.wolfram.com/09.29.16.0013.01:
191 ;;
192 ;;
193 ;; dn((1+sqrt(m))*z, 4*sqrt(m)/(1+sqrt(m))^2)
194 ;; = (1-sqrt(m)*sn(z, m)^2)/(1+sqrt(m)*sn(z,m)^2)
195 ;;
196 ;; So
197 ;;
198 ;; dn(y, mu) = (1-sqrt(m)*sn(z, m)^2)/(1+sqrt(m)*sn(z,m)^2)
199 ;;
200 ;; where z = y/(1+sqrt(m)) and mu=4*sqrt(m)/(1+sqrt(m))^2.
201 ;;
202 ;; Solve for m, and we get
203 ;;
204 ;; sqrt(m) = -(mu+2*sqrt(1-mu)-2)/mu or (-mu+2*sqrt(1-mu)+2)/mu.
205 ;;
206 ;; I don't think it matters which sqrt we use, so I (rtoy)
207 ;; arbitrarily choose the first one above.
208 ;;
209 ;; Note that (1-sqrt(1-mu))/(1+sqrt(1-mu)) is the same as
210 ;; -(mu+2*sqrt(1-mu)-2)/mu. Also, the former is more
211 ;; accurate for small mu.
212 (let* ((root (let ((root-1-m (sqrt (- 1 m))))
213 (/ (- 1 root-1-m)
214 (+ 1 root-1-m))))
215 (z (/ u (+ 1 root)))
216 (s (elliptic-sn-descending z (* root root)))
217 (p (* root s s )))
218 (/ (- 1 p)
219 (+ 1 p))))))
220
221 (defun cn (u m)
222 (cond ((zerop m)
223 ;; jacobi_cn(u,0) = cos(u). Should we use A&S 16.13.2 for
224 ;; small m?
225 ;;
226 ;; cn(u,m) = cos(u) + 1/4*m*(u-sin(u)*cos(u))*sin(u)
227 (cos u))
228 ((= m 1)
229 ;; jacobi_cn(u,1) = sech(u). Should we use A&S 16.15.3 if m
230 ;; is close enough to 1?
231 ;;
232 ;; cn(u,m) = sech(u) - 1/4*(1-m)*(sinh(u)*cosh(u)-u)*tanh(u)*sech(u)
233 (/ (cosh u)))
234 (t
235 ;; Use the ascending Landen transformation, A&S 16.14.3.
236 (multiple-value-bind (v mu root-mu1)
237 (ascending-transform u m)
238 (let ((d (dn v mu)))
239 (* (/ (+ 1 root-mu1) mu)
240 (/ (- (* d d) root-mu1)
241 d)))))))
242
243 (in-package :maxima)
244
245 ;; Tell maxima what the derivatives are.
246 ;;
247 ;; Lawden says the derivative wrt to k but that's not what we want.
248 ;;
249 ;; Here's the derivation we used, based on how Lawden get's his results.
250 ;;
251 ;; Let
252 ;;
253 ;; diff(sn(u,m),m) = s
254 ;; diff(cn(u,m),m) = p
255 ;; diff(dn(u,m),m) = q
256 ;;
257 ;; From the derivatives of sn, cn, dn wrt to u, we have
258 ;;
259 ;; diff(sn(u,m),u) = cn(u)*dn(u)
260 ;; diff(cn(u,m),u) = -cn(u)*dn(u)
261 ;; diff(dn(u,m),u) = -m*sn(u)*cn(u)
262 ;;
263
264 ;; Differentiate these wrt to m:
265 ;;
266 ;; diff(s,u) = p*dn + cn*q
267 ;; diff(p,u) = -p*dn - q*dn
268 ;; diff(q,u) = -sn*cn - m*s*cn - m*sn*q
269 ;;
270 ;; Also recall that
271 ;;
272 ;; sn(u)^2 + cn(u)^2 = 1
273 ;; dn(u)^2 + m*sn(u)^2 = 1
274 ;;
275 ;; Differentiate these wrt to m:
276 ;;
277 ;; sn*s + cn*p = 0
278 ;; 2*dn*q + sn^2 + 2*m*sn*s = 0
279 ;;
280 ;; Thus,
281 ;;
282 ;; p = -s*sn/cn
283 ;; q = -m*s*sn/dn - sn^2/dn/2
284 ;;
285 ;; So
286 ;; diff(s,u) = -s*sn*dn/cn - m*s*sn*cn/dn - sn^2*cn/dn/2
287 ;;
288 ;; or
289 ;;
290 ;; diff(s,u) + s*(sn*dn/cn + m*sn*cn/dn) = -1/2*sn^2*cn/dn
291 ;;
292 ;; diff(s,u) + s*sn/cn/dn*(dn^2 + m*cn^2) = -1/2*sn^2*cn/dn
293 ;;
294 ;; Multiply through by the integrating factor 1/cn/dn:
295 ;;
296 ;; diff(s/cn/dn, u) = -1/2*sn^2/dn^2 = -1/2*sd^2.
297 ;;
298 ;; Interate this to get
299 ;;
300 ;; s/cn/dn = C + -1/2*int sd^2
301 ;;
302 ;; It can be shown that C is zero.
303 ;;
304 ;; We know that (by differentiating this expression)
305 ;;
306 ;; int dn^2 = (1-m)*u+m*sn*cd + m*(1-m)*int sd^2
307 ;;
308 ;; or
309 ;;
310 ;; int sd^2 = 1/m/(1-m)*int dn^2 - u/m - sn*cd/(1-m)
311 ;;
312 ;; Thus, we get
313 ;;
314 ;; s/cn/dn = u/(2*m) + sn*cd/(2*(1-m)) - 1/2/m/(1-m)*int dn^2
315 ;;
316 ;; or
317 ;;
318 ;; s = 1/(2*m)*u*cn*dn + 1/(2*(1-m))*sn*cn^2 - 1/2/(m*(1-m))*cn*dn*E(u)
319 ;;
320 ;; where E(u) = int dn^2 = elliptic_e(am(u)) = elliptic_e(asin(sn(u)))
321 ;;
322 ;; This is our desired result:
323 ;;
324 ;; s = 1/(2*m)*cn*dn*[u - elliptic_e(asin(sn(u)),m)/(1-m)] + sn*cn^2/2/(1-m)
325 ;;
326 ;;
327 ;; Since diff(cn(u,m),m) = p = -s*sn/cn, we have
328 ;;
329 ;; p = -1/(2*m)*sn*dn[u - elliptic_e(asin(sn(u)),m)/(1-m)] - sn^2*cn/2/(1-m)
330 ;;
331 ;; diff(dn(u,m),m) = q = -m*s*sn/dn - sn^2/dn/2
332 ;;
333 ;; q = -1/2*sn*cn*[u-elliptic_e(asin(sn),m)/(1-m)] - m*sn^2*cn^2/dn/2/(1-m)
334 ;;
335 ;; - sn^2/dn/2
336 ;;
337 ;; = -1/2*sn*cn*[u-elliptic_e(asin(sn),m)/(1-m)] + dn*sn^2/2/(m-1)
338 ;;
339 (defprop %jacobi_sn
340 ((u m)
341 ((mtimes) ((%jacobi_cn) u m) ((%jacobi_dn) u m))
342 ((mplus simp)
343 ((mtimes simp) ((rat simp) 1 2)
344 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m)) -1)
345 ((mexpt simp) ((%jacobi_cn simp) u m) 2) ((%jacobi_sn simp) u m))
346 ((mtimes simp) ((rat simp) 1 2) ((mexpt simp) m -1)
347 ((%jacobi_cn simp) u m) ((%jacobi_dn simp) u m)
348 ((mplus simp) u
349 ((mtimes simp) -1 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m)) -1)
350 ((%elliptic_e simp) ((%asin simp) ((%jacobi_sn simp) u m)) m))))))
351 grad)
352
353 (defprop %jacobi_cn
354 ((u m)
355 ((mtimes simp) -1 ((%jacobi_sn simp) u m) ((%jacobi_dn simp) u m))
356 ((mplus simp)
357 ((mtimes simp) ((rat simp) -1 2)
358 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m)) -1)
359 ((%jacobi_cn simp) u m) ((mexpt simp) ((%jacobi_sn simp) u m) 2))
360 ((mtimes simp) ((rat simp) -1 2) ((mexpt simp) m -1)
361 ((%jacobi_dn simp) u m) ((%jacobi_sn simp) u m)
362 ((mplus simp) u
363 ((mtimes simp) -1 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m)) -1)
364 ((%elliptic_e simp) ((%asin simp) ((%jacobi_sn simp) u m)) m))))))
365 grad)
366
367 (defprop %jacobi_dn
368 ((u m)
369 ((mtimes) -1 m ((%jacobi_sn) u m) ((%jacobi_cn) u m))
370 ((mplus simp)
371 ((mtimes simp) ((rat simp) -1 2)
372 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m)) -1)
373 ((%jacobi_dn simp) u m) ((mexpt simp) ((%jacobi_sn simp) u m) 2))
374 ((mtimes simp) ((rat simp) -1 2) ((%jacobi_cn simp) u m)
375 ((%jacobi_sn simp) u m)
376 ((mplus simp) u
377 ((mtimes simp) -1
378 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m)) -1)
379 ((%elliptic_e simp) ((%asin simp) ((%jacobi_sn simp) u m)) m))))))
380 grad)
381
382 ;; The inverse elliptic functions.
383 ;;
384 ;; F(phi|m) = asn(sin(phi),m)
385 ;;
386 ;; so asn(u,m) = F(asin(u)|m)
387 (defprop %inverse_jacobi_sn
388 ((x m)
389 ;; Lawden 3.1.2:
390 ;; inverse_jacobi_sn(x) = integrate(1/sqrt(1-t^2)/sqrt(1-m*t^2),t,0,x)
391 ;; -> 1/sqrt(1-x^2)/sqrt(1-m*x^2)
392 ((mtimes simp)
393 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 ((mexpt simp) x 2)))
394 ((rat simp) -1 2))
395 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m ((mexpt simp) x 2)))
396 ((rat simp) -1 2)))
397 ;; diff(F(asin(u)|m),m)
398 ((mtimes simp) ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m)) -1)
399 ((mplus simp)
400 ((mtimes simp) -1 x
401 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 ((mexpt simp) x 2)))
402 ((rat simp) 1 2))
403 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m ((mexpt simp) x 2)))
404 ((rat simp) -1 2)))
405 ((mtimes simp) ((mexpt simp) m -1)
406 ((mplus simp) ((%elliptic_e simp) ((%asin simp) x) m)
407 ((mtimes simp) -1 ((mplus simp) 1 ((mtimes simp) -1 m))
408 ((%elliptic_f simp) ((%asin simp) x) m)))))))
409 grad)
410
411 ;; Let u = inverse_jacobi_cn(x). Then jacobi_cn(u) = x or
412 ;; sqrt(1-jacobi_sn(u)^2) = x. Or
413 ;;
414 ;; jacobi_sn(u) = sqrt(1-x^2)
415 ;;
416 ;; So u = inverse_jacobi_sn(sqrt(1-x^2),m) = inverse_jacob_cn(x,m)
417 ;;
418 (defprop %inverse_jacobi_cn
419 ((x m)
420 ;; Whittaker and Watson, 22.121
421 ;; inverse_jacobi_cn(u,m) = integrate(1/sqrt(1-t^2)/sqrt(1-m+m*t^2), t, u, 1)
422 ;; -> -1/sqrt(1-x^2)/sqrt(1-m+m*x^2)
423 ((mtimes simp) -1
424 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 ((mexpt simp) x 2)))
425 ((rat simp) -1 2))
426 ((mexpt simp)
427 ((mplus simp) 1 ((mtimes simp) -1 m)
428 ((mtimes simp) m ((mexpt simp) x 2)))
429 ((rat simp) -1 2)))
430 ((mtimes simp) ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m)) -1)
431 ((mplus simp)
432 ((mtimes simp) -1
433 ((mexpt simp)
434 ((mplus simp) 1
435 ((mtimes simp) -1 m ((mplus simp) 1 ((mtimes simp) -1 ((mexpt simp) x 2)))))
436 ((rat simp) -1 2))
437 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 ((mexpt simp) x 2))) ((rat simp) 1 2))
438 ((mabs simp) x))
439 ((mtimes simp) ((mexpt simp) m -1)
440 ((mplus simp)
441 ((%elliptic_e simp)
442 ((%asin simp)
443 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 ((mexpt simp) x 2))) ((rat simp) 1 2)))
444 m)
445 ((mtimes simp) -1 ((mplus simp) 1 ((mtimes simp) -1 m))
446 ((%elliptic_f simp)
447 ((%asin simp)
448 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 ((mexpt simp) x 2))) ((rat simp) 1 2)))
449 m)))))))
450 grad)
451
452 ;; Let u = inverse_jacobi_dn(x). Then
453 ;;
454 ;; jacobi_dn(u) = x or
455 ;;
456 ;; x^2 = jacobi_dn(u)^2 = 1 - m*jacobi_sn(u)^2
457 ;;
458 ;; so jacobi_sn(u) = sqrt(1-x^2)/sqrt(m)
459 ;;
460 ;; or u = inverse_jacobi_sn(sqrt(1-x^2)/sqrt(m))
461 (defprop %inverse_jacobi_dn
462 ((x m)
463 ;; Whittaker and Watson, 22.121
464 ;; inverse_jacobi_dn(u,m) = integrate(1/sqrt(1-t^2)/sqrt(t^2-(1-m)), t, u, 1)
465 ;; -> -1/sqrt(1-x^2)/sqrt(x^2+m-1)
466 ((mtimes simp)
467 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 ((mexpt simp) x 2)))
468 ((rat simp) -1 2))
469 ((mexpt simp) ((mplus simp) -1 m ((mexpt simp) x 2)) ((rat simp) -1 2)))
470 ((mplus simp)
471 ((mtimes simp) ((rat simp) -1 2) ((mexpt simp) m ((rat simp) -3 2))
472 ((mexpt simp)
473 ((mplus simp) 1
474 ((mtimes simp) -1 ((mexpt simp) m -1)
475 ((mplus simp) 1 ((mtimes simp) -1 ((mexpt simp) x 2)))))
476 ((rat simp) -1 2))
477 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 ((mexpt simp) x 2)))
478 ((rat simp) 1 2))
479 ((mexpt simp) ((mabs simp) x) -1))
480 ((mtimes simp) ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m)) -1)
481 ((mplus simp)
482 ((mtimes simp) -1 ((mexpt simp) m ((rat simp) -1 2))
483 ((mexpt simp)
484 ((mplus simp) 1
485 ((mtimes simp) -1 ((mexpt simp) m -1)
486 ((mplus simp) 1 ((mtimes simp) -1 ((mexpt simp) x 2)))))
487 ((rat simp) 1 2))
488 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 ((mexpt simp) x 2)))
489 ((rat simp) 1 2))
490 ((mexpt simp) ((mabs simp) x) -1))
491 ((mtimes simp) ((mexpt simp) m -1)
492 ((mplus simp)
493 ((%elliptic_e simp)
494 ((%asin simp)
495 ((mtimes simp) ((mexpt simp) m ((rat simp) -1 2))
496 ((mexpt simp) ((mplus simp) 1
497 ((mtimes simp) -1 ((mexpt simp) x 2)))
498 ((rat simp) 1 2))))
499 m)
500 ((mtimes simp) -1 ((mplus simp) 1 ((mtimes simp) -1 m))
501 ((%elliptic_f simp)
502 ((%asin simp)
503 ((mtimes simp) ((mexpt simp) m ((rat simp) -1 2))
504 ((mexpt simp) ((mplus simp) 1
505 ((mtimes simp) -1 ((mexpt simp) x 2)))
506 ((rat simp) 1 2))))
507 m))))))))
508 grad)
509
510
511 ;; Possible forms of a complex number:
512 ;;
513 ;; 2.3
514 ;; $%i
515 ;; ((mplus simp) 2.3 ((mtimes simp) 2.3 $%i))
516 ;; ((mplus simp) 2.3 $%i))
517 ;; ((mtimes simp) 2.3 $%i)
518 ;;
519
520
521 ;; Is argument u a complex number with real and imagpart satisfying predicate ntypep?
522 (defun complex-number-p (u &optional (ntypep 'numberp))
523 (let ((R 0) (I 0))
524 (labels ((a1 (x) (cadr x))
525 (a2 (x) (caddr x))
526 (a3+ (x) (cdddr x))
527 (N (x) (funcall ntypep x)) ; N
528 (i (x) (and (eq x '$%i) (N 1))) ; %i
529 (N+i (x) (and (null (a3+ x)) ; mplus test is precondition
530 (N (setq R (a1 x)))
531 (or (and (i (a2 x)) (setq I 1) t)
532 (and (mtimesp (a2 x)) (N*i (a2 x))))))
533 (N*i (x) (and (null (a3+ x)) ; mtimes test is precondition
534 (N (setq I (a1 x)))
535 (eq (a2 x) '$%i))))
536 (declare (inline a1 a2 a3+ N i N+i N*i))
537 (cond ((N u) (values t u 0)) ;2.3
538 ((atom u) (if (i u) (values t 0 1))) ;%i
539 ((mplusp u) (if (N+i u) (values t R I))) ;N+%i, N+N*%i
540 ((mtimesp u) (if (N*i u) (values t R I))) ;N*%i
541 (t nil)))))
542
543 (defun complexify (x)
544 ;; Convert a Lisp number to a maxima number
545 (cond ((realp x) x)
546 ((complexp x) (add (realpart x) (mul '$%i (imagpart x))))
547 (t (merror (intl:gettext "COMPLEXIFY: argument must be a Lisp real or complex number.~%COMPLEXIFY: found: ~:M") x))))
548
549 (defun kc-arg (exp m)
550 ;; Replace elliptic_kc(m) in the expression with sym. Check to see
551 ;; if the resulting expression is linear in sym and the constant
552 ;; term is zero. If so, return the coefficient of sym, i.e, the
553 ;; coefficient of elliptic_kc(m).
554 (let* ((sym (gensym))
555 (arg (maxima-substitute sym `((%elliptic_kc) ,m) exp)))
556 (if (and (not (equalp arg exp))
557 (linearp arg sym)
558 (zerop1 (coefficient arg sym 0)))
559 (coefficient arg sym 1)
560 nil)))
561
562 (defun kc-arg2 (exp m)
563 ;; Replace elliptic_kc(m) in the expression with sym. Check to see
564 ;; if the resulting expression is linear in sym and the constant
565 ;; term is zero. If so, return the coefficient of sym, i.e, the
566 ;; coefficient of elliptic_kc(m), and the constant term. Otherwise,
567 ;; return NIL.
568 (let* ((sym (gensym))
569 (arg (maxima-substitute sym `((%elliptic_kc) ,m) exp)))
570 (if (and (not (equalp arg exp))
571 (linearp arg sym))
572 (list (coefficient arg sym 1)
573 (coefficient arg sym 0))
574 nil)))
575
576 ;; Tell maxima how to simplify the functions
577
578 (def-simplifier jacobi_sn (u m)
579 (let (coef args)
580 (cond
581 ((float-numerical-eval-p u m)
582 (to (bigfloat::sn (bigfloat:to ($float u)) (bigfloat:to ($float m)))))
583 ((setf args (complex-float-numerical-eval-p u m))
584 (destructuring-bind (u m)
585 args
586 (to (bigfloat::sn (bigfloat:to ($float u)) (bigfloat:to ($float m))))))
587 ((bigfloat-numerical-eval-p u m)
588 (to (bigfloat::sn (bigfloat:to ($bfloat u)) (bigfloat:to ($bfloat m)))))
589 ((setf args (complex-bigfloat-numerical-eval-p u m))
590 (destructuring-bind (u m)
591 args
592 (to (bigfloat::sn (bigfloat:to ($bfloat u)) (bigfloat:to ($bfloat m))))))
593 ((zerop1 u)
594 ;; A&S 16.5.1
595 0)
596 ((zerop1 m)
597 ;; A&S 16.6.1
598 (ftake '%sin u))
599 ((onep1 m)
600 ;; A&S 16.6.1
601 (ftake '%tanh u))
602 ((and $trigsign (mminusp* u))
603 (neg (ftake* '%jacobi_sn (neg u) m)))
604 ((and $triginverses
605 (listp u)
606 (member (caar u) '(%inverse_jacobi_sn
607 %inverse_jacobi_ns
608 %inverse_jacobi_cn
609 %inverse_jacobi_nc
610 %inverse_jacobi_dn
611 %inverse_jacobi_nd
612 %inverse_jacobi_sc
613 %inverse_jacobi_cs
614 %inverse_jacobi_sd
615 %inverse_jacobi_ds
616 %inverse_jacobi_cd
617 %inverse_jacobi_dc))
618 (alike1 (third u) m))
619 (let ((inv-arg (second u)))
620 (ecase (caar u)
621 (%inverse_jacobi_sn
622 ;; jacobi_sn(inverse_jacobi_sn(u,m), m) = u
623 inv-arg)
624 (%inverse_jacobi_ns
625 ;; inverse_jacobi_ns(u,m) = inverse_jacobi_sn(1/u,m)
626 (div 1 inv-arg))
627 (%inverse_jacobi_cn
628 ;; sn(x)^2 + cn(x)^2 = 1 so sn(x) = sqrt(1-cn(x)^2)
629 (power (sub 1 (mul inv-arg inv-arg)) 1//2))
630 (%inverse_jacobi_nc
631 ;; inverse_jacobi_nc(u) = inverse_jacobi_cn(1/u)
632 (ftake '%jacobi_sn (ftake '%inverse_jacobi_cn (div 1 inv-arg) m)
633 m))
634 (%inverse_jacobi_dn
635 ;; dn(x)^2 + m*sn(x)^2 = 1 so
636 ;; sn(x) = 1/sqrt(m)*sqrt(1-dn(x)^2)
637 (mul (div 1 (power m 1//2))
638 (power (sub 1 (mul inv-arg inv-arg)) 1//2)))
639 (%inverse_jacobi_nd
640 ;; inverse_jacobi_nd(u) = inverse_jacobi_dn(1/u)
641 (ftake '%jacobi_sn (ftake '%inverse_jacobi_dn (div 1 inv-arg) m)
642 m))
643 (%inverse_jacobi_sc
644 ;; See below for inverse_jacobi_sc.
645 (div inv-arg (power (add 1 (mul inv-arg inv-arg)) 1//2)))
646 (%inverse_jacobi_cs
647 ;; inverse_jacobi_cs(u) = inverse_jacobi_sc(1/u)
648 (ftake '%jacobi_sn (ftake '%inverse_jacobi_sc (div 1 inv-arg) m)
649 m))
650 (%inverse_jacobi_sd
651 ;; See below for inverse_jacobi_sd
652 (div inv-arg (power (add 1 (mul m (mul inv-arg inv-arg))) 1//2)))
653 (%inverse_jacobi_ds
654 ;; inverse_jacobi_ds(u) = inverse_jacobi_sd(1/u)
655 (ftake '%jacobi_sn (ftake '%inverse_jacobi_sd (div 1 inv-arg) m)
656 m))
657 (%inverse_jacobi_cd
658 ;; See below
659 (div (power (sub 1 (mul inv-arg inv-arg)) 1//2)
660 (power (sub 1 (mul m (mul inv-arg inv-arg))) 1//2)))
661 (%inverse_jacobi_dc
662 (ftake '%jacobi_sn (ftake '%inverse_jacobi_cd (div 1 inv-arg) m) m)))))
663 ;; A&S 16.20.1 (Jacobi's Imaginary transformation)
664 ((and $%iargs (multiplep u '$%i))
665 (mul '$%i
666 (ftake* '%jacobi_sc (coeff u '$%i 1) (add 1 (neg m)))))
667 ((setq coef (kc-arg2 u m))
668 ;; sn(m*K+u)
669 ;;
670 ;; A&S 16.8.1
671 (destructuring-bind (lin const)
672 coef
673 (cond ((integerp lin)
674 (ecase (mod lin 4)
675 (0
676 ;; sn(4*m*K + u) = sn(u), sn(0) = 0
677 (if (zerop1 const)
678 0
679 (ftake '%jacobi_sn const m)))
680 (1
681 ;; sn(4*m*K + K + u) = sn(K+u) = cd(u)
682 ;; sn(K) = 1
683 (if (zerop1 const)
684 1
685 (ftake '%jacobi_cd const m)))
686 (2
687 ;; sn(4*m*K+2*K + u) = sn(2*K+u) = -sn(u)
688 ;; sn(2*K) = 0
689 (if (zerop1 const)
690 0
691 (neg (ftake '%jacobi_sn const m))))
692 (3
693 ;; sn(4*m*K+3*K+u) = sn(2*K + K + u) = -sn(K+u) = -cd(u)
694 ;; sn(3*K) = -1
695 (if (zerop1 const)
696 -1
697 (neg (ftake '%jacobi_cd const m))))))
698 ((and (alike1 lin 1//2)
699 (zerop1 const))
700 ;; A&S 16.5.2
701 ;;
702 ;; sn(1/2*K) = 1/sqrt(1+sqrt(1-m))
703 (div 1
704 (power (add 1 (power (sub 1 m) 1//2))
705 1//2)))
706 ((and (alike1 lin 3//2)
707 (zerop1 const))
708 ;; A&S 16.5.2
709 ;;
710 ;; sn(1/2*K + K) = cd(1/2*K,m)
711 (ftake '%jacobi_cd (mul 1//2
712 (ftake '%elliptic_kc m))
713 m))
714 (t
715 (give-up)))))
716 (t
717 ;; Nothing to do
718 (give-up)))))
719
720 (def-simplifier jacobi_cn (u m)
721 (let (coef args)
722 (cond
723 ((float-numerical-eval-p u m)
724 (to (bigfloat::cn (bigfloat:to ($float u)) (bigfloat:to ($float m)))))
725 ((setf args (complex-float-numerical-eval-p u m))
726 (destructuring-bind (u m)
727 args
728 (to (bigfloat::cn (bigfloat:to ($float u)) (bigfloat:to ($float m))))))
729 ((bigfloat-numerical-eval-p u m)
730 (to (bigfloat::cn (bigfloat:to ($bfloat u)) (bigfloat:to ($bfloat m)))))
731 ((setf args (complex-bigfloat-numerical-eval-p u m))
732 (destructuring-bind (u m)
733 args
734 (to (bigfloat::cn (bigfloat:to ($bfloat u)) (bigfloat:to ($bfloat m))))))
735 ((zerop1 u)
736 ;; A&S 16.5.1
737 1)
738 ((zerop1 m)
739 ;; A&S 16.6.2
740 (ftake '%cos u))
741 ((onep1 m)
742 ;; A&S 16.6.2
743 (ftake '%sech u))
744 ((and $trigsign (mminusp* u))
745 (ftake* '%jacobi_cn (neg u) m))
746 ((and $triginverses
747 (listp u)
748 (member (caar u) '(%inverse_jacobi_sn
749 %inverse_jacobi_ns
750 %inverse_jacobi_cn
751 %inverse_jacobi_nc
752 %inverse_jacobi_dn
753 %inverse_jacobi_nd
754 %inverse_jacobi_sc
755 %inverse_jacobi_cs
756 %inverse_jacobi_sd
757 %inverse_jacobi_ds
758 %inverse_jacobi_cd
759 %inverse_jacobi_dc))
760 (alike1 (third u) m))
761 (cond ((eq (caar u) '%inverse_jacobi_cn)
762 (second u))
763 (t
764 ;; I'm lazy. Use cn(x) = sqrt(1-sn(x)^2). Hope
765 ;; this is right.
766 (power (sub 1 (power (ftake '%jacobi_sn u (third u)) 2))
767 1//2))))
768 ;; A&S 16.20.2 (Jacobi's Imaginary transformation)
769 ((and $%iargs (multiplep u '$%i))
770 (ftake* '%jacobi_nc (coeff u '$%i 1) (add 1 (neg m))))
771 ((setq coef (kc-arg2 u m))
772 ;; cn(m*K+u)
773 ;;
774 ;; A&S 16.8.2
775 (destructuring-bind (lin const)
776 coef
777 (cond ((integerp lin)
778 (ecase (mod lin 4)
779 (0
780 ;; cn(4*m*K + u) = cn(u),
781 ;; cn(0) = 1
782 (if (zerop1 const)
783 1
784 (ftake '%jacobi_cn const m)))
785 (1
786 ;; cn(4*m*K + K + u) = cn(K+u) = -sqrt(m1)*sd(u)
787 ;; cn(K) = 0
788 (if (zerop1 const)
789 0
790 (neg (mul (power (sub 1 m) 1//2)
791 (ftake '%jacobi_sd const m)))))
792 (2
793 ;; cn(4*m*K + 2*K + u) = cn(2*K+u) = -cn(u)
794 ;; cn(2*K) = -1
795 (if (zerop1 const)
796 -1
797 (neg (ftake '%jacobi_cn const m))))
798 (3
799 ;; cn(4*m*K + 3*K + u) = cn(2*K + K + u) =
800 ;; -cn(K+u) = sqrt(m1)*sd(u)
801 ;;
802 ;; cn(3*K) = 0
803 (if (zerop1 const)
804 0
805 (mul (power (sub 1 m) 1//2)
806 (ftake '%jacobi_sd const m))))))
807 ((and (alike1 lin 1//2)
808 (zerop1 const))
809 ;; A&S 16.5.2
810 ;; cn(1/2*K) = (1-m)^(1/4)/sqrt(1+sqrt(1-m))
811 (mul (power (sub 1 m) (div 1 4))
812 (power (add 1
813 (power (sub 1 m)
814 1//2))
815 1//2)))
816 (t
817 (give-up)))))
818 (t
819 (give-up)))))
820
821 (def-simplifier jacobi_dn (u m)
822 (let (coef args)
823 (cond
824 ((float-numerical-eval-p u m)
825 (to (bigfloat::dn (bigfloat:to ($float u)) (bigfloat:to ($float m)))))
826 ((setf args (complex-float-numerical-eval-p u m))
827 (destructuring-bind (u m)
828 args
829 (to (bigfloat::dn (bigfloat:to ($float u)) (bigfloat:to ($float m))))))
830 ((bigfloat-numerical-eval-p u m)
831 (to (bigfloat::dn (bigfloat:to ($bfloat u)) (bigfloat:to ($bfloat m)))))
832 ((setf args (complex-bigfloat-numerical-eval-p u m))
833 (destructuring-bind (u m)
834 args
835 (to (bigfloat::dn (bigfloat:to ($bfloat u)) (bigfloat:to ($bfloat m))))))
836 ((zerop1 u)
837 ;; A&S 16.5.1
838 1)
839 ((zerop1 m)
840 ;; A&S 16.6.3
841 1)
842 ((onep1 m)
843 ;; A&S 16.6.3
844 (ftake '%sech u))
845 ((and $trigsign (mminusp* u))
846 (ftake* '%jacobi_dn (neg u) m))
847 ((and $triginverses
848 (listp u)
849 (member (caar u) '(%inverse_jacobi_sn
850 %inverse_jacobi_ns
851 %inverse_jacobi_cn
852 %inverse_jacobi_nc
853 %inverse_jacobi_dn
854 %inverse_jacobi_nd
855 %inverse_jacobi_sc
856 %inverse_jacobi_cs
857 %inverse_jacobi_sd
858 %inverse_jacobi_ds
859 %inverse_jacobi_cd
860 %inverse_jacobi_dc))
861 (alike1 (third u) m))
862 (cond ((eq (caar u) '%inverse_jacobi_dn)
863 ;; jacobi_dn(inverse_jacobi_dn(u,m), m) = u
864 (second u))
865 (t
866 ;; Express in terms of sn:
867 ;; dn(x) = sqrt(1-m*sn(x)^2)
868 (power (sub 1 (mul m
869 (power (ftake '%jacobi_sn u m) 2)))
870 1//2))))
871 ((zerop1 ($ratsimp (sub u (power (sub 1 m) 1//2))))
872 ;; A&S 16.5.3
873 ;; dn(sqrt(1-m),m) = K(m)
874 (ftake '%elliptic_kc m))
875 ;; A&S 16.20.2 (Jacobi's Imaginary transformation)
876 ((and $%iargs (multiplep u '$%i))
877 (ftake* '%jacobi_dc (coeff u '$%i 1)
878 (add 1 (neg m))))
879 ((setq coef (kc-arg2 u m))
880 ;; A&S 16.8.3
881 ;;
882 ;; dn(m*K+u) has period 2K
883 ;;
884 (destructuring-bind (lin const)
885 coef
886 (cond ((integerp lin)
887 (ecase (mod lin 2)
888 (0
889 ;; dn(2*m*K + u) = dn(u)
890 ;; dn(0) = 1
891 (if (zerop1 const)
892 1
893 ;; dn(4*m*K+2*K + u) = dn(2*K+u) = dn(u)
894 (ftake '%jacobi_dn const m)))
895 (1
896 ;; dn(2*m*K + K + u) = dn(K + u) = sqrt(1-m)*nd(u)
897 ;; dn(K) = sqrt(1-m)
898 (if (zerop1 const)
899 (power (sub 1 m) 1//2)
900 (mul (power (sub 1 m) 1//2)
901 (ftake '%jacobi_nd const m))))))
902 ((and (alike1 lin 1//2)
903 (zerop1 const))
904 ;; A&S 16.5.2
905 ;; dn(1/2*K) = (1-m)^(1/4)
906 (power (sub 1 m)
907 (div 1 4)))
908 (t
909 (give-up)))))
910 (t (give-up)))))
911
912 ;; Should we simplify the inverse elliptic functions into the
913 ;; appropriate incomplete elliptic integral? I think we should leave
914 ;; it, but perhaps allow some way to do that transformation if
915 ;; desired.
916
917 (def-simplifier inverse_jacobi_sn (u m)
918 (let (args)
919 ;; To numerically evaluate inverse_jacobi_sn (asn), use
920 ;;
921 ;; asn(x,m) = F(asin(x),m)
922 ;;
923 ;; But F(phi,m) = sin(phi)*rf(cos(phi)^2, 1-m*sin(phi)^2,1). Thus
924 ;;
925 ;; asn(x,m) = F(asin(x),m)
926 ;; = x*rf(1-x^2,1-m*x^2,1)
927 ;;
928 ;; I (rtoy) am not 100% about the first identity above for all
929 ;; complex values of x and m, but tests seem to indicate that it
930 ;; produces the correct value as verified by verifying
931 ;; jacobi_sn(inverse_jacobi_sn(x,m),m) = x.
932 (cond ((float-numerical-eval-p u m)
933 (let ((uu (bigfloat:to ($float u)))
934 (mm (bigfloat:to ($float m))))
935 (complexify
936 (* uu
937 (bigfloat::bf-rf (bigfloat:to (- 1 (* uu uu)))
938 (bigfloat:to (- 1 (* mm uu uu)))
939 1)))))
940 ((setf args (complex-float-numerical-eval-p u m))
941 (destructuring-bind (u m)
942 args
943 (let ((uu (bigfloat:to ($float u)))
944 (mm (bigfloat:to ($float m))))
945 (complexify (* uu (bigfloat::bf-rf (- 1 (* uu uu))
946 (- 1 (* mm uu uu))
947 1))))))
948 ((bigfloat-numerical-eval-p u m)
949 (let ((uu (bigfloat:to u))
950 (mm (bigfloat:to m)))
951 (to (bigfloat:* uu
952 (bigfloat::bf-rf (bigfloat:- 1 (bigfloat:* uu uu))
953 (bigfloat:- 1 (bigfloat:* mm uu uu))
954 1)))))
955 ((setf args (complex-bigfloat-numerical-eval-p u m))
956 (destructuring-bind (u m)
957 args
958 (let ((uu (bigfloat:to u))
959 (mm (bigfloat:to m)))
960 (to (bigfloat:* uu
961 (bigfloat::bf-rf (bigfloat:- 1 (bigfloat:* uu uu))
962 (bigfloat:- 1 (bigfloat:* mm uu uu))
963 1))))))
964 ((zerop1 u)
965 ;; asn(0,m) = 0
966 0)
967 ((onep1 u)
968 ;; asn(1,m) = elliptic_kc(m)
969 (ftake '%elliptic_kc m))
970 ((and (numberp u) (onep1 (- u)))
971 ;; asn(-1,m) = -elliptic_kc(m)
972 (mul -1 (ftake '%elliptic_kc m)))
973 ((zerop1 m)
974 ;; asn(x,0) = F(asin(x),0) = asin(x)
975 (ftake '%asin u))
976 ((onep1 m)
977 ;; asn(x,1) = F(asin(x),1) = log(tan(pi/4+asin(x)/2))
978 (ftake '%elliptic_f (ftake '%asin u) 1))
979 ((and (eq $triginverses '$all)
980 (listp u)
981 (eq (caar u) '%jacobi_sn)
982 (alike1 (third u) m))
983 ;; inverse_jacobi_sn(sn(u)) = u
984 (second u))
985 (t
986 ;; Nothing to do
987 (give-up)))))
988
989 (def-simplifier inverse_jacobi_cn (u m)
990 (let (args)
991 (cond ((float-numerical-eval-p u m)
992 ;; Numerically evaluate acn
993 ;;
994 ;; acn(x,m) = F(acos(x),m)
995 (to (elliptic-f (cl:acos ($float u)) ($float m))))
996 ((setf args (complex-float-numerical-eval-p u m))
997 (destructuring-bind (u m)
998 args
999 (to (elliptic-f (cl:acos (bigfloat:to ($float u)))
1000 (bigfloat:to ($float m))))))
1001 ((bigfloat-numerical-eval-p u m)
1002 (to (bigfloat::bf-elliptic-f (bigfloat:acos (bigfloat:to u))
1003 (bigfloat:to m))))
1004 ((setf args (complex-bigfloat-numerical-eval-p u m))
1005 (destructuring-bind (u m)
1006 args
1007 (to (bigfloat::bf-elliptic-f (bigfloat:acos (bigfloat:to u))
1008 (bigfloat:to m)))))
1009 ((zerop1 m)
1010 ;; asn(x,0) = F(acos(x),0) = acos(x)
1011 (ftake '%elliptic_f (ftake '%acos u) 0))
1012 ((onep1 m)
1013 ;; asn(x,1) = F(asin(x),1) = log(tan(pi/2+asin(x)/2))
1014 (ftake '%elliptic_f (ftake '%acos u) 1))
1015 ((zerop1 u)
1016 (ftake '%elliptic_kc m))
1017 ((onep1 u)
1018 0)
1019 ((and (eq $triginverses '$all)
1020 (listp u)
1021 (eq (caar u) '%jacobi_cn)
1022 (alike1 (third u) m))
1023 ;; inverse_jacobi_cn(cn(u)) = u
1024 (second u))
1025 (t
1026 ;; Nothing to do
1027 (give-up)))))
1028
1029 (def-simplifier inverse_jacobi_dn (u m)
1030 (let (args)
1031 (cond ((float-numerical-eval-p u m)
1032 (to (bigfloat::bf-inverse-jacobi-dn (bigfloat:to (float u))
1033 (bigfloat:to (float m)))))
1034 ((setf args (complex-float-numerical-eval-p u m))
1035 (destructuring-bind (u m)
1036 args
1037 (let ((uu (bigfloat:to ($float u)))
1038 (mm (bigfloat:to ($float m))))
1039 (to (bigfloat::bf-inverse-jacobi-dn uu mm)))))
1040 ((bigfloat-numerical-eval-p u m)
1041 (let ((uu (bigfloat:to u))
1042 (mm (bigfloat:to m)))
1043 (to (bigfloat::bf-inverse-jacobi-dn uu mm))))
1044 ((setf args (complex-bigfloat-numerical-eval-p u m))
1045 (destructuring-bind (u m)
1046 args
1047 (to (bigfloat::bf-inverse-jacobi-dn (bigfloat:to u) (bigfloat:to m)))))
1048 ((onep1 m)
1049 ;; x = dn(u,1) = sech(u). so u = asech(x)
1050 (ftake '%asech u))
1051 ((onep1 u)
1052 ;; jacobi_dn(0,m) = 1
1053 0)
1054 ((zerop1 ($ratsimp (sub u (power (sub 1 m) 1//2))))
1055 ;; jacobi_dn(K(m),m) = sqrt(1-m) so
1056 ;; inverse_jacobi_dn(sqrt(1-m),m) = K(m)
1057 (ftake '%elliptic_kc m))
1058 ((and (eq $triginverses '$all)
1059 (listp u)
1060 (eq (caar u) '%jacobi_dn)
1061 (alike1 (third u) m))
1062 ;; inverse_jacobi_dn(dn(u)) = u
1063 (second u))
1064 (t
1065 ;; Nothing to do
1066 (give-up)))))
1067
1068 ;;;; Elliptic integrals
1069
1070 (let ((errtol (expt (* 4 flonum-epsilon) 1/6))
1071 (c1 (float 1/24))
1072 (c2 (float 3/44))
1073 (c3 (float 1/14)))
1074 (declare (type flonum errtol c1 c2 c3))
1075 (defun crf (x y z)
1076 "Compute Carlson's incomplete or complete elliptic integral of the
1077 first kind:
1078
1079 INF
1080 /
1081 [ 1
1082 RF(x, y, z) = I ----------------------------------- dt
1083 ] SQRT(x + t) SQRT(y + t) SQRT(z + t)
1084 /
1085 0
1086
1087 x, y, and z may be complex.
1088 "
1089 (declare (number x y z))
1090 (let ((x (coerce x '(complex flonum)))
1091 (y (coerce y '(complex flonum)))
1092 (z (coerce z '(complex flonum))))
1093 (declare (type (complex flonum) x y z))
1094 (loop
1095 (let* ((mu (/ (+ x y z) 3))
1096 (x-dev (- 2 (/ (+ mu x) mu)))
1097 (y-dev (- 2 (/ (+ mu y) mu)))
1098 (z-dev (- 2 (/ (+ mu z) mu))))
1099 (when (< (max (abs x-dev) (abs y-dev) (abs z-dev)) errtol)
1100 (let ((e2 (- (* x-dev y-dev) (* z-dev z-dev)))
1101 (e3 (* x-dev y-dev z-dev)))
1102 (return (/ (+ 1
1103 (* e2 (- (* c1 e2)
1104 1/10
1105 (* c2 e3)))
1106 (* c3 e3))
1107 (sqrt mu)))))
1108 (let* ((x-root (sqrt x))
1109 (y-root (sqrt y))
1110 (z-root (sqrt z))
1111 (lam (+ (* x-root (+ y-root z-root)) (* y-root z-root))))
1112 (setf x (* (+ x lam) 1/4))
1113 (setf y (* (+ y lam) 1/4))
1114 (setf z (* (+ z lam) 1/4))))))))
1115
1116 ;; Elliptic integral of the first kind (Legendre's form):
1117 ;;
1118 ;;
1119 ;; phi
1120 ;; /
1121 ;; [ 1
1122 ;; I ------------------- ds
1123 ;; ] 2
1124 ;; / SQRT(1 - m SIN (s))
1125 ;; 0
1126
1127 (defun elliptic-f (phi-arg m-arg)
1128 (flet ((base (phi-arg m-arg)
1129 (cond ((and (realp m-arg) (realp phi-arg))
1130 (let ((phi (float phi-arg))
1131 (m (float m-arg)))
1132 (cond ((> m 1)
1133 ;; A&S 17.4.15
1134 ;;
1135 ;; F(phi|m) = 1/sqrt(m)*F(theta|1/m)
1136 ;;
1137 ;; with sin(theta) = sqrt(m)*sin(phi)
1138 (/ (elliptic-f (cl:asin (* (sqrt m) (sin phi))) (/ m))
1139 (sqrt m)))
1140 ((< m 0)
1141 ;; A&S 17.4.17
1142 (let* ((m (- m))
1143 (m+1 (+ 1 m))
1144 (root (sqrt m+1))
1145 (m/m+1 (/ m m+1)))
1146 (- (/ (elliptic-f (float (/ pi 2)) m/m+1)
1147 root)
1148 (/ (elliptic-f (- (float (/ pi 2)) phi) m/m+1)
1149 root))))
1150 ((= m 0)
1151 ;; A&S 17.4.19
1152 phi)
1153 ((= m 1)
1154 ;; A&S 17.4.21
1155 ;;
1156 1 ;; F(phi,1) = log(sec(phi)+tan(phi))
1157 ;; = log(tan(pi/4+pi/2))
1158 (log (cl:tan (+ (/ phi 2) (float (/ pi 4))))))
1159 ((minusp phi)
1160 (- (elliptic-f (- phi) m)))
1161 ((> phi pi)
1162 ;; A&S 17.4.3
1163 (multiple-value-bind (s phi-rem)
1164 (truncate phi (float pi))
1165 (+ (* 2 s (elliptic-k m))
1166 (elliptic-f phi-rem m))))
1167 ((<= phi (/ pi 2))
1168 (let ((sin-phi (sin phi))
1169 (cos-phi (cos phi))
1170 (k (sqrt m)))
1171 (* sin-phi
1172 (bigfloat::bf-rf (* cos-phi cos-phi)
1173 (* (- 1 (* k sin-phi))
1174 (+ 1 (* k sin-phi)))
1175 1.0))))
1176 ((< phi pi)
1177 (+ (* 2 (elliptic-k m))
1178 (elliptic-f (- phi (float pi)) m)))
1179 (t
1180 (error "Shouldn't happen! Unhandled case in elliptic-f: ~S ~S~%"
1181 phi-arg m-arg)))))
1182 (t
1183 (let ((phi (coerce phi-arg '(complex flonum)))
1184 (m (coerce m-arg '(complex flonum))))
1185 (let ((sin-phi (sin phi))
1186 (cos-phi (cos phi))
1187 (k (sqrt m)))
1188 (* sin-phi
1189 (crf (* cos-phi cos-phi)
1190 (* (- 1 (* k sin-phi))
1191 (+ 1 (* k sin-phi)))
1192 1.0))))))))
1193 ;; Elliptic F is quasi-periodic wrt to z:
1194 ;;
1195 ;; F(z|m) = F(z - pi*round(Re(z)/pi)|m) + 2*round(Re(z)/pi)*K(m)
1196 (let ((period (round (realpart phi-arg) pi)))
1197 (+ (base (- phi-arg (* pi period)) m-arg)
1198 (if (zerop period)
1199 0
1200 (* 2 period
1201 (bigfloat:to (elliptic-k m-arg))))))))
1202
1203 ;; Complete elliptic integral of the first kind
1204 (defun elliptic-k (m)
1205 (cond ((realp m)
1206 (cond ((< m 0)
1207 ;; A&S 17.4.17
1208 (let* ((m (- m))
1209 (m+1 (+ 1 m))
1210 (root (sqrt m+1))
1211 (m/m+1 (/ m m+1)))
1212 (- (/ (elliptic-k m/m+1)
1213 root)
1214 (/ (elliptic-f 0.0 m/m+1)
1215 root))))
1216 ((= m 0)
1217 ;; A&S 17.4.19
1218 (float (/ pi 2)))
1219 ((= m 1)
1220 (maxima::merror
1221 (intl:gettext "elliptic_kc: elliptic_kc(1) is undefined.")))
1222 (t
1223 (bigfloat::bf-rf 0.0 (- 1 m)
1224 1.0))))
1225 (t
1226 (bigfloat::bf-rf 0.0 (- 1 m)
1227 1.0))))
1228
1229 ;; Elliptic integral of the second kind (Legendre's form):
1230 ;;
1231 ;;
1232 ;; phi
1233 ;; /
1234 ;; [ 2
1235 ;; I SQRT(1 - m SIN (s)) ds
1236 ;; ]
1237 ;; /
1238 ;; 0
1239
1240 (defun elliptic-e (phi m)
1241 (declare (type flonum phi m))
1242 (flet ((base (phi m)
1243 (cond ((= m 0)
1244 ;; A&S 17.4.23
1245 phi)
1246 ((= m 1)
1247 ;; A&S 17.4.25
1248 (sin phi))
1249 (t
1250 (let* ((sin-phi (sin phi))
1251 (cos-phi (cos phi))
1252 (k (sqrt m))
1253 (y (* (- 1 (* k sin-phi))
1254 (+ 1 (* k sin-phi)))))
1255 (to (- (* sin-phi
1256 (bigfloat::bf-rf (* cos-phi cos-phi) y 1.0))
1257 (* (/ m 3)
1258 (expt sin-phi 3)
1259 (bigfloat::bf-rd (* cos-phi cos-phi) y 1.0)))))))))
1260 ;; Elliptic E is quasi-periodic wrt to phi:
1261 ;;
1262 ;; E(z|m) = E(z - %pi*round(Re(z)/%pi)|m) + 2*round(Re(z)/%pi)*E(m)
1263 (let ((period (round (realpart phi) pi)))
1264 (+ (base (- phi (* pi period)) m)
1265 (* 2 period (elliptic-ec m))))))
1266
1267 ;; Complete version
1268 (defun elliptic-ec (m)
1269 (declare (type flonum m))
1270 (cond ((= m 0)
1271 ;; A&S 17.4.23
1272 (float (/ pi 2)))
1273 ((= m 1)
1274 ;; A&S 17.4.25
1275 1.0)
1276 (t
1277 (let* ((y (- 1 m)))
1278 (to (- (bigfloat::bf-rf 0.0 y 1.0)
1279 (* (/ m 3)
1280 (bigfloat::bf-rd 0.0 y 1.0))))))))
1281
1282
1283 ;; Define the elliptic integrals for maxima
1284 ;;
1285 ;; We use the definitions given in A&S 17.2.6 and 17.2.8. In particular:
1286 ;;
1287 ;; phi
1288 ;; /
1289 ;; [ 1
1290 ;; F(phi|m) = I ------------------- ds
1291 ;; ] 2
1292 ;; / SQRT(1 - m SIN (s))
1293 ;; 0
1294 ;;
1295 ;; and
1296 ;;
1297 ;; phi
1298 ;; /
1299 ;; [ 2
1300 ;; E(phi|m) = I SQRT(1 - m SIN (s)) ds
1301 ;; ]
1302 ;; /
1303 ;; 0
1304 ;;
1305 ;; That is, we do not use the modular angle, alpha, as the second arg;
1306 ;; the parameter m = sin(alpha)^2 is used.
1307 ;;
1308
1309
1310 ;; The derivative of F(phi|m) wrt to phi is easy. The derivative wrt
1311 ;; to m is harder. Here is a derivation. Hope I got it right.
1312 ;;
1313 ;; diff(integrate(1/sqrt(1-m*sin(x)^2),x,0,phi), m);
1314 ;;
1315 ;; PHI
1316 ;; / 2
1317 ;; [ SIN (x)
1318 ;; I ------------------ dx
1319 ;; ] 2 3/2
1320 ;; / (1 - m SIN (x))
1321 ;; 0
1322 ;; --------------------------
1323 ;; 2
1324 ;;
1325 ;;
1326 ;; Now use the following relationship that is easily verified:
1327 ;;
1328 ;; 2 2
1329 ;; (1 - m) SIN (x) COS (x) COS(x) SIN(x)
1330 ;; ------------------- = ------------------- - DIFF(-------------------, x)
1331 ;; 2 2 2
1332 ;; SQRT(1 - m SIN (x)) SQRT(1 - m SIN (x)) SQRT(1 - m SIN (x))
1333 ;;
1334 ;;
1335 ;; Now integrate this to get:
1336 ;;
1337 ;;
1338 ;; PHI
1339 ;; / 2
1340 ;; [ SIN (x)
1341 ;; (1 - m) I ------------------- dx =
1342 ;; ] 2
1343 ;; / SQRT(1 - m SIN (x))
1344 ;; 0
1345
1346 ;;
1347 ;; PHI
1348 ;; / 2
1349 ;; [ COS (x)
1350 ;; + I ------------------- dx
1351 ;; ] 2
1352 ;; / SQRT(1 - m SIN (x))
1353 ;; 0
1354 ;; COS(PHI) SIN(PHI)
1355 ;; - ---------------------
1356 ;; 2
1357 ;; SQRT(1 - m SIN (PHI))
1358 ;;
1359 ;; Use the fact that cos(x)^2 = 1 - sin(x)^2 to show that this
1360 ;; integral on the RHS is:
1361 ;;
1362 ;;
1363 ;; (1 - m) elliptic_F(PHI, m) + elliptic_E(PHI, m)
1364 ;; -------------------------------------------
1365 ;; m
1366 ;; So, finally, we have
1367 ;;
1368 ;;
1369 ;;
1370 ;; d
1371 ;; 2 -- (elliptic_F(PHI, m)) =
1372 ;; dm
1373 ;;
1374 ;; elliptic_E(PHI, m) - (1 - m) elliptic_F(PHI, m) COS(PHI) SIN(PHI)
1375 ;; ---------------------------------------------- - ---------------------
1376 ;; m 2
1377 ;; SQRT(1 - m SIN (PHI))
1378 ;; ----------------------------------------------------------------------
1379 ;; 1 - m
1380
1381 (defprop %elliptic_f
1382 ((phi m)
1383 ;; diff wrt phi
1384 ;; 1/sqrt(1-m*sin(phi)^2)
1385 ((mexpt simp)
1386 ((mplus simp) 1 ((mtimes simp) -1 m ((mexpt simp) ((%sin simp) phi) 2)))
1387 ((rat simp) -1 2))
1388 ;; diff wrt m
1389 ((mtimes simp) ((rat simp) 1 2)
1390 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m)) -1)
1391 ((mplus simp)
1392 ((mtimes simp) ((mexpt simp) m -1)
1393 ((mplus simp) ((%elliptic_e simp) phi m)
1394 ((mtimes simp) -1 ((mplus simp) 1 ((mtimes simp) -1 m))
1395 ((%elliptic_f simp) phi m))))
1396 ((mtimes simp) -1 ((%cos simp) phi) ((%sin simp) phi)
1397 ((mexpt simp)
1398 ((mplus simp) 1
1399 ((mtimes simp) -1 m ((mexpt simp) ((%sin simp) phi) 2)))
1400 ((rat simp) -1 2))))))
1401 grad)
1402
1403 ;;
1404 ;; The derivative of E(phi|m) wrt to m is much simpler to derive than F(phi|m).
1405 ;;
1406 ;; Take the derivative of the definition to get
1407 ;;
1408 ;; PHI
1409 ;; / 2
1410 ;; [ SIN (x)
1411 ;; I ------------------- dx
1412 ;; ] 2
1413 ;; / SQRT(1 - m SIN (x))
1414 ;; 0
1415 ;; - ---------------------------
1416 ;; 2
1417 ;;
1418 ;; It is easy to see that
1419 ;;
1420 ;; PHI
1421 ;; / 2
1422 ;; [ SIN (x)
1423 ;; elliptic_F(PHI, m) - m I ------------------- dx = elliptic_E(PHI, m)
1424 ;; ] 2
1425 ;; / SQRT(1 - m SIN (x))
1426 ;; 0
1427 ;;
1428 ;; So we finally have
1429 ;;
1430 ;; d elliptic_E(PHI, m) - elliptic_F(PHI, m)
1431 ;; -- (elliptic_E(PHI, m)) = ---------------------------------------
1432 ;; dm 2 m
1433
1434 (defprop %elliptic_e
1435 ((phi m)
1436 ;; sqrt(1-m*sin(phi)^2)
1437 ((mexpt simp)
1438 ((mplus simp) 1 ((mtimes simp) -1 m ((mexpt simp) ((%sin simp) phi) 2)))
1439 ((rat simp) 1 2))
1440 ;; diff wrt m
1441 ((mtimes simp) ((rat simp) 1 2) ((mexpt simp) m -1)
1442 ((mplus simp) ((%elliptic_e simp) phi m)
1443 ((mtimes simp) -1 ((%elliptic_f simp) phi m)))))
1444 grad)
1445
1446 (def-simplifier elliptic_f (phi m)
1447 (let (args)
1448 (cond ((float-numerical-eval-p phi m)
1449 ;; Numerically evaluate it
1450 (to (elliptic-f ($float phi) ($float m))))
1451 ((setf args (complex-float-numerical-eval-p phi m))
1452 (destructuring-bind (phi m)
1453 args
1454 (to (elliptic-f (bigfloat:to ($float phi))
1455 (bigfloat:to ($float m))))))
1456 ((bigfloat-numerical-eval-p phi m)
1457 (to (bigfloat::bf-elliptic-f (bigfloat:to ($bfloat phi))
1458 (bigfloat:to ($bfloat m)))))
1459 ((setf args (complex-bigfloat-numerical-eval-p phi m))
1460 (destructuring-bind (phi m)
1461 args
1462 (to (bigfloat::bf-elliptic-f (bigfloat:to ($bfloat phi))
1463 (bigfloat:to ($bfloat m))))))
1464 ((zerop1 phi)
1465 0)
1466 ((zerop1 m)
1467 ;; A&S 17.4.19
1468 phi)
1469 ((onep1 m)
1470 ;; A&S 17.4.21. Let's pick the log tan form. But this
1471 ;; isn't right if we know that abs(phi) > %pi/2, where
1472 ;; elliptic_f is undefined (or infinity).
1473 (cond ((not (eq '$pos (csign (sub ($abs phi) (div '$%pi 2)))))
1474 (ftake '%log
1475 (ftake '%tan
1476 (add (mul '$%pi (div 1 4))
1477 (mul 1//2 phi)))))
1478 (t
1479 (merror (intl:gettext "elliptic_f: elliptic_f(~:M, ~:M) is undefined.")
1480 phi m))))
1481 ((alike1 phi '((mtimes) ((rat) 1 2) $%pi))
1482 ;; Complete elliptic integral
1483 (ftake '%elliptic_kc m))
1484 (t
1485 ;; Nothing to do
1486 (give-up)))))
1487
1488 (def-simplifier elliptic_e (phi m)
1489 (let (args)
1490 (cond ((float-numerical-eval-p phi m)
1491 ;; Numerically evaluate it
1492 (elliptic-e ($float phi) ($float m)))
1493 ((complex-float-numerical-eval-p phi m)
1494 (complexify (bigfloat::bf-elliptic-e (complex ($float ($realpart phi)) ($float ($imagpart phi)))
1495 (complex ($float ($realpart m)) ($float ($imagpart m))))))
1496 ((bigfloat-numerical-eval-p phi m)
1497 (to (bigfloat::bf-elliptic-e (bigfloat:to ($bfloat phi))
1498 (bigfloat:to ($bfloat m)))))
1499 ((setf args (complex-bigfloat-numerical-eval-p phi m))
1500 (destructuring-bind (phi m)
1501 args
1502 (to (bigfloat::bf-elliptic-e (bigfloat:to ($bfloat phi))
1503 (bigfloat:to ($bfloat m))))))
1504 ((zerop1 phi)
1505 0)
1506 ((zerop1 m)
1507 ;; A&S 17.4.23
1508 phi)
1509 ((onep1 m)
1510 ;; A&S 17.4.25, but handle periodicity:
1511 ;; elliptic_e(x,m) = elliptic_e(x-%pi*round(x/%pi), m)
1512 ;; + 2*round(x/%pi)*elliptic_ec(m)
1513 ;;
1514 ;; Or
1515 ;;
1516 ;; elliptic_e(x,1) = sin(phi) + 2*round(x/%pi)*elliptic_ec(m)
1517 ;;
1518 (add (ftake '%sin phi)
1519 (mul 2
1520 (mul (ftake '%round (div phi '$%pi))
1521 (ftake '%elliptic_ec m)))))
1522 ((alike1 phi '((mtimes) ((rat) 1 2) $%pi))
1523 ;; Complete elliptic integral
1524 (ftake '%elliptic_ec m))
1525 ((and ($numberp phi)
1526 (let ((r ($round (div phi '$%pi))))
1527 (and ($numberp r)
1528 (not (zerop1 r)))))
1529 ;; Handle the case where phi is a number where we can apply
1530 ;; the periodicity property without blowing up the
1531 ;; expression.
1532 (add (ftake '%elliptic_e
1533 (add phi
1534 (mul (mul -1 '$%pi)
1535 (ftake '%round (div phi '$%pi))))
1536 m)
1537 (mul 2
1538 (mul (ftake '%round (div phi '$%pi))
1539 (ftake '%elliptic_ec m)))))
1540 (t
1541 ;; Nothing to do
1542 (give-up)))))
1543
1544 ;; Complete elliptic integrals
1545 ;;
1546 ;; elliptic_kc(m) = elliptic_f(%pi/2, m)
1547 ;;
1548 ;; elliptic_ec(m) = elliptic_e(%pi/2, m)
1549 ;;
1550
1551 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
1552
1553 ;;; We support a simplim%function. The function is looked up in simplimit and
1554 ;;; handles specific values of the function.
1555
1556 (defprop %elliptic_kc simplim%elliptic_kc simplim%function)
1557
1558 (defun simplim%elliptic_kc (expr var val)
1559 ;; Look for the limit of the argument
1560 (let ((m (limit (cadr expr) var val 'think)))
1561 (cond ((onep1 m)
1562 ;; For an argument 1 return $infinity.
1563 '$infinity)
1564 (t
1565 ;; All other cases are handled by the simplifier of the function.
1566 (simplify (list '(%elliptic_kc) m))))))
1567
1568 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
1569
1570 (def-simplifier elliptic_kc (m)
1571 (let (args)
1572 (cond ((onep1 m)
1573 ;; elliptic_kc(1) is complex infinity. Maxima can not handle
1574 ;; infinities correctly, throw a Maxima error.
1575 (merror
1576 (intl:gettext "elliptic_kc: elliptic_kc(~:M) is undefined.")
1577 m))
1578 ((float-numerical-eval-p m)
1579 ;; Numerically evaluate it
1580 (to (elliptic-k ($float m))))
1581 ((complex-float-numerical-eval-p m)
1582 (complexify (bigfloat::bf-elliptic-k (complex ($float ($realpart m)) ($float ($imagpart m))))))
1583 ((setf args (complex-bigfloat-numerical-eval-p m))
1584 (destructuring-bind (m)
1585 args
1586 (to (bigfloat::bf-elliptic-k (bigfloat:to ($bfloat m))))))
1587 ((zerop1 m)
1588 '((mtimes) ((rat) 1 2) $%pi))
1589 ((alike1 m 1//2)
1590 ;; http://functions.wolfram.com/EllipticIntegrals/EllipticK/03/01/
1591 ;;
1592 ;; elliptic_kc(1/2) = 8*%pi^(3/2)/gamma(-1/4)^2
1593 (div (mul 8 (power '$%pi (div 3 2)))
1594 (power (gm (div -1 4)) 2)))
1595 ((eql -1 m)
1596 ;; elliptic_kc(-1) = gamma(1/4)^2/(4*sqrt(2*%pi))
1597 (div (power (gm (div 1 4)) 2)
1598 (mul 4 (power (mul 2 '$%pi) 1//2))))
1599 ((alike1 m (add 17 (mul -12 (power 2 1//2))))
1600 ;; elliptic_kc(17-12*sqrt(2)) = 2*(2+sqrt(2))*%pi^(3/2)/gamma(-1/4)^2
1601 (div (mul 2 (mul (add 2 (power 2 1//2))
1602 (power '$%pi (div 3 2))))
1603 (power (gm (div -1 4)) 2)))
1604 (t
1605 ;; Nothing to do
1606 (give-up)))))
1607
1608 (defprop %elliptic_kc
1609 ((m)
1610 ;; diff wrt m
1611 ((mtimes)
1612 ((rat) 1 2)
1613 ((mplus) ((%elliptic_ec) m)
1614 ((mtimes) -1
1615 ((%elliptic_kc) m)
1616 ((mplus) 1 ((mtimes) -1 m))))
1617 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
1618 ((mexpt) m -1)))
1619 grad)
1620
1621 (def-simplifier elliptic_ec (m)
1622 (let (args)
1623 (cond ((float-numerical-eval-p m)
1624 ;; Numerically evaluate it
1625 (elliptic-ec ($float m)))
1626 ((setf args (complex-float-numerical-eval-p m))
1627 (destructuring-bind (m)
1628 args
1629 (complexify (bigfloat::bf-elliptic-ec (bigfloat:to ($float m))))))
1630 ((setf args (complex-bigfloat-numerical-eval-p m))
1631 (destructuring-bind (m)
1632 args
1633 (to (bigfloat::bf-elliptic-ec (bigfloat:to ($bfloat m))))))
1634 ;; Some special cases we know about.
1635 ((zerop1 m)
1636 '((mtimes) ((rat) 1 2) $%pi))
1637 ((onep1 m)
1638 1)
1639 ((alike1 m 1//2)
1640 ;; elliptic_ec(1/2). Use the identity
1641 ;;
1642 ;; elliptic_ec(z)*elliptic_kc(1-z) - elliptic_kc(z)*elliptic_kc(1-z)
1643 ;; + elliptic_ec(1-z)*elliptic_kc(z) = %pi/2;
1644 ;;
1645 ;; Let z = 1/2 to get
1646 ;;
1647 ;; %pi^(3/2)*'elliptic_ec(1/2)/gamma(3/4)^2-%pi^3/(4*gamma(3/4)^4) = %pi/2
1648 ;;
1649 ;; since we know that elliptic_kc(1/2) = %pi^(3/2)/(2*gamma(3/4)^2). Hence
1650 ;;
1651 ;; elliptic_ec(1/2)
1652 ;; = (2*%pi*gamma(3/4)^4+%pi^3)/(4*%pi^(3/2)*gamma(3/4)^2)
1653 ;; = gamma(3/4)^2/(2*sqrt(%pi))+%pi^(3/2)/(4*gamma(3/4)^2)
1654 ;;
1655 (add (div (power (ftake '%gamma (div 3 4)) 2)
1656 (mul 2 (power '$%pi 1//2)))
1657 (div (power '$%pi (div 3 2))
1658 (mul 4 (power (ftake '%gamma (div 3 4)) 2)))))
1659 ((zerop1 (add 1 m))
1660 ;; elliptic_ec(-1). Use the identity
1661 ;; http://functions.wolfram.com/08.01.17.0002.01
1662 ;;
1663 ;;
1664 ;; elliptic_ec(z) = sqrt(1 - z)*elliptic_ec(z/(z-1))
1665 ;;
1666 ;; Let z = -1 to get
1667 ;;
1668 ;; elliptic_ec(-1) = sqrt(2)*elliptic_ec(1/2)
1669 ;;
1670 ;; Should we expand out elliptic_ec(1/2) using the above result?
1671 (mul (power 2 1//2)
1672 (ftake '%elliptic_ec 1//2)))
1673 (t
1674 ;; Nothing to do
1675 (give-up)))))
1676
1677 (defprop %elliptic_ec
1678 ((m)
1679 ((mtimes) ((rat) 1 2)
1680 ((mplus) ((%elliptic_ec) m)
1681 ((mtimes) -1 ((%elliptic_kc)
1682 m)))
1683 ((mexpt) m -1)))
1684 grad)
1685
1686 ;;
1687 ;; Elliptic integral of the third kind:
1688 ;;
1689 ;; (A&S 17.2.14)
1690 ;;
1691 ;; phi
1692 ;; /
1693 ;; [ 1
1694 ;; PI(n;phi|m) = I ----------------------------------- ds
1695 ;; ] 2 2
1696 ;; / SQRT(1 - m SIN (s)) (1 - n SIN (s))
1697 ;; 0
1698 ;;
1699 ;; As with E and F, we do not use the modular angle alpha but the
1700 ;; parameter m = sin(alpha)^2.
1701 ;;
1702 (def-simplifier elliptic_pi (n phi m)
1703 (let (args)
1704 (cond
1705 ((float-numerical-eval-p n phi m)
1706 ;; Numerically evaluate it
1707 (elliptic-pi ($float n) ($float phi) ($float m)))
1708 ((setf args (complex-float-numerical-eval-p n phi m))
1709 (destructuring-bind (n phi m)
1710 args
1711 (elliptic-pi (bigfloat:to ($float n))
1712 (bigfloat:to ($float phi))
1713 (bigfloat:to ($float m)))))
1714 ((bigfloat-numerical-eval-p n phi m)
1715 (to (bigfloat::bf-elliptic-pi (bigfloat:to n)
1716 (bigfloat:to phi)
1717 (bigfloat:to m))))
1718 ((setq args (complex-bigfloat-numerical-eval-p n phi m))
1719 (destructuring-bind (n phi m)
1720 args
1721 (to (bigfloat::bf-elliptic-pi (bigfloat:to n)
1722 (bigfloat:to phi)
1723 (bigfloat:to m)))))
1724 ((zerop1 n)
1725 (ftake '%elliptic_f phi m))
1726 ((zerop1 m)
1727 ;; 3 cases depending on n < 1, n > 1, or n = 1.
1728 (let ((s (asksign (add -1 n))))
1729 (case s
1730 ($positive
1731 (div (ftake '%atanh (mul (power (add n -1) 1//2)
1732 (ftake '%tan phi)))
1733 (power (add n -1) 1//2)))
1734 ($negative
1735 (div (ftake '%atan (mul (power (sub 1 n) 1//2)
1736 (ftake '%tan phi)))
1737 (power (sub 1 n) 1//2)))
1738 ($zero
1739 (ftake '%tan phi)))))
1740 (t
1741 ;; Nothing to do
1742 (give-up)))))
1743
1744 ;; Complete elliptic-pi. That is phi = %pi/2. Then
1745 ;; elliptic_pi(n,m)
1746 ;; = Rf(0, 1-m,1) + Rj(0,1-m,1-n)*n/3;
1747 (defun elliptic-pi-complete (n m)
1748 (to (bigfloat:+ (bigfloat::bf-rf 0 (- 1 m) 1)
1749 (bigfloat:* 1/3 n (bigfloat::bf-rj 0 (- 1 m) 1 (- 1 n))))))
1750
1751 ;; To compute elliptic_pi for all z, we use the property
1752 ;; (http://functions.wolfram.com/08.06.16.0002.01)
1753 ;;
1754 ;; elliptic_pi(n, z + %pi*k, m)
1755 ;; = 2*k*elliptic_pi(n, %pi/2, m) + elliptic_pi(n, z, m)
1756 ;;
1757 ;; So we are left with computing the integral for 0 <= z < %pi. Using
1758 ;; Carlson's formulation produces the wrong values for %pi/2 < z <
1759 ;; %pi. How to do that?
1760 ;;
1761 ;; Let
1762 ;;
1763 ;; I(a,b) = integrate(1/(1-n*sin(x)^2)/sqrt(1 - m*sin(x)^2), x, a, b)
1764 ;;
1765 ;; That is, I(a,b) is the integral for the elliptic_pi function but
1766 ;; with a lower limit of a and an upper limit of b.
1767 ;;
1768 ;; Then, we want to compute I(0, z), with %pi <= z < %pi. Let w = z +
1769 ;; %pi/2, 0 <= w < %pi/2. Then
1770 ;;
1771 ;; I(0, w+%pi/2) = I(0, %pi/2) + I(%pi/2, w+%pi/2)
1772 ;;
1773 ;; To evaluate I(%pi/2, w+%pi/2), use a change of variables:
1774 ;;
1775 ;; changevar('integrate(1/(1-n*sin(x)^2)/sqrt(1 - m*sin(x)^2), x, %pi/2, w + %pi/2),
1776 ;; x-%pi+u,u,x)
1777 ;;
1778 ;; = integrate(-1/(sqrt(1-m*sin(u)^2)*(1-n*sin(u)^2)),u,%pi/2-w,%pi/2)
1779 ;; = I(%pi/2-w,%pi/2)
1780 ;; = I(0,%pi/2) - I(0,%pi/2-w)
1781 ;;
1782 ;; Thus,
1783 ;;
1784 ;; I(0,%pi/2+w) = 2*I(0,%pi/2) - I(0,%pi/2-w)
1785 ;;
1786 ;; This allows us to compute the general result with 0 <= z < %pi
1787 ;;
1788 ;; I(0, k*%pi + z) = 2*k*I(0,%pi/2) + I(0,z);
1789 ;;
1790 ;; If 0 <= z < %pi/2, then the we are done. If %pi/2 <= z < %pi, let
1791 ;; z = w+%pi/2. Then
1792 ;;
1793 ;; I(0,z) = 2*I(0,%pi/2) - I(0,%pi/2-w)
1794 ;;
1795 ;; Or, since w = z-%pi/2:
1796 ;;
1797 ;; I(0,z) = 2*I(0,%pi/2) - I(0,%pi-z)
1798
1799 (defun elliptic-pi (n phi m)
1800 ;; elliptic_pi(n, -phi, m) = -elliptic_pi(n, phi, m). That is, it
1801 ;; is an odd function of phi.
1802 (when (minusp (realpart phi))
1803 (return-from elliptic-pi (- (elliptic-pi n (- phi) m))))
1804
1805 ;; Note: Carlson's DRJ has n defined as the negative of the n given
1806 ;; in A&S.
1807 (flet ((base (n phi m)
1808 ;; elliptic_pi(n,phi,m) =
1809 ;; sin(phi)*Rf(cos(phi)^2, 1-m*sin(phi)^2, 1)
1810 ;; - (-n / 3) * sin(phi)^3
1811 ;; * Rj(cos(phi)^2, 1-m*sin(phi)^2, 1, 1-n*sin(phi)^2)
1812 (let* ((nn (- n))
1813 (sin-phi (sin phi))
1814 (cos-phi (cos phi))
1815 (k (sqrt m))
1816 (k2sin (* (- 1 (* k sin-phi))
1817 (+ 1 (* k sin-phi)))))
1818 (- (* sin-phi (bigfloat::bf-rf (expt cos-phi 2) k2sin 1.0))
1819 (* (/ nn 3) (expt sin-phi 3)
1820 (bigfloat::bf-rj (expt cos-phi 2) k2sin 1.0
1821 (- 1 (* n (expt sin-phi 2)))))))))
1822 ;; FIXME: Reducing the arg by pi has significant round-off.
1823 ;; Consider doing something better.
1824 (let* ((cycles (round (realpart phi) pi))
1825 (rem (- phi (* cycles pi))))
1826 (let ((complete (elliptic-pi-complete n m)))
1827 (to (+ (* 2 cycles complete)
1828 (base n rem m)))))))
1829
1830 ;;; Deriviatives from functions.wolfram.com
1831 ;;; http://functions.wolfram.com/EllipticIntegrals/EllipticPi3/20/
1832 (defprop %elliptic_pi
1833 ((n z m)
1834 ;Derivative wrt first argument
1835 ((mtimes) ((rat) 1 2)
1836 ((mexpt) ((mplus) m ((mtimes) -1 n)) -1)
1837 ((mexpt) ((mplus) -1 n) -1)
1838 ((mplus)
1839 ((mtimes) ((mexpt) n -1)
1840 ((mplus) ((mtimes) -1 m) ((mexpt) n 2))
1841 ((%elliptic_pi) n z m))
1842 ((%elliptic_e) z m)
1843 ((mtimes) ((mplus) m ((mtimes) -1 n)) ((mexpt) n -1)
1844 ((%elliptic_f) z m))
1845 ((mtimes) ((rat) -1 2) n
1846 ((mexpt)
1847 ((mplus) 1 ((mtimes) -1 m ((mexpt) ((%sin) z) 2)))
1848 ((rat) 1 2))
1849 ((mexpt)
1850 ((mplus) 1 ((mtimes) -1 n ((mexpt) ((%sin) z) 2)))
1851 -1)
1852 ((%sin) ((mtimes) 2 z)))))
1853 ;derivative wrt second argument
1854 ((mtimes)
1855 ((mexpt)
1856 ((mplus) 1 ((mtimes) -1 m ((mexpt) ((%sin) z) 2)))
1857 ((rat) -1 2))
1858 ((mexpt)
1859 ((mplus) 1 ((mtimes) -1 n ((mexpt) ((%sin) z) 2))) -1))
1860 ;Derivative wrt third argument
1861 ((mtimes) ((rat) 1 2)
1862 ((mexpt) ((mplus) ((mtimes) -1 m) n) -1)
1863 ((mplus) ((%elliptic_pi) n z m)
1864 ((mtimes) ((mexpt) ((mplus) -1 m) -1)
1865 ((%elliptic_e) z m))
1866 ((mtimes) ((rat) -1 2) ((mexpt) ((mplus) -1 m) -1) m
1867 ((mexpt)
1868 ((mplus) 1 ((mtimes) -1 m ((mexpt) ((%sin) z) 2)))
1869 ((rat) -1 2))
1870 ((%sin) ((mtimes) 2 z))))))
1871 grad)
1872
1873 (in-package #-gcl #:bigfloat #+gcl "BIGFLOAT")
1874 ;; Translation of Jim FitzSimons' bigfloat implementation of elliptic
1875 ;; integrals from http://www.getnet.com/~cherry/elliptbf3.mac.
1876 ;;
1877 ;; The algorithms are based on B.C. Carlson's "Numerical Computation
1878 ;; of Real or Complex Elliptic Integrals". These are updated to the
1879 ;; algorithms in Journal of Computational and Applied Mathematics 118
1880 ;; (2000) 71-85 "Reduction Theorems for Elliptic Integrands with the
1881 ;; Square Root of two quadritic factors"
1882 ;;
1883
1884 ;; NOTE: Despite the names indicating these are for bigfloat numbers,
1885 ;; the algorithms and routines are generic and will work with floats
1886 ;; and bigfloats.
1887
1888 (defun bferrtol (&rest args)
1889 ;; Compute error tolerance as sqrt(2^(-fpprec)). Not sure this is
1890 ;; quite right, but it makes the routines more accurate as fpprec
1891 ;; increases.
1892 (sqrt (reduce #'min (mapcar #'(lambda (x)
1893 (if (rationalp (realpart x))
1894 maxima::flonum-epsilon
1895 (epsilon x)))
1896 args))))
1897
1898 ;; rc(x,y) = integrate(1/2*(t+x)^(-1/2)/(t+y), t, 0, inf)
1899 ;;
1900 ;; log(x) = (x-1)*rc(((1+x)/2)^2, x), x > 0
1901 ;; asin(x) = x * rc(1-x^2, 1), |x|<= 1
1902 ;; acos(x) = sqrt(1-x^2)*rc(x^2,1), 0 <= x <=1
1903 ;; atan(x) = x * rc(1,1+x^2)
1904 ;; asinh(x) = x * rc(1+x^2,1)
1905 ;; acosh(x) = sqrt(x^2-1) * rc(x^2,1), x >= 1
1906 ;; atanh(x) = x * rc(1,1-x^2), |x|<=1
1907
1908 (defun bf-rc (x y)
1909 (let ((yn y)
1910 xn z w a an pwr4 n epslon lambda sn s)
1911 (cond ((and (zerop (imagpart yn))
1912 (minusp (realpart yn)))
1913 (setf xn (- x y))
1914 (setf yn (- yn))
1915 (setf z yn)
1916 (setf w (sqrt (/ x xn))))
1917 (t
1918 (setf xn x)
1919 (setf z yn)
1920 (setf w 1)))
1921 (setf a (/ (+ xn yn yn) 3))
1922 (setf epslon (/ (abs (- a xn)) (bferrtol x y)))
1923 (setf an a)
1924 (setf pwr4 1)
1925 (setf n 0)
1926 (loop while (> (* epslon pwr4) (abs an))
1927 do
1928 (setf pwr4 (/ pwr4 4))
1929 (setf lambda (+ (* 2 (sqrt xn) (sqrt yn)) yn))
1930 (setf an (/ (+ an lambda) 4))
1931 (setf xn (/ (+ xn lambda) 4))
1932 (setf yn (/ (+ yn lambda) 4))
1933 (incf n))
1934 ;; c2=3/10,c3=1/7,c4=3/8,c5=9/22,c6=159/208,c7=9/8
1935 (setf sn (/ (* pwr4 (- z a)) an))
1936 (setf s (* sn sn (+ 3/10
1937 (* sn (+ 1/7
1938 (* sn (+ 3/8
1939 (* sn (+ 9/22
1940 (* sn (+ 159/208
1941 (* sn 9/8))))))))))))
1942 (/ (* w (+ 1 s))
1943 (sqrt an))))
1944
1945
1946
1947 ;; See https://dlmf.nist.gov/19.16.E5:
1948 ;;
1949 ;; rd(x,y,z) = integrate(3/2/sqrt(t+x)/sqrt(t+y)/sqrt(t+z)/(t+z), t, 0, inf)
1950 ;;
1951 ;; rd(1,1,1) = 1
1952 ;; E(K) = rf(0, 1-K^2, 1) - (K^2/3)*rd(0,1-K^2,1)
1953 ;;
1954 ;; B = integrate(s^2/sqrt(1-s^4), s, 0 ,1)
1955 ;; = beta(3/4,1/2)/4
1956 ;; = sqrt(%pi)*gamma(3/4)/gamma(1/4)
1957 ;; = 1/3*rd(0,2,1)
1958
1959 (defun bf-rd (x y z)
1960 (let* ((xn x)
1961 (yn y)
1962 (zn z)
1963 (a (/ (+ xn yn (* 3 zn)) 5))
1964 (epslon (/ (max (abs (- a xn))
1965 (abs (- a yn))
1966 (abs (- a zn)))
1967 (bferrtol x y z)))
1968 (an a)
1969 (sigma 0)
1970 (power4 1)
1971 (n 0)
1972 xnroot ynroot znroot lam)
1973 (loop while (> (* power4 epslon) (abs an))
1974 do
1975 (setf xnroot (sqrt xn))
1976 (setf ynroot (sqrt yn))
1977 (setf znroot (sqrt zn))
1978 (setf lam (+ (* xnroot ynroot)
1979 (* xnroot znroot)
1980 (* ynroot znroot)))
1981 (setf sigma (+ sigma (/ power4
1982 (* znroot (+ zn lam)))))
1983 (setf power4 (* power4 1/4))
1984 (setf xn (* (+ xn lam) 1/4))
1985 (setf yn (* (+ yn lam) 1/4))
1986 (setf zn (* (+ zn lam) 1/4))
1987 (setf an (* (+ an lam) 1/4))
1988 (incf n))
1989 ;; c1=-3/14,c2=1/6,c3=9/88,c4=9/22,c5=-3/22,c6=-9/52,c7=3/26
1990 (let* ((xndev (/ (* (- a x) power4) an))
1991 (yndev (/ (* (- a y) power4) an))
1992 (zndev (- (* (+ xndev yndev) 1/3)))
1993 (ee2 (- (* xndev yndev) (* 6 zndev zndev)))
1994 (ee3 (* (- (* 3 xndev yndev)
1995 (* 8 zndev zndev))
1996 zndev))
1997 (ee4 (* 3 (- (* xndev yndev) (* zndev zndev)) zndev zndev))
1998 (ee5 (* xndev yndev zndev zndev zndev))
1999 (s (+ 1
2000 (* -3/14 ee2)
2001 (* 1/6 ee3)
2002 (* 9/88 ee2 ee2)
2003 (* -3/22 ee4)
2004 (* -9/52 ee2 ee3)
2005 (* 3/26 ee5)
2006 (* -1/16 ee2 ee2 ee2)
2007 (* 3/10 ee3 ee3)
2008 (* 3/20 ee2 ee4)
2009 (* 45/272 ee2 ee2 ee3)
2010 (* -9/68 (+ (* ee2 ee5) (* ee3 ee4))))))
2011 (+ (* 3 sigma)
2012 (/ (* power4 s)
2013 (expt an 3/2))))))
2014
2015 ;; See https://dlmf.nist.gov/19.16.E1
2016 ;;
2017 ;; rf(x,y,z) = 1/2*integrate(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+z)), t, 0, inf);
2018 ;;
2019 ;; rf(1,1,1) = 1
2020 (defun bf-rf (x y z)
2021 (let* ((xn x)
2022 (yn y)
2023 (zn z)
2024 (a (/ (+ xn yn zn) 3))
2025 (epslon (/ (max (abs (- a xn))
2026 (abs (- a yn))
2027 (abs (- a zn)))
2028 (bferrtol x y z)))
2029 (an a)
2030 (power4 1)
2031 (n 0)
2032 xnroot ynroot znroot lam)
2033 (loop while (> (* power4 epslon) (abs an))
2034 do
2035 (setf xnroot (sqrt xn))
2036 (setf ynroot (sqrt yn))
2037 (setf znroot (sqrt zn))
2038 (setf lam (+ (* xnroot ynroot)
2039 (* xnroot znroot)
2040 (* ynroot znroot)))
2041 (setf power4 (* power4 1/4))
2042 (setf xn (* (+ xn lam) 1/4))
2043 (setf yn (* (+ yn lam) 1/4))
2044 (setf zn (* (+ zn lam) 1/4))
2045 (setf an (* (+ an lam) 1/4))
2046 (incf n))
2047 ;; c1=-3/14,c2=1/6,c3=9/88,c4=9/22,c5=-3/22,c6=-9/52,c7=3/26
2048 (let* ((xndev (/ (* (- a x) power4) an))
2049 (yndev (/ (* (- a y) power4) an))
2050 (zndev (- (+ xndev yndev)))
2051 (ee2 (- (* xndev yndev) (* 6 zndev zndev)))
2052 (ee3 (* xndev yndev zndev))
2053 (s (+ 1
2054 (* -1/10 ee2)
2055 (* 1/14 ee3)
2056 (* 1/24 ee2 ee2)
2057 (* -3/44 ee2 ee3))))
2058 (/ s (sqrt an)))))
2059
2060 (defun bf-rj1 (x y z p)
2061 (let* ((xn x)
2062 (yn y)
2063 (zn z)
2064 (pn p)
2065 (en (* (- pn xn)
2066 (- pn yn)
2067 (- pn zn)))
2068 (sigma 0)
2069 (power4 1)
2070 (k 0)
2071 (a (/ (+ xn yn zn pn pn) 5))
2072 (epslon (/ (max (abs (- a xn))
2073 (abs (- a yn))
2074 (abs (- a zn))
2075 (abs (- a pn)))
2076 (bferrtol x y z p)))
2077 (an a)
2078 xnroot ynroot znroot pnroot lam dn)
2079 (loop while (> (* power4 epslon) (abs an))
2080 do
2081 (setf xnroot (sqrt xn))
2082 (setf ynroot (sqrt yn))
2083 (setf znroot (sqrt zn))
2084 (setf pnroot (sqrt pn))
2085 (setf lam (+ (* xnroot ynroot)
2086 (* xnroot znroot)
2087 (* ynroot znroot)))
2088 (setf dn (* (+ pnroot xnroot)
2089 (+ pnroot ynroot)
2090 (+ pnroot znroot)))
2091 (setf sigma (+ sigma
2092 (/ (* power4
2093 (bf-rc 1 (+ 1 (/ en (* dn dn)))))
2094 dn)))
2095 (setf power4 (* power4 1/4))
2096 (setf en (/ en 64))
2097 (setf xn (* (+ xn lam) 1/4))
2098 (setf yn (* (+ yn lam) 1/4))
2099 (setf zn (* (+ zn lam) 1/4))
2100 (setf pn (* (+ pn lam) 1/4))
2101 (setf an (* (+ an lam) 1/4))
2102 (incf k))
2103 (let* ((xndev (/ (* (- a x) power4) an))
2104 (yndev (/ (* (- a y) power4) an))
2105 (zndev (/ (* (- a z) power4) an))
2106 (pndev (* -0.5 (+ xndev yndev zndev)))
2107 (ee2 (+ (* xndev yndev)
2108 (* xndev zndev)
2109 (* yndev zndev)
2110 (* -3 pndev pndev)))
2111 (ee3 (+ (* xndev yndev zndev)
2112 (* 2 ee2 pndev)
2113 (* 4 pndev pndev pndev)))
2114 (ee4 (* (+ (* 2 xndev yndev zndev)
2115 (* ee2 pndev)
2116 (* 3 pndev pndev pndev))
2117 pndev))
2118 (ee5 (* xndev yndev zndev pndev pndev))
2119 (s (+ 1
2120 (* -3/14 ee2)
2121 (* 1/6 ee3)
2122 (* 9/88 ee2 ee2)
2123 (* -3/22 ee4)
2124 (* -9/52 ee2 ee3)
2125 (* 3/26 ee5)
2126 (* -1/16 ee2 ee2 ee2)
2127 (* 3/10 ee3 ee3)
2128 (* 3/20 ee2 ee4)
2129 (* 45/272 ee2 ee2 ee3)
2130 (* -9/68 (+ (* ee2 ee5) (* ee3 ee4))))))
2131 (+ (* 6 sigma)
2132 (/ (* power4 s)
2133 (sqrt (* an an an)))))))
2134
2135 (defun bf-rj (x y z p)
2136 (let* ((xn x)
2137 (yn y)
2138 (zn z)
2139 (qn (- p)))
2140 (cond ((and (and (zerop (imagpart xn)) (>= (realpart xn) 0))
2141 (and (zerop (imagpart yn)) (>= (realpart yn) 0))
2142 (and (zerop (imagpart zn)) (>= (realpart zn) 0))
2143 (and (zerop (imagpart qn)) (> (realpart qn) 0)))
2144 (destructuring-bind (xn yn zn)
2145 (sort (list xn yn zn) #'<)
2146 (let* ((pn (+ yn (* (- zn yn) (/ (- yn xn) (+ yn qn)))))
2147 (s (- (* (- pn yn) (bf-rj1 xn yn zn pn))
2148 (* 3 (bf-rf xn yn zn)))))
2149 (setf s (+ s (* 3 (sqrt (/ (* xn yn zn)
2150 (+ (* xn zn) (* pn qn))))
2151 (bf-rc (+ (* xn zn) (* pn qn)) (* pn qn)))))
2152 (/ s (+ yn qn)))))
2153 (t
2154 (bf-rj1 x y z p)))))
2155
2156 (defun bf-rg (x y z)
2157 (* 0.5
2158 (+ (* z (bf-rf x y z))
2159 (* (- z x)
2160 (- y z)
2161 (bf-rd x y z)
2162 1/3)
2163 (sqrt (/ (* x y) z)))))
2164
2165 ;; elliptic_f(phi,m) = sin(phi)*rf(cos(phi)^2, 1-m*sin(phi)^2,1)
2166 (defun bf-elliptic-f (phi m)
2167 (flet ((base (phi m)
2168 (cond ((= m 1)
2169 ;; F(z|1) = log(tan(z/2+%pi/4))
2170 (log (tan (+ (/ phi 2) (/ (%pi phi) 4)))))
2171 (t
2172 (let ((s (sin phi))
2173 (c (cos phi)))
2174 (* s (bf-rf (* c c) (- 1 (* m s s)) 1)))))))
2175 ;; Handle periodicity (see elliptic-f)
2176 (let* ((bfpi (%pi phi))
2177 (period (round (realpart phi) bfpi)))
2178 (+ (base (- phi (* bfpi period)) m)
2179 (if (zerop period)
2180 0
2181 (* 2 period (bf-elliptic-k m)))))))
2182
2183 ;; elliptic_kc(k) = rf(0, 1-k^2,1)
2184 ;;
2185 ;; or
2186 ;; elliptic_kc(m) = rf(0, 1-m,1)
2187
2188 (defun bf-elliptic-k (m)
2189 (cond ((= m 0)
2190 (if (maxima::$bfloatp m)
2191 (maxima::$bfloat (maxima::div 'maxima::$%pi 2))
2192 (float (/ pi 2) 1e0)))
2193 ((= m 1)
2194 (maxima::merror
2195 (intl:gettext "elliptic_kc: elliptic_kc(1) is undefined.")))
2196 (t
2197 (bf-rf 0 (- 1 m) 1))))
2198
2199 ;; elliptic_e(phi, k) = sin(phi)*rf(cos(phi)^2,1-k^2*sin(phi)^2,1)
2200 ;; - (k^2/3)*sin(phi)^3*rd(cos(phi)^2, 1-k^2*sin(phi)^2,1)
2201 ;;
2202 ;;
2203 ;; or
2204 ;; elliptic_e(phi, m) = sin(phi)*rf(cos(phi)^2,1-m*sin(phi)^2,1)
2205 ;; - (m/3)*sin(phi)^3*rd(cos(phi)^2, 1-m*sin(phi)^2,1)
2206 ;;
2207 (defun bf-elliptic-e (phi m)
2208 (flet ((base (phi m)
2209 (let* ((s (sin phi))
2210 (c (cos phi))
2211 (c2 (* c c))
2212 (s2 (- 1 (* m s s))))
2213 (- (* s (bf-rf c2 s2 1))
2214 (* (/ m 3) (* s s s) (bf-rd c2 s2 1))))))
2215 ;; Elliptic E is quasi-periodic wrt to phi:
2216 ;;
2217 ;; E(z|m) = E(z - %pi*round(Re(z)/%pi)|m) + 2*round(Re(z)/%pi)*E(m)
2218 (let* ((bfpi (%pi phi))
2219 (period (round (realpart phi) bfpi)))
2220 (+ (base (- phi (* bfpi period)) m)
2221 (* 2 period (bf-elliptic-ec m))))))
2222
2223
2224 ;; elliptic_ec(k) = rf(0,1-k^2,1) - (k^2/3)*rd(0,1-k^2,1);
2225 ;;
2226 ;; or
2227 ;; elliptic_ec(m) = rf(0,1-m,1) - (m/3)*rd(0,1-m,1);
2228
2229 (defun bf-elliptic-ec (m)
2230 (cond ((= m 0)
2231 (if (typep m 'bigfloat)
2232 (bigfloat (maxima::$bfloat (maxima::div 'maxima::$%pi 2)))
2233 (float (/ pi 2) 1e0)))
2234 ((= m 1)
2235 (if (typep m 'bigfloat)
2236 (bigfloat 1)
2237 1e0))
2238 (t
2239 (let ((m1 (- 1 m)))
2240 (- (bf-rf 0 m1 1)
2241 (* m 1/3 (bf-rd 0 m1 1)))))))
2242
2243 (defun bf-elliptic-pi-complete (n m)
2244 (+ (bf-rf 0 (- 1 m) 1)
2245 (* 1/3 n (bf-rj 0 (- 1 m) 1 (- 1 n)))))
2246
2247 (defun bf-elliptic-pi (n phi m)
2248 ;; Note: Carlson's DRJ has n defined as the negative of the n given
2249 ;; in A&S.
2250 (flet ((base (n phi m)
2251 (let* ((nn (- n))
2252 (sin-phi (sin phi))
2253 (cos-phi (cos phi))
2254 (k (sqrt m))
2255 (k2sin (* (- 1 (* k sin-phi))
2256 (+ 1 (* k sin-phi)))))
2257 (- (* sin-phi (bf-rf (expt cos-phi 2) k2sin 1.0))
2258 (* (/ nn 3) (expt sin-phi 3)
2259 (bf-rj (expt cos-phi 2) k2sin 1.0
2260 (- 1 (* n (expt sin-phi 2)))))))))
2261 ;; FIXME: Reducing the arg by pi has significant round-off.
2262 ;; Consider doing something better.
2263 (let* ((bf-pi (%pi (realpart phi)))
2264 (cycles (round (realpart phi) bf-pi))
2265 (rem (- phi (* cycles bf-pi))))
2266 (let ((complete (bf-elliptic-pi-complete n m)))
2267 (+ (* 2 cycles complete)
2268 (base n rem m))))))
2269
2270 ;; Compute inverse_jacobi_sn, for float or bigfloat args.
2271 (defun bf-inverse-jacobi-sn (u m)
2272 (* u (bf-rf (- 1 (* u u))
2273 (- 1 (* m u u))
2274 1)))
2275
2276 ;; Compute inverse_jacobi_dn. We use the following identity
2277 ;; from Gradshteyn & Ryzhik, 8.153.6
2278 ;;
2279 ;; w = dn(z|m) = cn(sqrt(m)*z, 1/m)
2280 ;;
2281 ;; Solve for z to get
2282 ;;
2283 ;; z = inverse_jacobi_dn(w,m)
2284 ;; = 1/sqrt(m) * inverse_jacobi_cn(w, 1/m)
2285 (defun bf-inverse-jacobi-dn (w m)
2286 (cond ((= w 1)
2287 (float 0 w))
2288 ((= m 1)
2289 ;; jacobi_dn(x,1) = sech(x) so the inverse is asech(x)
2290 (maxima::take '(maxima::%asech) (maxima::to w)))
2291 (t
2292 ;; We should do something better to make sure that things
2293 ;; that should be real are real.
2294 (/ (to (maxima::take '(maxima::%inverse_jacobi_cn)
2295 (maxima::to w)
2296 (maxima::to (/ m))))
2297 (sqrt m)))))
2298
2299 (in-package :maxima)
2300
2301 ;; Define Carlson's elliptic integrals.
2302
2303 (def-simplifier carlson_rc (x y)
2304 (let (args)
2305 (flet ((calc (x y)
2306 (flet ((floatify (z)
2307 ;; If z is a complex rational, convert to a
2308 ;; complex double-float. Otherwise, leave it as
2309 ;; is. If we don't do this, %i is handled as
2310 ;; #c(0 1), which makes bf-rc use single-float
2311 ;; arithmetic instead of the desired
2312 ;; double-float.
2313 (if (and (complexp z) (rationalp (realpart z)))
2314 (complex (float (realpart z))
2315 (float (imagpart z)))
2316 z)))
2317 (to (bigfloat::bf-rc (floatify (bigfloat:to x))
2318 (floatify (bigfloat:to y)))))))
2319 ;; See comments from bf-rc
2320 (cond ((float-numerical-eval-p x y)
2321 (calc ($float x) ($float y)))
2322 ((bigfloat-numerical-eval-p x y)
2323 (calc ($bfloat x) ($bfloat y)))
2324 ((setf args (complex-float-numerical-eval-p x y))
2325 (destructuring-bind (x y)
2326 args
2327 (calc ($float x) ($float y))))
2328 ((setf args (complex-bigfloat-numerical-eval-p x y))
2329 (destructuring-bind (x y)
2330 args
2331 (calc ($bfloat x) ($bfloat y))))
2332 ((and (zerop1 x)
2333 (onep1 y))
2334 ;; rc(0, 1) = %pi/2
2335 (div '$%pi 2))
2336 ((and (zerop1 x)
2337 (alike1 y (div 1 4)))
2338 ;; rc(0,1/4) = %pi
2339 '$%pi)
2340 ((and (eql x 2)
2341 (onep1 y))
2342 ;; rc(2,1) = 1/2*integrate(1/sqrt(t+2)/(t+1), t, 0, inf)
2343 ;; = (log(sqrt(2)+1)-log(sqrt(2)-1))/2
2344 ;; ratsimp(logcontract(%)),algebraic:
2345 ;; = -log(3-2^(3/2))/2
2346 ;; = -log(sqrt(3-2^(3/2)))
2347 ;; = -log(sqrt(2)-1)
2348 ;; = log(1/(sqrt(2)-1))
2349 ;; ratsimp(%),algebraic;
2350 ;; = log(sqrt(2)+1)
2351 (ftake '%log (add 1 (power 2 1//2))))
2352 ((and (alike x '$%i)
2353 (alike y (add 1 '$%i)))
2354 ;; rc(%i, %i+1) = 1/2*integrate(1/sqrt(t+%i)/(t+%i+1), t, 0, inf)
2355 ;; = %pi/2-atan((-1)^(1/4))
2356 ;; ratsimp(logcontract(ratsimp(rectform(%o42)))),algebraic;
2357 ;; = (%i*log(3-2^(3/2))+%pi)/4
2358 ;; = (%i*log(3-2^(3/2)))/4+%pi/4
2359 ;; = %i*log(sqrt(3-2^(3/2)))/2+%pi/4
2360 ;; sqrtdenest(%);
2361 ;; = %pi/4 + %i*log(sqrt(2)-1)/2
2362 (add (div '$%pi 4)
2363 (mul '$%i
2364 1//2
2365 (ftake '%log (sub (power 2 1//2) 1)))))
2366 ((and (zerop1 x)
2367 (alike1 y '$%i))
2368 ;; rc(0,%i) = 1/2*integrate(1/(sqrt(t)*(t+%i)), t, 0, inf)
2369 ;; = -((sqrt(2)*%i-sqrt(2))*%pi)/4
2370 ;; = ((1-%i)*%pi)/2^(3/2)
2371 (div (mul (sub 1 '$%i)
2372 '$%pi)
2373 (power 2 3//2)))
2374 ((and (alike1 x y)
2375 (eq ($sign ($realpart x)) '$pos))
2376 ;; carlson_rc(x,x) = 1/2*integrate(1/sqrt(t+x)/(t+x), t, 0, inf)
2377 ;; = 1/sqrt(x)
2378 (power x -1//2))
2379 ((and (alike1 x (power (div (add 1 y) 2) 2))
2380 (eq ($sign ($realpart y)) '$pos))
2381 ;; Rc(((1+x)/2)^2,x) = log(x)/(x-1) for x > 0.
2382 ;;
2383 ;; This is done by looking at Rc(x,y) and seeing if
2384 ;; ((1+y)/2)^2 is the same as x.
2385 (div (ftake '%log y)
2386 (sub y 1)))
2387 (t
2388 (give-up))))))
2389
2390 (def-simplifier carlson_rd (x y z)
2391 (let (args)
2392 (flet ((calc (x y z)
2393 (to (bigfloat::bf-rd (bigfloat:to x)
2394 (bigfloat:to y)
2395 (bigfloat:to z)))))
2396 ;; See https://dlmf.nist.gov/19.20.E18
2397 (cond ((and (eql x 1)
2398 (eql x y)
2399 (eql y z))
2400 ;; Rd(1,1,1) = 1
2401 1)
2402 ((and (alike1 x y)
2403 (alike1 y z))
2404 ;; Rd(x,x,x) = x^(-3/2)
2405 (power x (div -3 2)))
2406 ((and (zerop1 x)
2407 (alike1 y z))
2408 ;; Rd(0,y,y) = 3/4*%pi*y^(-3/2)
2409 (mul (div 3 4)
2410 '$%pi
2411 (power y (div -3 2))))
2412 ((alike1 y z)
2413 ;; Rd(x,y,y) = 3/(2*(y-x))*(Rc(x, y) - sqrt(x)/y)
2414 (mul (div 3 (mul 2 (sub y x)))
2415 (sub (ftake '%carlson_rc x y)
2416 (div (power x 1//2)
2417 y))))
2418 ((alike1 x y)
2419 ;; Rd(x,x,z) = 3/(z-x)*(Rc(z,x) - 1/sqrt(z))
2420 (mul (div 3 (sub z x))
2421 (sub (ftake '%carlson_rc z x)
2422 (div 1 (power z 1//2)))))
2423 ((and (eql z 1)
2424 (or (and (eql x 0)
2425 (eql y 2))
2426 (and (eql x 2)
2427 (eql y 0))))
2428 ;; Rd(0,2,1) = 3*(gamma(3/4)^2)/sqrt(2*%pi)
2429 ;; See https://dlmf.nist.gov/19.20.E22.
2430 ;;
2431 ;; But that's the same as
2432 ;; 3*sqrt(%pi)*gamma(3/4)/gamma(1/4). We can see this by
2433 ;; taking the ratio to get
2434 ;; gamma(1/4)*gamma(3/4)/sqrt(2)*%pi. But
2435 ;; gamma(1/4)*gamma(3/4) = beta(1/4,3/4) = sqrt(2)*%pi.
2436 ;; Hence, the ratio is 1.
2437 ;;
2438 ;; Note also that Rd(x,y,z) = Rd(y,x,z)
2439 (mul 3
2440 (power '$%pi 1//2)
2441 (div (ftake '%gamma (div 3 4))
2442 (ftake '%gamma (div 1 4)))))
2443 ((and (or (eql x 0) (eql y 0))
2444 (eql z 1))
2445 ;; 1/3*m*Rd(0,1-m,1) = K(m) - E(m).
2446 ;; See https://dlmf.nist.gov/19.25.E1
2447 ;;
2448 ;; Thus, Rd(0,y,1) = 3/(1-y)*(K(1-y) - E(1-y))
2449 ;;
2450 ;; Note that Rd(x,y,z) = Rd(y,x,z).
2451 (let ((m (sub 1 y)))
2452 (mul (div 3 m)
2453 (sub (ftake '%elliptic_kc m)
2454 (ftake '%elliptic_ec m)))))
2455 ((or (and (eql x 0)
2456 (eql y 1))
2457 (and (eql x 1)
2458 (eql y 0)))
2459 ;; 1/3*m*(1-m)*Rd(0,1,1-m) = E(m) - (1-m)*K(m)
2460 ;; See https://dlmf.nist.gov/19.25.E1
2461 ;;
2462 ;; Thus
2463 ;; Rd(0,1,z) = 3/(z*(1-z))*(E(1-z) - z*K(1-z))
2464 ;; Recall that Rd(x,y,z) = Rd(y,x,z).
2465 (mul (div 3 (mul z (sub 1 z)))
2466 (sub (ftake '%elliptic_ec (sub 1 z))
2467 (mul z
2468 (ftake '%elliptic_kc (sub 1 z))))))
2469 ((float-numerical-eval-p x y z)
2470 (calc ($float x) ($float y) ($float z)))
2471 ((bigfloat-numerical-eval-p x y z)
2472 (calc ($bfloat x) ($bfloat y) ($bfloat z)))
2473 ((setf args (complex-float-numerical-eval-p x y z))
2474 (destructuring-bind (x y z)
2475 args
2476 (calc ($float x) ($float y) ($float z))))
2477 ((setf args (complex-bigfloat-numerical-eval-p x y z))
2478 (destructuring-bind (x y z)
2479 args
2480 (calc ($bfloat x) ($bfloat y) ($bfloat z))))
2481 (t
2482 (give-up))))))
2483
2484 (def-simplifier carlson_rf (x y z)
2485 (let (args)
2486 (flet ((calc (x y z)
2487 (to (bigfloat::bf-rf (bigfloat:to x)
2488 (bigfloat:to y)
2489 (bigfloat:to z)))))
2490 ;; See https://dlmf.nist.gov/19.20.i
2491 (cond ((and (alike1 x y)
2492 (alike1 y z))
2493 ;; Rf(x,x,x) = x^(-1/2)
2494 (power x -1//2))
2495 ((and (zerop1 x)
2496 (alike1 y z))
2497 ;; Rf(0,y,y) = 1/2*%pi*y^(-1/2)
2498 (mul 1//2 '$%pi
2499 (power y -1//2)))
2500 ((alike1 y z)
2501 (ftake '%carlson_rc x y))
2502 ((some #'(lambda (args)
2503 (destructuring-bind (x y z)
2504 args
2505 (and (zerop1 x)
2506 (eql y 1)
2507 (eql z 2))))
2508 (list (list x y z)
2509 (list x z y)
2510 (list y x z)
2511 (list y z x)
2512 (list z x y)
2513 (list z y x)))
2514 ;; Rf(0,1,2) = (gamma(1/4))^2/(4*sqrt(2*%pi))
2515 ;;
2516 ;; And Rf is symmetric in all the args, so check every
2517 ;; permutation too. This could probably be simplified
2518 ;; without consing all the lists, but I'm lazy.
2519 (div (power (ftake '%gamma (div 1 4)) 2)
2520 (mul 4 (power (mul 2 '$%pi) 1//2))))
2521 ((some #'(lambda (args)
2522 (destructuring-bind (x y z)
2523 args
2524 (and (alike1 x '$%i)
2525 (alike1 y (mul -1 '$%i))
2526 (eql z 0))))
2527 (list (list x y z)
2528 (list x z y)
2529 (list y x z)
2530 (list y z x)
2531 (list z x y)
2532 (list z y x)))
2533 ;; rf(%i, -%i, 0)
2534 ;; = 1/2*integrate(1/sqrt(t^2+1)/sqrt(t),t,0,inf)
2535 ;; = beta(1/4,1/4)/4;
2536 ;; makegamma(%)
2537 ;; = gamma(1/4)^2/(4*sqrt(%pi))
2538 ;;
2539 ;; Rf is symmetric, so check all the permutations too.
2540 (div (power (ftake '%gamma (div 1 4)) 2)
2541 (mul 4 (power '$%pi 1//2))))
2542 ((setf args
2543 (some #'(lambda (args)
2544 (destructuring-bind (x y z)
2545 args
2546 ;; Check that x = 0 and z = 1, and
2547 ;; return y.
2548 (and (zerop1 x)
2549 (eql z 1)
2550 y)))
2551 (list (list x y z)
2552 (list x z y)
2553 (list y x z)
2554 (list y z x)
2555 (list z x y)
2556 (list z y x))))
2557 ;; Rf(0,1-m,1) = elliptic_kc(m).
2558 ;; See https://dlmf.nist.gov/19.25.E1
2559 (ftake '%elliptic_kc (sub 1 args)))
2560 ((some #'(lambda (args)
2561 (destructuring-bind (x y z)
2562 args
2563 (and (alike1 x '$%i)
2564 (alike1 y (mul -1 '$%i))
2565 (eql z 0))))
2566 (list (list x y z)
2567 (list x z y)
2568 (list y x z)
2569 (list y z x)
2570 (list z x y)
2571 (list z y x)))
2572 ;; rf(%i, -%i, 0)
2573 ;; = 1/2*integrate(1/sqrt(t^2+1)/sqrt(t),t,0,inf)
2574 ;; = beta(1/4,1/4)/4;
2575 ;; makegamma(%)
2576 ;; = gamma(1/4)^2/(4*sqrt(%pi))
2577 ;;
2578 ;; Rf is symmetric, so check all the permutations too.
2579 (div (power (ftake '%gamma (div 1 4)) 2)
2580 (mul 4 (power '$%pi 1//2))))
2581 ((float-numerical-eval-p x y z)
2582 (calc ($float x) ($float y) ($float z)))
2583 ((bigfloat-numerical-eval-p x y z)
2584 (calc ($bfloat x) ($bfloat y) ($bfloat z)))
2585 ((setf args (complex-float-numerical-eval-p x y z))
2586 (destructuring-bind (x y z)
2587 args
2588 (calc ($float x) ($float y) ($float z))))
2589 ((setf args (complex-bigfloat-numerical-eval-p x y z))
2590 (destructuring-bind (x y z)
2591 args
2592 (calc ($bfloat x) ($bfloat y) ($bfloat z))))
2593 (t
2594 (give-up))))))
2595
2596 (def-simplifier carlson_rj (x y z p)
2597 (let (args)
2598 (flet ((calc (x y z p)
2599 (to (bigfloat::bf-rj (bigfloat:to x)
2600 (bigfloat:to y)
2601 (bigfloat:to z)
2602 (bigfloat:to p)))))
2603 ;; See https://dlmf.nist.gov/19.20.iii
2604 (cond ((and (alike1 x y)
2605 (alike1 y z)
2606 (alike1 z p))
2607 ;; Rj(x,x,x,x) = x^(-3/2)
2608 (power x (div -3 2)))
2609 ((alike1 z p)
2610 ;; Rj(x,y,z,z) = Rd(x,y,z)
2611 (ftake '%carlson_rd x y z))
2612 ((and (zerop1 x)
2613 (alike1 y z))
2614 ;; Rj(0,y,y,p) = 3*%pi/(2*(y*sqrt(p)+p*sqrt(y)))
2615 (div (mul 3 '$%pi)
2616 (mul 2
2617 (add (mul y (power p 1//2))
2618 (mul p (power y 1//2))))))
2619 ((alike1 y z)
2620 ;; Rj(x,y,y,p) = 3/(p-y)*(Rc(x,y) - Rc(x,p))
2621 (mul (div 3 (sub p y))
2622 (sub (ftake '%carlson_rc x y)
2623 (ftake '%carlson_rc x p))))
2624 ((and (alike1 y z)
2625 (alike1 y p))
2626 ;; Rj(x,y,y,y) = Rd(x,y,y)
2627 (ftake '%carlson_rd x y y))
2628 ((float-numerical-eval-p x y z p)
2629 (calc ($float x) ($float y) ($float z) ($float p)))
2630 ((bigfloat-numerical-eval-p x y z p)
2631 (calc ($bfloat x) ($bfloat y) ($bfloat z) ($bfloat p)))
2632 ((setf args (complex-float-numerical-eval-p x y z p))
2633 (destructuring-bind (x y z p)
2634 args
2635 (calc ($float x) ($float y) ($float z) ($float p))))
2636 ((setf args (complex-bigfloat-numerical-eval-p x y z p))
2637 (destructuring-bind (x y z p)
2638 args
2639 (calc ($bfloat x) ($bfloat y) ($bfloat z) ($bfloat p))))
2640 (t
2641 (give-up))))))
2642
2643 ;;; Other Jacobian elliptic functions
2644
2645 ;; jacobi_ns(u,m) = 1/jacobi_sn(u,m)
2646 (defprop %jacobi_ns
2647 ((u m)
2648 ;; diff wrt u
2649 ((mtimes) -1 ((%jacobi_cn) u m) ((%jacobi_dn) u m)
2650 ((mexpt) ((%jacobi_sn) u m) -2))
2651 ;; diff wrt m
2652 ((mtimes) -1 ((mexpt) ((%jacobi_sn) u m) -2)
2653 ((mplus)
2654 ((mtimes) ((rat) 1 2)
2655 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
2656 ((mexpt) ((%jacobi_cn) u m) 2)
2657 ((%jacobi_sn) u m))
2658 ((mtimes) ((rat) 1 2) ((mexpt) m -1)
2659 ((%jacobi_cn) u m) ((%jacobi_dn) u m)
2660 ((mplus) u
2661 ((mtimes) -1
2662 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
2663 ((%elliptic_e) ((%asin) ((%jacobi_sn) u m))
2664 m)))))))
2665 grad)
2666
2667 (def-simplifier jacobi_ns (u m)
2668 (let (coef args)
2669 (cond
2670 ((float-numerical-eval-p u m)
2671 (to (bigfloat:/ (bigfloat::sn (bigfloat:to ($float u))
2672 (bigfloat:to ($float m))))))
2673 ((setf args (complex-float-numerical-eval-p u m))
2674 (destructuring-bind (u m)
2675 args
2676 (to (bigfloat:/ (bigfloat::sn (bigfloat:to ($float u))
2677 (bigfloat:to ($float m)))))))
2678 ((bigfloat-numerical-eval-p u m)
2679 (let ((uu (bigfloat:to ($bfloat u)))
2680 (mm (bigfloat:to ($bfloat m))))
2681 (to (bigfloat:/ (bigfloat::sn uu mm)))))
2682 ((setf args (complex-bigfloat-numerical-eval-p u m))
2683 (destructuring-bind (u m)
2684 args
2685 (let ((uu (bigfloat:to ($bfloat u)))
2686 (mm (bigfloat:to ($bfloat m))))
2687 (to (bigfloat:/ (bigfloat::sn uu mm))))))
2688 ((zerop1 m)
2689 ;; A&S 16.6.10
2690 (ftake '%csc u))
2691 ((onep1 m)
2692 ;; A&S 16.6.10
2693 (ftake '%coth u))
2694 ((zerop1 u)
2695 (dbz-err1 'jacobi_ns))
2696 ((and $trigsign (mminusp* u))
2697 ;; ns is odd
2698 (neg (ftake* '%jacobi_ns (neg u) m)))
2699 ((and $triginverses
2700 (listp u)
2701 (member (caar u) '(%inverse_jacobi_sn
2702 %inverse_jacobi_ns
2703 %inverse_jacobi_cn
2704 %inverse_jacobi_nc
2705 %inverse_jacobi_dn
2706 %inverse_jacobi_nd
2707 %inverse_jacobi_sc
2708 %inverse_jacobi_cs
2709 %inverse_jacobi_sd
2710 %inverse_jacobi_ds
2711 %inverse_jacobi_cd
2712 %inverse_jacobi_dc))
2713 (alike1 (third u) m))
2714 (cond ((eq (caar u) '%inverse_jacobi_ns)
2715 (second u))
2716 (t
2717 ;; Express in terms of sn:
2718 ;; ns(x) = 1/sn(x)
2719 (div 1 (ftake '%jacobi_sn u m)))))
2720 ;; A&S 16.20 (Jacobi's Imaginary transformation)
2721 ((and $%iargs (multiplep u '$%i))
2722 ;; ns(i*u) = 1/sn(i*u) = -i/sc(u,m1) = -i*cs(u,m1)
2723 (neg (mul '$%i
2724 (ftake* '%jacobi_cs (coeff u '$%i 1) (add 1 (neg m))))))
2725 ((setq coef (kc-arg2 u m))
2726 ;; A&S 16.8.10
2727 ;;
2728 ;; ns(m*K+u) = 1/sn(m*K+u)
2729 ;;
2730 (destructuring-bind (lin const)
2731 coef
2732 (cond ((integerp lin)
2733 (ecase (mod lin 4)
2734 (0
2735 ;; ns(4*m*K+u) = ns(u)
2736 ;; ns(0) = infinity
2737 (if (zerop1 const)
2738 (dbz-err1 'jacobi_ns)
2739 (ftake '%jacobi_ns const m)))
2740 (1
2741 ;; ns(4*m*K + K + u) = ns(K+u) = dc(u)
2742 ;; ns(K) = 1
2743 (if (zerop1 const)
2744 1
2745 (ftake '%jacobi_dc const m)))
2746 (2
2747 ;; ns(4*m*K+2*K + u) = ns(2*K+u) = -ns(u)
2748 ;; ns(2*K) = infinity
2749 (if (zerop1 const)
2750 (dbz-err1 'jacobi_ns)
2751 (neg (ftake '%jacobi_ns const m))))
2752 (3
2753 ;; ns(4*m*K+3*K+u) = ns(2*K + K + u) = -ns(K+u) = -dc(u)
2754 ;; ns(3*K) = -1
2755 (if (zerop1 const)
2756 -1
2757 (neg (ftake '%jacobi_dc const m))))))
2758 ((and (alike1 lin 1//2)
2759 (zerop1 const))
2760 (div 1 (ftake '%jacobi_sn u m)))
2761 (t
2762 (give-up)))))
2763 (t
2764 ;; Nothing to do
2765 (give-up)))))
2766
2767 ;; jacobi_nc(u,m) = 1/jacobi_cn(u,m)
2768 (defprop %jacobi_nc
2769 ((u m)
2770 ;; wrt u
2771 ((mtimes) ((mexpt) ((%jacobi_cn) u m) -2)
2772 ((%jacobi_dn) u m) ((%jacobi_sn) u m))
2773 ;; wrt m
2774 ((mtimes) -1 ((mexpt) ((%jacobi_cn) u m) -2)
2775 ((mplus)
2776 ((mtimes) ((rat) -1 2)
2777 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
2778 ((%jacobi_cn) u m) ((mexpt) ((%jacobi_sn) u m) 2))
2779 ((mtimes) ((rat) -1 2) ((mexpt) m -1)
2780 ((%jacobi_dn) u m) ((%jacobi_sn) u m)
2781 ((mplus) u
2782 ((mtimes) -1 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
2783 ((%elliptic_e) ((%asin) ((%jacobi_sn) u m)) m)))))))
2784 grad)
2785
2786 (def-simplifier jacobi_nc (u m)
2787 (let (coef args)
2788 (cond
2789 ((float-numerical-eval-p u m)
2790 (to (bigfloat:/ (bigfloat::cn (bigfloat:to ($float u))
2791 (bigfloat:to ($float m))))))
2792 ((setf args (complex-float-numerical-eval-p u m))
2793 (destructuring-bind (u m)
2794 args
2795 (to (bigfloat:/ (bigfloat::cn (bigfloat:to ($float u))
2796 (bigfloat:to ($float m)))))))
2797 ((bigfloat-numerical-eval-p u m)
2798 (let ((uu (bigfloat:to ($bfloat u)))
2799 (mm (bigfloat:to ($bfloat m))))
2800 (to (bigfloat:/ (bigfloat::cn uu mm)))))
2801 ((setf args (complex-bigfloat-numerical-eval-p u m))
2802 (destructuring-bind (u m)
2803 args
2804 (let ((uu (bigfloat:to ($bfloat u)))
2805 (mm (bigfloat:to ($bfloat m))))
2806 (to (bigfloat:/ (bigfloat::cn uu mm))))))
2807 ((zerop1 u)
2808 1)
2809 ((zerop1 m)
2810 ;; A&S 16.6.8
2811 (ftake '%sec u))
2812 ((onep1 m)
2813 ;; A&S 16.6.8
2814 (ftake '%cosh u))
2815 ((and $trigsign (mminusp* u))
2816 ;; nc is even
2817 (ftake* '%jacobi_nc (neg u) m))
2818 ((and $triginverses
2819 (listp u)
2820 (member (caar u) '(%inverse_jacobi_sn
2821 %inverse_jacobi_ns
2822 %inverse_jacobi_cn
2823 %inverse_jacobi_nc
2824 %inverse_jacobi_dn
2825 %inverse_jacobi_nd
2826 %inverse_jacobi_sc
2827 %inverse_jacobi_cs
2828 %inverse_jacobi_sd
2829 %inverse_jacobi_ds
2830 %inverse_jacobi_cd
2831 %inverse_jacobi_dc))
2832 (alike1 (third u) m))
2833 (cond ((eq (caar u) '%inverse_jacobi_nc)
2834 (second u))
2835 (t
2836 ;; Express in terms of cn:
2837 ;; nc(x) = 1/cn(x)
2838 (div 1 (ftake '%jacobi_cn u m)))))
2839 ;; A&S 16.20 (Jacobi's Imaginary transformation)
2840 ((and $%iargs (multiplep u '$%i))
2841 ;; nc(i*u) = 1/cn(i*u) = 1/nc(u,1-m) = cn(u,1-m)
2842 (ftake* '%jacobi_cn (coeff u '$%i 1) (add 1 (neg m))))
2843 ((setq coef (kc-arg2 u m))
2844 ;; A&S 16.8.8
2845 ;;
2846 ;; nc(u) = 1/cn(u)
2847 ;;
2848 (destructuring-bind (lin const)
2849 coef
2850 (cond ((integerp lin)
2851 (ecase (mod lin 4)
2852 (0
2853 ;; nc(4*m*K+u) = nc(u)
2854 ;; nc(0) = 1
2855 (if (zerop1 const)
2856 1
2857 (ftake '%jacobi_nc const m)))
2858 (1
2859 ;; nc(4*m*K+K+u) = nc(K+u) = -ds(u)/sqrt(1-m)
2860 ;; nc(K) = infinity
2861 (if (zerop1 const)
2862 (dbz-err1 'jacobi_nc)
2863 (neg (div (ftake '%jacobi_ds const m)
2864 (power (sub 1 m) 1//2)))))
2865 (2
2866 ;; nc(4*m*K+2*K+u) = nc(2*K+u) = -nc(u)
2867 ;; nc(2*K) = -1
2868 (if (zerop1 const)
2869 -1
2870 (neg (ftake '%jacobi_nc const m))))
2871 (3
2872 ;; nc(4*m*K+3*K+u) = nc(3*K+u) = nc(2*K+K+u) =
2873 ;; -nc(K+u) = ds(u)/sqrt(1-m)
2874 ;;
2875 ;; nc(3*K) = infinity
2876 (if (zerop1 const)
2877 (dbz-err1 'jacobi_nc)
2878 (div (ftake '%jacobi_ds const m)
2879 (power (sub 1 m) 1//2))))))
2880 ((and (alike1 1//2 lin)
2881 (zerop1 const))
2882 (div 1 (ftake '%jacobi_cn u m)))
2883 (t
2884 (give-up)))))
2885 (t
2886 ;; Nothing to do
2887 (give-up)))))
2888
2889 ;; jacobi_nd(u,m) = 1/jacobi_dn(u,m)
2890 (defprop %jacobi_nd
2891 ((u m)
2892 ;; wrt u
2893 ((mtimes) m ((%jacobi_cn) u m)
2894 ((mexpt) ((%jacobi_dn) u m) -2) ((%jacobi_sn) u m))
2895 ;; wrt m
2896 ((mtimes) -1 ((mexpt) ((%jacobi_dn) u m) -2)
2897 ((mplus)
2898 ((mtimes) ((rat) -1 2)
2899 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
2900 ((%jacobi_dn) u m)
2901 ((mexpt) ((%jacobi_sn) u m) 2))
2902 ((mtimes) ((rat) -1 2) ((%jacobi_cn) u m)
2903 ((%jacobi_sn) u m)
2904 ((mplus) u
2905 ((mtimes) -1
2906 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
2907 ((%elliptic_e) ((%asin) ((%jacobi_sn) u m))
2908 m)))))))
2909 grad)
2910
2911 (def-simplifier jacobi_nd (u m)
2912 (let (coef args)
2913 (cond
2914 ((float-numerical-eval-p u m)
2915 (to (bigfloat:/ (bigfloat::dn (bigfloat:to ($float u))
2916 (bigfloat:to ($float m))))))
2917 ((setf args (complex-float-numerical-eval-p u m))
2918 (destructuring-bind (u m)
2919 args
2920 (to (bigfloat:/ (bigfloat::dn (bigfloat:to ($float u))
2921 (bigfloat:to ($float m)))))))
2922 ((bigfloat-numerical-eval-p u m)
2923 (let ((uu (bigfloat:to ($bfloat u)))
2924 (mm (bigfloat:to ($bfloat m))))
2925 (to (bigfloat:/ (bigfloat::dn uu mm)))))
2926 ((setf args (complex-bigfloat-numerical-eval-p u m))
2927 (destructuring-bind (u m)
2928 args
2929 (let ((uu (bigfloat:to ($bfloat u)))
2930 (mm (bigfloat:to ($bfloat m))))
2931 (to (bigfloat:/ (bigfloat::dn uu mm))))))
2932 ((zerop1 u)
2933 1)
2934 ((zerop1 m)
2935 ;; A&S 16.6.6
2936 1)
2937 ((onep1 m)
2938 ;; A&S 16.6.6
2939 (ftake '%cosh u))
2940 ((and $trigsign (mminusp* u))
2941 ;; nd is even
2942 (ftake* '%jacobi_nd (neg u) m))
2943 ((and $triginverses
2944 (listp u)
2945 (member (caar u) '(%inverse_jacobi_sn
2946 %inverse_jacobi_ns
2947 %inverse_jacobi_cn
2948 %inverse_jacobi_nc
2949 %inverse_jacobi_dn
2950 %inverse_jacobi_nd
2951 %inverse_jacobi_sc
2952 %inverse_jacobi_cs
2953 %inverse_jacobi_sd
2954 %inverse_jacobi_ds
2955 %inverse_jacobi_cd
2956 %inverse_jacobi_dc))
2957 (alike1 (third u) m))
2958 (cond ((eq (caar u) '%inverse_jacobi_nd)
2959 (second u))
2960 (t
2961 ;; Express in terms of dn:
2962 ;; nd(x) = 1/dn(x)
2963 (div 1 (ftake '%jacobi_dn u m)))))
2964 ;; A&S 16.20 (Jacobi's Imaginary transformation)
2965 ((and $%iargs (multiplep u '$%i))
2966 ;; nd(i*u) = 1/dn(i*u) = 1/dc(u,1-m) = cd(u,1-m)
2967 (ftake* '%jacobi_cd (coeff u '$%i 1) (add 1 (neg m))))
2968 ((setq coef (kc-arg2 u m))
2969 ;; A&S 16.8.6
2970 ;;
2971 (destructuring-bind (lin const)
2972 coef
2973 (cond ((integerp lin)
2974 ;; nd has period 2K
2975 (ecase (mod lin 2)
2976 (0
2977 ;; nd(2*m*K+u) = nd(u)
2978 ;; nd(0) = 1
2979 (if (zerop1 const)
2980 1
2981 (ftake '%jacobi_nd const m)))
2982 (1
2983 ;; nd(2*m*K+K+u) = nd(K+u) = dn(u)/sqrt(1-m)
2984 ;; nd(K) = 1/sqrt(1-m)
2985 (if (zerop1 const)
2986 (power (sub 1 m) -1//2)
2987 (div (ftake '%jacobi_nd const m)
2988 (power (sub 1 m) 1//2))))))
2989 (t
2990 (give-up)))))
2991 (t
2992 ;; Nothing to do
2993 (give-up)))))
2994
2995 ;; jacobi_sc(u,m) = jacobi_sn/jacobi_cn
2996 (defprop %jacobi_sc
2997 ((u m)
2998 ;; wrt u
2999 ((mtimes) ((mexpt) ((%jacobi_cn) u m) -2)
3000 ((%jacobi_dn) u m))
3001 ;; wrt m
3002 ((mplus)
3003 ((mtimes) ((mexpt) ((%jacobi_cn) u m) -1)
3004 ((mplus)
3005 ((mtimes) ((rat) 1 2)
3006 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3007 ((mexpt) ((%jacobi_cn) u m) 2)
3008 ((%jacobi_sn) u m))
3009 ((mtimes) ((rat) 1 2) ((mexpt) m -1)
3010 ((%jacobi_cn) u m) ((%jacobi_dn) u m)
3011 ((mplus) u
3012 ((mtimes) -1
3013 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3014 ((%elliptic_e) ((%asin) ((%jacobi_sn) u m))
3015 m))))))
3016 ((mtimes) -1 ((mexpt) ((%jacobi_cn) u m) -2)
3017 ((%jacobi_sn) u m)
3018 ((mplus)
3019 ((mtimes) ((rat) -1 2)
3020 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3021 ((%jacobi_cn) u m)
3022 ((mexpt) ((%jacobi_sn) u m) 2))
3023 ((mtimes) ((rat) -1 2) ((mexpt) m -1)
3024 ((%jacobi_dn) u m) ((%jacobi_sn) u m)
3025 ((mplus) u
3026 ((mtimes) -1
3027 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3028 ((%elliptic_e) ((%asin) ((%jacobi_sn) u m))
3029 m))))))))
3030 grad)
3031
3032 (def-simplifier jacobi_sc (u m)
3033 (let (coef args)
3034 (cond
3035 ((float-numerical-eval-p u m)
3036 (let ((fu (bigfloat:to ($float u)))
3037 (fm (bigfloat:to ($float m))))
3038 (to (bigfloat:/ (bigfloat::sn fu fm) (bigfloat::cn fu fm)))))
3039 ((setf args (complex-float-numerical-eval-p u m))
3040 (destructuring-bind (u m)
3041 args
3042 (let ((fu (bigfloat:to ($float u)))
3043 (fm (bigfloat:to ($float m))))
3044 (to (bigfloat:/ (bigfloat::sn fu fm) (bigfloat::cn fu fm))))))
3045 ((bigfloat-numerical-eval-p u m)
3046 (let ((uu (bigfloat:to ($bfloat u)))
3047 (mm (bigfloat:to ($bfloat m))))
3048 (to (bigfloat:/ (bigfloat::sn uu mm)
3049 (bigfloat::cn uu mm)))))
3050 ((setf args (complex-bigfloat-numerical-eval-p u m))
3051 (destructuring-bind (u m)
3052 args
3053 (let ((uu (bigfloat:to ($bfloat u)))
3054 (mm (bigfloat:to ($bfloat m))))
3055 (to (bigfloat:/ (bigfloat::sn uu mm)
3056 (bigfloat::cn uu mm))))))
3057 ((zerop1 u)
3058 0)
3059 ((zerop1 m)
3060 ;; A&S 16.6.9
3061 (ftake '%tan u))
3062 ((onep1 m)
3063 ;; A&S 16.6.9
3064 (ftake '%sinh u))
3065 ((and $trigsign (mminusp* u))
3066 ;; sc is odd
3067 (neg (ftake* '%jacobi_sc (neg u) m)))
3068 ((and $triginverses
3069 (listp u)
3070 (member (caar u) '(%inverse_jacobi_sn
3071 %inverse_jacobi_ns
3072 %inverse_jacobi_cn
3073 %inverse_jacobi_nc
3074 %inverse_jacobi_dn
3075 %inverse_jacobi_nd
3076 %inverse_jacobi_sc
3077 %inverse_jacobi_cs
3078 %inverse_jacobi_sd
3079 %inverse_jacobi_ds
3080 %inverse_jacobi_cd
3081 %inverse_jacobi_dc))
3082 (alike1 (third u) m))
3083 (cond ((eq (caar u) '%inverse_jacobi_sc)
3084 (second u))
3085 (t
3086 ;; Express in terms of sn and cn
3087 ;; sc(x) = sn(x)/cn(x)
3088 (div (ftake '%jacobi_sn u m)
3089 (ftake '%jacobi_cn u m)))))
3090 ;; A&S 16.20 (Jacobi's Imaginary transformation)
3091 ((and $%iargs (multiplep u '$%i))
3092 ;; sc(i*u) = sn(i*u)/cn(i*u) = i*sc(u,m1)/nc(u,m1) = i*sn(u,m1)
3093 (mul '$%i
3094 (ftake* '%jacobi_sn (coeff u '$%i 1) (add 1 (neg m)))))
3095 ((setq coef (kc-arg2 u m))
3096 ;; A&S 16.8.9
3097 ;; sc(2*m*K+u) = sc(u)
3098 (destructuring-bind (lin const)
3099 coef
3100 (cond ((integerp lin)
3101 (ecase (mod lin 2)
3102 (0
3103 ;; sc(2*m*K+ u) = sc(u)
3104 ;; sc(0) = 0
3105 (if (zerop1 const)
3106 1
3107 (ftake '%jacobi_sc const m)))
3108 (1
3109 ;; sc(2*m*K + K + u) = sc(K+u)= - cs(u)/sqrt(1-m)
3110 ;; sc(K) = infinity
3111 (if (zerop1 const)
3112 (dbz-err1 'jacobi_sc)
3113 (mul -1
3114 (div (ftake* '%jacobi_cs const m)
3115 (power (sub 1 m) 1//2)))))))
3116 ((and (alike1 lin 1//2)
3117 (zerop1 const))
3118 ;; From A&S 16.3.3 and 16.5.2:
3119 ;; sc(1/2*K) = 1/(1-m)^(1/4)
3120 (power (sub 1 m) (div -1 4)))
3121 (t
3122 (give-up)))))
3123 (t
3124 ;; Nothing to do
3125 (give-up)))))
3126
3127 ;; jacobi_sd(u,m) = jacobi_sn/jacobi_dn
3128 (defprop %jacobi_sd
3129 ((u m)
3130 ;; wrt u
3131 ((mtimes) ((%jacobi_cn) u m)
3132 ((mexpt) ((%jacobi_dn) u m) -2))
3133 ;; wrt m
3134 ((mplus)
3135 ((mtimes) ((mexpt) ((%jacobi_dn) u m) -1)
3136 ((mplus)
3137 ((mtimes) ((rat) 1 2)
3138 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3139 ((mexpt) ((%jacobi_cn) u m) 2)
3140 ((%jacobi_sn) u m))
3141 ((mtimes) ((rat) 1 2) ((mexpt) m -1)
3142 ((%jacobi_cn) u m) ((%jacobi_dn) u m)
3143 ((mplus) u
3144 ((mtimes) -1
3145 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3146 ((%elliptic_e) ((%asin) ((%jacobi_sn) u m))
3147 m))))))
3148 ((mtimes) -1 ((mexpt) ((%jacobi_dn) u m) -2)
3149 ((%jacobi_sn) u m)
3150 ((mplus)
3151 ((mtimes) ((rat) -1 2)
3152 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3153 ((%jacobi_dn) u m)
3154 ((mexpt) ((%jacobi_sn) u m) 2))
3155 ((mtimes) ((rat) -1 2) ((%jacobi_cn) u m)
3156 ((%jacobi_sn) u m)
3157 ((mplus) u
3158 ((mtimes) -1
3159 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3160 ((%elliptic_e) ((%asin) ((%jacobi_sn) u m))
3161 m))))))))
3162 grad)
3163
3164 (def-simplifier jacobi_sd (u m)
3165 (let (coef args)
3166 (cond
3167 ((float-numerical-eval-p u m)
3168 (let ((fu (bigfloat:to ($float u)))
3169 (fm (bigfloat:to ($float m))))
3170 (to (bigfloat:/ (bigfloat::sn fu fm) (bigfloat::dn fu fm)))))
3171 ((setf args (complex-float-numerical-eval-p u m))
3172 (destructuring-bind (u m)
3173 args
3174 (let ((fu (bigfloat:to ($float u)))
3175 (fm (bigfloat:to ($float m))))
3176 (to (bigfloat:/ (bigfloat::sn fu fm) (bigfloat::dn fu fm))))))
3177 ((bigfloat-numerical-eval-p u m)
3178 (let ((uu (bigfloat:to ($bfloat u)))
3179 (mm (bigfloat:to ($bfloat m))))
3180 (to (bigfloat:/ (bigfloat::sn uu mm)
3181 (bigfloat::dn uu mm)))))
3182 ((setf args (complex-bigfloat-numerical-eval-p u m))
3183 (destructuring-bind (u m)
3184 args
3185 (let ((uu (bigfloat:to ($bfloat u)))
3186 (mm (bigfloat:to ($bfloat m))))
3187 (to (bigfloat:/ (bigfloat::sn uu mm)
3188 (bigfloat::dn uu mm))))))
3189 ((zerop1 u)
3190 0)
3191 ((zerop1 m)
3192 ;; A&S 16.6.5
3193 (ftake '%sin u))
3194 ((onep1 m)
3195 ;; A&S 16.6.5
3196 (ftake '%sinh u))
3197 ((and $trigsign (mminusp* u))
3198 ;; sd is odd
3199 (neg (ftake* '%jacobi_sd (neg u) m)))
3200 ((and $triginverses
3201 (listp u)
3202 (member (caar u) '(%inverse_jacobi_sn
3203 %inverse_jacobi_ns
3204 %inverse_jacobi_cn
3205 %inverse_jacobi_nc
3206 %inverse_jacobi_dn
3207 %inverse_jacobi_nd
3208 %inverse_jacobi_sc
3209 %inverse_jacobi_cs
3210 %inverse_jacobi_sd
3211 %inverse_jacobi_ds
3212 %inverse_jacobi_cd
3213 %inverse_jacobi_dc))
3214 (alike1 (third u) m))
3215 (cond ((eq (caar u) '%inverse_jacobi_sd)
3216 (second u))
3217 (t
3218 ;; Express in terms of sn and dn
3219 (div (ftake '%jacobi_sn u m)
3220 (ftake '%jacobi_dn u m)))))
3221 ;; A&S 16.20 (Jacobi's Imaginary transformation)
3222 ((and $%iargs (multiplep u '$%i))
3223 ;; sd(i*u) = sn(i*u)/dn(i*u) = i*sc(u,m1)/dc(u,m1) = i*sd(u,m1)
3224 (mul '$%i
3225 (ftake* '%jacobi_sd (coeff u '$%i 1) (add 1 (neg m)))))
3226 ((setq coef (kc-arg2 u m))
3227 ;; A&S 16.8.5
3228 ;; sd(4*m*K+u) = sd(u)
3229 (destructuring-bind (lin const)
3230 coef
3231 (cond ((integerp lin)
3232 (ecase (mod lin 4)
3233 (0
3234 ;; sd(4*m*K+u) = sd(u)
3235 ;; sd(0) = 0
3236 (if (zerop1 const)
3237 0
3238 (ftake '%jacobi_sd const m)))
3239 (1
3240 ;; sd(4*m*K+K+u) = sd(K+u) = cn(u)/sqrt(1-m)
3241 ;; sd(K) = 1/sqrt(m1)
3242 (if (zerop1 const)
3243 (power (sub 1 m) 1//2)
3244 (div (ftake '%jacobi_cn const m)
3245 (power (sub 1 m) 1//2))))
3246 (2
3247 ;; sd(4*m*K+2*K+u) = sd(2*K+u) = -sd(u)
3248 ;; sd(2*K) = 0
3249 (if (zerop1 const)
3250 0
3251 (neg (ftake '%jacobi_sd const m))))
3252 (3
3253 ;; sd(4*m*K+3*K+u) = sd(3*K+u) = sd(2*K+K+u) =
3254 ;; -sd(K+u) = -cn(u)/sqrt(1-m)
3255 ;; sd(3*K) = -1/sqrt(m1)
3256 (if (zerop1 const)
3257 (neg (power (sub 1 m) -1//2))
3258 (neg (div (ftake '%jacobi_cn const m)
3259 (power (sub 1 m) 1//2)))))))
3260 ((and (alike1 lin 1//2)
3261 (zerop1 const))
3262 ;; jacobi_sn/jacobi_dn
3263 (div (ftake '%jacobi_sn
3264 (mul 1//2
3265 (ftake '%elliptic_kc m))
3266 m)
3267 (ftake '%jacobi_dn
3268 (mul 1//2
3269 (ftake '%elliptic_kc m))
3270 m)))
3271 (t
3272 (give-up)))))
3273 (t
3274 ;; Nothing to do
3275 (give-up)))))
3276
3277 ;; jacobi_cs(u,m) = jacobi_cn/jacobi_sn
3278 (defprop %jacobi_cs
3279 ((u m)
3280 ;; wrt u
3281 ((mtimes) -1 ((%jacobi_dn) u m)
3282 ((mexpt) ((%jacobi_sn) u m) -2))
3283 ;; wrt m
3284 ((mplus)
3285 ((mtimes) -1 ((%jacobi_cn) u m)
3286 ((mexpt) ((%jacobi_sn) u m) -2)
3287 ((mplus)
3288 ((mtimes) ((rat) 1 2)
3289 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3290 ((mexpt) ((%jacobi_cn) u m) 2)
3291 ((%jacobi_sn) u m))
3292 ((mtimes) ((rat) 1 2) ((mexpt) m -1)
3293 ((%jacobi_cn) u m) ((%jacobi_dn) u m)
3294 ((mplus) u
3295 ((mtimes) -1
3296 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3297 ((%elliptic_e) ((%asin) ((%jacobi_sn) u m))
3298 m))))))
3299 ((mtimes) ((mexpt) ((%jacobi_sn) u m) -1)
3300 ((mplus)
3301 ((mtimes) ((rat) -1 2)
3302 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3303 ((%jacobi_cn) u m)
3304 ((mexpt) ((%jacobi_sn) u m) 2))
3305 ((mtimes) ((rat) -1 2) ((mexpt) m -1)
3306 ((%jacobi_dn) u m) ((%jacobi_sn) u m)
3307 ((mplus) u
3308 ((mtimes) -1
3309 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3310 ((%elliptic_e) ((%asin) ((%jacobi_sn) u m))
3311 m))))))))
3312 grad)
3313
3314 (def-simplifier jacobi_cs (u m)
3315 (let (coef args)
3316 (cond
3317 ((float-numerical-eval-p u m)
3318 (let ((fu (bigfloat:to ($float u)))
3319 (fm (bigfloat:to ($float m))))
3320 (to (bigfloat:/ (bigfloat::cn fu fm) (bigfloat::sn fu fm)))))
3321 ((setf args (complex-float-numerical-eval-p u m))
3322 (destructuring-bind (u m)
3323 args
3324 (let ((fu (bigfloat:to ($float u)))
3325 (fm (bigfloat:to ($float m))))
3326 (to (bigfloat:/ (bigfloat::cn fu fm) (bigfloat::sn fu fm))))))
3327 ((bigfloat-numerical-eval-p u m)
3328 (let ((uu (bigfloat:to ($bfloat u)))
3329 (mm (bigfloat:to ($bfloat m))))
3330 (to (bigfloat:/ (bigfloat::cn uu mm)
3331 (bigfloat::sn uu mm)))))
3332 ((setf args (complex-bigfloat-numerical-eval-p u m))
3333 (destructuring-bind (u m)
3334 args
3335 (let ((uu (bigfloat:to ($bfloat u)))
3336 (mm (bigfloat:to ($bfloat m))))
3337 (to (bigfloat:/ (bigfloat::cn uu mm)
3338 (bigfloat::sn uu mm))))))
3339 ((zerop1 m)
3340 ;; A&S 16.6.12
3341 (ftake '%cot u))
3342 ((onep1 m)
3343 ;; A&S 16.6.12
3344 (ftake '%csch u))
3345 ((zerop1 u)
3346 (dbz-err1 'jacobi_cs))
3347 ((and $trigsign (mminusp* u))
3348 ;; cs is odd
3349 (neg (ftake* '%jacobi_cs (neg u) m)))
3350 ((and $triginverses
3351 (listp u)
3352 (member (caar u) '(%inverse_jacobi_sn
3353 %inverse_jacobi_ns
3354 %inverse_jacobi_cn
3355 %inverse_jacobi_nc
3356 %inverse_jacobi_dn
3357 %inverse_jacobi_nd
3358 %inverse_jacobi_sc
3359 %inverse_jacobi_cs
3360 %inverse_jacobi_sd
3361 %inverse_jacobi_ds
3362 %inverse_jacobi_cd
3363 %inverse_jacobi_dc))
3364 (alike1 (third u) m))
3365 (cond ((eq (caar u) '%inverse_jacobi_cs)
3366 (second u))
3367 (t
3368 ;; Express in terms of cn an sn
3369 (div (ftake '%jacobi_cn u m)
3370 (ftake '%jacobi_sn u m)))))
3371 ;; A&S 16.20 (Jacobi's Imaginary transformation)
3372 ((and $%iargs (multiplep u '$%i))
3373 ;; cs(i*u) = cn(i*u)/sn(i*u) = -i*nc(u,m1)/sc(u,m1) = -i*ns(u,m1)
3374 (neg (mul '$%i
3375 (ftake* '%jacobi_ns (coeff u '$%i 1) (add 1 (neg m))))))
3376 ((setq coef (kc-arg2 u m))
3377 ;; A&S 16.8.12
3378 ;;
3379 ;; cs(2*m*K + u) = cs(u)
3380 (destructuring-bind (lin const)
3381 coef
3382 (cond ((integerp lin)
3383 (ecase (mod lin 2)
3384 (0
3385 ;; cs(2*m*K + u) = cs(u)
3386 ;; cs(0) = infinity
3387 (if (zerop1 const)
3388 (dbz-err1 'jacobi_cs)
3389 (ftake '%jacobi_cs const m)))
3390 (1
3391 ;; cs(K+u) = -sqrt(1-m)*sc(u)
3392 ;; cs(K) = 0
3393 (if (zerop1 const)
3394 0
3395 (neg (mul (power (sub 1 m) 1//2)
3396 (ftake '%jacobi_sc const m)))))))
3397 ((and (alike1 lin 1//2)
3398 (zerop1 const))
3399 ;; 1/jacobi_sc
3400 (div 1
3401 (ftake '%jacobi_sc (mul 1//2
3402 (ftake '%elliptic_kc m))
3403 m)))
3404 (t
3405 (give-up)))))
3406 (t
3407 ;; Nothing to do
3408 (give-up)))))
3409
3410 ;; jacobi_cd(u,m) = jacobi_cn/jacobi_dn
3411 (defprop %jacobi_cd
3412 ((u m)
3413 ;; wrt u
3414 ((mtimes) ((mplus) -1 m)
3415 ((mexpt) ((%jacobi_dn) u m) -2)
3416 ((%jacobi_sn) u m))
3417 ;; wrt m
3418 ((mplus)
3419 ((mtimes) -1 ((%jacobi_cn) u m)
3420 ((mexpt) ((%jacobi_dn) u m) -2)
3421 ((mplus)
3422 ((mtimes) ((rat) -1 2)
3423 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3424 ((%jacobi_dn) u m)
3425 ((mexpt) ((%jacobi_sn) u m) 2))
3426 ((mtimes) ((rat) -1 2) ((%jacobi_cn) u m)
3427 ((%jacobi_sn) u m)
3428 ((mplus) u
3429 ((mtimes) -1
3430 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3431 ((%elliptic_e) ((%asin) ((%jacobi_sn) u m))
3432 m))))))
3433 ((mtimes) ((mexpt) ((%jacobi_dn) u m) -1)
3434 ((mplus)
3435 ((mtimes) ((rat) -1 2)
3436 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3437 ((%jacobi_cn) u m)
3438 ((mexpt) ((%jacobi_sn) u m) 2))
3439 ((mtimes) ((rat) -1 2) ((mexpt) m -1)
3440 ((%jacobi_dn) u m) ((%jacobi_sn) u m)
3441 ((mplus) u
3442 ((mtimes) -1
3443 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3444 ((%elliptic_e) ((%asin) ((%jacobi_sn) u m))
3445 m))))))))
3446 grad)
3447
3448 (def-simplifier jacobi_cd (u m)
3449 (let (coef args)
3450 (cond
3451 ((float-numerical-eval-p u m)
3452 (let ((fu (bigfloat:to ($float u)))
3453 (fm (bigfloat:to ($float m))))
3454 (to (bigfloat:/ (bigfloat::cn fu fm) (bigfloat::dn fu fm)))))
3455 ((setf args (complex-float-numerical-eval-p u m))
3456 (destructuring-bind (u m)
3457 args
3458 (let ((fu (bigfloat:to ($float u)))
3459 (fm (bigfloat:to ($float m))))
3460 (to (bigfloat:/ (bigfloat::cn fu fm) (bigfloat::dn fu fm))))))
3461 ((bigfloat-numerical-eval-p u m)
3462 (let ((uu (bigfloat:to ($bfloat u)))
3463 (mm (bigfloat:to ($bfloat m))))
3464 (to (bigfloat:/ (bigfloat::cn uu mm) (bigfloat::dn uu mm)))))
3465 ((setf args (complex-bigfloat-numerical-eval-p u m))
3466 (destructuring-bind (u m)
3467 args
3468 (let ((uu (bigfloat:to ($bfloat u)))
3469 (mm (bigfloat:to ($bfloat m))))
3470 (to (bigfloat:/ (bigfloat::cn uu mm) (bigfloat::dn uu mm))))))
3471 ((zerop1 u)
3472 1)
3473 ((zerop1 m)
3474 ;; A&S 16.6.4
3475 (ftake '%cos u))
3476 ((onep1 m)
3477 ;; A&S 16.6.4
3478 1)
3479 ((and $trigsign (mminusp* u))
3480 ;; cd is even
3481 (ftake* '%jacobi_cd (neg u) m))
3482 ((and $triginverses
3483 (listp u)
3484 (member (caar u) '(%inverse_jacobi_sn
3485 %inverse_jacobi_ns
3486 %inverse_jacobi_cn
3487 %inverse_jacobi_nc
3488 %inverse_jacobi_dn
3489 %inverse_jacobi_nd
3490 %inverse_jacobi_sc
3491 %inverse_jacobi_cs
3492 %inverse_jacobi_sd
3493 %inverse_jacobi_ds
3494 %inverse_jacobi_cd
3495 %inverse_jacobi_dc))
3496 (alike1 (third u) m))
3497 (cond ((eq (caar u) '%inverse_jacobi_cd)
3498 (second u))
3499 (t
3500 ;; Express in terms of cn and dn
3501 (div (ftake '%jacobi_cn u m)
3502 (ftake '%jacobi_dn u m)))))
3503 ;; A&S 16.20 (Jacobi's Imaginary transformation)
3504 ((and $%iargs (multiplep u '$%i))
3505 ;; cd(i*u) = cn(i*u)/dn(i*u) = nc(u,m1)/dc(u,m1) = nd(u,m1)
3506 (ftake* '%jacobi_nd (coeff u '$%i 1) (add 1 (neg m))))
3507 ((setf coef (kc-arg2 u m))
3508 ;; A&S 16.8.4
3509 ;;
3510 (destructuring-bind (lin const)
3511 coef
3512 (cond ((integerp lin)
3513 (ecase (mod lin 4)
3514 (0
3515 ;; cd(4*m*K + u) = cd(u)
3516 ;; cd(0) = 1
3517 (if (zerop1 const)
3518 1
3519 (ftake '%jacobi_cd const m)))
3520 (1
3521 ;; cd(4*m*K + K + u) = cd(K+u) = -sn(u)
3522 ;; cd(K) = 0
3523 (if (zerop1 const)
3524 0
3525 (neg (ftake '%jacobi_sn const m))))
3526 (2
3527 ;; cd(4*m*K + 2*K + u) = cd(2*K+u) = -cd(u)
3528 ;; cd(2*K) = -1
3529 (if (zerop1 const)
3530 -1
3531 (neg (ftake '%jacobi_cd const m))))
3532 (3
3533 ;; cd(4*m*K + 3*K + u) = cd(2*K + K + u) =
3534 ;; -cd(K+u) = sn(u)
3535 ;; cd(3*K) = 0
3536 (if (zerop1 const)
3537 0
3538 (ftake '%jacobi_sn const m)))))
3539 ((and (alike1 lin 1//2)
3540 (zerop1 const))
3541 ;; jacobi_cn/jacobi_dn
3542 (div (ftake '%jacobi_cn
3543 (mul 1//2
3544 (ftake '%elliptic_kc m))
3545 m)
3546 (ftake '%jacobi_dn
3547 (mul 1//2
3548 (ftake '%elliptic_kc m))
3549 m)))
3550 (t
3551 ;; Nothing to do
3552 (give-up)))))
3553 (t
3554 ;; Nothing to do
3555 (give-up)))))
3556
3557 ;; jacobi_ds(u,m) = jacobi_dn/jacobi_sn
3558 (defprop %jacobi_ds
3559 ((u m)
3560 ;; wrt u
3561 ((mtimes) -1 ((%jacobi_cn) u m)
3562 ((mexpt) ((%jacobi_sn) u m) -2))
3563 ;; wrt m
3564 ((mplus)
3565 ((mtimes) -1 ((%jacobi_dn) u m)
3566 ((mexpt) ((%jacobi_sn) u m) -2)
3567 ((mplus)
3568 ((mtimes) ((rat) 1 2)
3569 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3570 ((mexpt) ((%jacobi_cn) u m) 2)
3571 ((%jacobi_sn) u m))
3572 ((mtimes) ((rat) 1 2) ((mexpt) m -1)
3573 ((%jacobi_cn) u m) ((%jacobi_dn) u m)
3574 ((mplus) u
3575 ((mtimes) -1
3576 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3577 ((%elliptic_e) ((%asin) ((%jacobi_sn) u m))
3578 m))))))
3579 ((mtimes) ((mexpt) ((%jacobi_sn) u m) -1)
3580 ((mplus)
3581 ((mtimes) ((rat) -1 2)
3582 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3583 ((%jacobi_dn) u m)
3584 ((mexpt) ((%jacobi_sn) u m) 2))
3585 ((mtimes) ((rat) -1 2) ((%jacobi_cn) u m)
3586 ((%jacobi_sn) u m)
3587 ((mplus) u
3588 ((mtimes) -1
3589 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3590 ((%elliptic_e) ((%asin) ((%jacobi_sn) u m))
3591 m))))))))
3592 grad)
3593
3594 (def-simplifier jacobi_ds (u m)
3595 (let (coef args)
3596 (cond
3597 ((float-numerical-eval-p u m)
3598 (let ((fu (bigfloat:to ($float u)))
3599 (fm (bigfloat:to ($float m))))
3600 (to (bigfloat:/ (bigfloat::dn fu fm) (bigfloat::sn fu fm)))))
3601 ((setf args (complex-float-numerical-eval-p u m))
3602 (destructuring-bind (u m)
3603 args
3604 (let ((fu (bigfloat:to ($float u)))
3605 (fm (bigfloat:to ($float m))))
3606 (to (bigfloat:/ (bigfloat::dn fu fm) (bigfloat::sn fu fm))))))
3607 ((bigfloat-numerical-eval-p u m)
3608 (let ((uu (bigfloat:to ($bfloat u)))
3609 (mm (bigfloat:to ($bfloat m))))
3610 (to (bigfloat:/ (bigfloat::dn uu mm)
3611 (bigfloat::sn uu mm)))))
3612 ((setf args (complex-bigfloat-numerical-eval-p u m))
3613 (destructuring-bind (u m)
3614 args
3615 (let ((uu (bigfloat:to ($bfloat u)))
3616 (mm (bigfloat:to ($bfloat m))))
3617 (to (bigfloat:/ (bigfloat::dn uu mm)
3618 (bigfloat::sn uu mm))))))
3619 ((zerop1 m)
3620 ;; A&S 16.6.11
3621 (ftake '%csc u))
3622 ((onep1 m)
3623 ;; A&S 16.6.11
3624 (ftake '%csch u))
3625 ((zerop1 u)
3626 (dbz-err1 'jacobi_ds))
3627 ((and $trigsign (mminusp* u))
3628 (neg (ftake* '%jacobi_ds (neg u) m)))
3629 ((and $triginverses
3630 (listp u)
3631 (member (caar u) '(%inverse_jacobi_sn
3632 %inverse_jacobi_ns
3633 %inverse_jacobi_cn
3634 %inverse_jacobi_nc
3635 %inverse_jacobi_dn
3636 %inverse_jacobi_nd
3637 %inverse_jacobi_sc
3638 %inverse_jacobi_cs
3639 %inverse_jacobi_sd
3640 %inverse_jacobi_ds
3641 %inverse_jacobi_cd
3642 %inverse_jacobi_dc))
3643 (alike1 (third u) m))
3644 (cond ((eq (caar u) '%inverse_jacobi_ds)
3645 (second u))
3646 (t
3647 ;; Express in terms of dn and sn
3648 (div (ftake '%jacobi_dn u m)
3649 (ftake '%jacobi_sn u m)))))
3650 ;; A&S 16.20 (Jacobi's Imaginary transformation)
3651 ((and $%iargs (multiplep u '$%i))
3652 ;; ds(i*u) = dn(i*u)/sn(i*u) = -i*dc(u,m1)/sc(u,m1) = -i*ds(u,m1)
3653 (neg (mul '$%i
3654 (ftake* '%jacobi_ds (coeff u '$%i 1) (add 1 (neg m))))))
3655 ((setf coef (kc-arg2 u m))
3656 ;; A&S 16.8.11
3657 (destructuring-bind (lin const)
3658 coef
3659 (cond ((integerp lin)
3660 (ecase (mod lin 4)
3661 (0
3662 ;; ds(4*m*K + u) = ds(u)
3663 ;; ds(0) = infinity
3664 (if (zerop1 const)
3665 (dbz-err1 'jacobi_ds)
3666 (ftake '%jacobi_ds const m)))
3667 (1
3668 ;; ds(4*m*K + K + u) = ds(K+u) = sqrt(1-m)*nc(u)
3669 ;; ds(K) = sqrt(1-m)
3670 (if (zerop1 const)
3671 (power (sub 1 m) 1//2)
3672 (mul (power (sub 1 m) 1//2)
3673 (ftake '%jacobi_nc const m))))
3674 (2
3675 ;; ds(4*m*K + 2*K + u) = ds(2*K+u) = -ds(u)
3676 ;; ds(0) = pole
3677 (if (zerop1 const)
3678 (dbz-err1 'jacobi_ds)
3679 (neg (ftake '%jacobi_ds const m))))
3680 (3
3681 ;; ds(4*m*K + 3*K + u) = ds(2*K + K + u) =
3682 ;; -ds(K+u) = -sqrt(1-m)*nc(u)
3683 ;; ds(3*K) = -sqrt(1-m)
3684 (if (zerop1 const)
3685 (neg (power (sub 1 m) 1//2))
3686 (neg (mul (power (sub 1 m) 1//2)
3687 (ftake '%jacobi_nc u m)))))))
3688 ((and (alike1 lin 1//2)
3689 (zerop1 const))
3690 ;; jacobi_dn/jacobi_sn
3691 (div
3692 (ftake '%jacobi_dn
3693 (mul 1//2 (ftake '%elliptic_kc m))
3694 m)
3695 (ftake '%jacobi_sn
3696 (mul 1//2 (ftake '%elliptic_kc m))
3697 m)))
3698 (t
3699 ;; Nothing to do
3700 (give-up)))))
3701 (t
3702 ;; Nothing to do
3703 (give-up)))))
3704
3705 ;; jacobi_dc(u,m) = jacobi_dn/jacobi_cn
3706 (defprop %jacobi_dc
3707 ((u m)
3708 ;; wrt u
3709 ((mtimes) ((mplus) 1 ((mtimes) -1 m))
3710 ((mexpt) ((%jacobi_cn) u m) -2)
3711 ((%jacobi_sn) u m))
3712 ;; wrt m
3713 ((mplus)
3714 ((mtimes) ((mexpt) ((%jacobi_cn) u m) -1)
3715 ((mplus)
3716 ((mtimes) ((rat) -1 2)
3717 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3718 ((%jacobi_dn) u m)
3719 ((mexpt) ((%jacobi_sn) u m) 2))
3720 ((mtimes) ((rat) -1 2) ((%jacobi_cn) u m)
3721 ((%jacobi_sn) u m)
3722 ((mplus) u
3723 ((mtimes) -1
3724 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3725 ((%elliptic_e) ((%asin) ((%jacobi_sn) u m))
3726 m))))))
3727 ((mtimes) -1 ((mexpt) ((%jacobi_cn) u m) -2)
3728 ((%jacobi_dn) u m)
3729 ((mplus)
3730 ((mtimes) ((rat) -1 2)
3731 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3732 ((%jacobi_cn) u m)
3733 ((mexpt) ((%jacobi_sn) u m) 2))
3734 ((mtimes) ((rat) -1 2) ((mexpt) m -1)
3735 ((%jacobi_dn) u m) ((%jacobi_sn) u m)
3736 ((mplus) u
3737 ((mtimes) -1
3738 ((mexpt) ((mplus) 1 ((mtimes) -1 m)) -1)
3739 ((%elliptic_e) ((%asin) ((%jacobi_sn) u m))
3740 m))))))))
3741 grad)
3742
3743 (def-simplifier jacobi_dc (u m)
3744 (let (coef args)
3745 (cond
3746 ((float-numerical-eval-p u m)
3747 (let ((fu (bigfloat:to ($float u)))
3748 (fm (bigfloat:to ($float m))))
3749 (to (bigfloat:/ (bigfloat::dn fu fm) (bigfloat::cn fu fm)))))
3750 ((setf args (complex-float-numerical-eval-p u m))
3751 (destructuring-bind (u m)
3752 args
3753 (let ((fu (bigfloat:to ($float u)))
3754 (fm (bigfloat:to ($float m))))
3755 (to (bigfloat:/ (bigfloat::dn fu fm) (bigfloat::cn fu fm))))))
3756 ((bigfloat-numerical-eval-p u m)
3757 (let ((uu (bigfloat:to ($bfloat u)))
3758 (mm (bigfloat:to ($bfloat m))))
3759 (to (bigfloat:/ (bigfloat::dn uu mm)
3760 (bigfloat::cn uu mm)))))
3761 ((setf args (complex-bigfloat-numerical-eval-p u m))
3762 (destructuring-bind (u m)
3763 args
3764 (let ((uu (bigfloat:to ($bfloat u)))
3765 (mm (bigfloat:to ($bfloat m))))
3766 (to (bigfloat:/ (bigfloat::dn uu mm)
3767 (bigfloat::cn uu mm))))))
3768 ((zerop1 u)
3769 1)
3770 ((zerop1 m)
3771 ;; A&S 16.6.7
3772 (ftake '%sec u))
3773 ((onep1 m)
3774 ;; A&S 16.6.7
3775 1)
3776 ((and $trigsign (mminusp* u))
3777 (ftake* '%jacobi_dc (neg u) m))
3778 ((and $triginverses
3779 (listp u)
3780 (member (caar u) '(%inverse_jacobi_sn
3781 %inverse_jacobi_ns
3782 %inverse_jacobi_cn
3783 %inverse_jacobi_nc
3784 %inverse_jacobi_dn
3785 %inverse_jacobi_nd
3786 %inverse_jacobi_sc
3787 %inverse_jacobi_cs
3788 %inverse_jacobi_sd
3789 %inverse_jacobi_ds
3790 %inverse_jacobi_cd
3791 %inverse_jacobi_dc))
3792 (alike1 (third u) m))
3793 (cond ((eq (caar u) '%inverse_jacobi_dc)
3794 (second u))
3795 (t
3796 ;; Express in terms of dn and cn
3797 (div (ftake '%jacobi_dn u m)
3798 (ftake '%jacobi_cn u m)))))
3799 ;; A&S 16.20 (Jacobi's Imaginary transformation)
3800 ((and $%iargs (multiplep u '$%i))
3801 ;; dc(i*u) = dn(i*u)/cn(i*u) = dc(u,m1)/nc(u,m1) = dn(u,m1)
3802 (ftake* '%jacobi_dn (coeff u '$%i 1) (add 1 (neg m))))
3803 ((setf coef (kc-arg2 u m))
3804 ;; See A&S 16.8.7
3805 (destructuring-bind (lin const)
3806 coef
3807 (cond ((integerp lin)
3808 (ecase (mod lin 4)
3809 (0
3810 ;; dc(4*m*K + u) = dc(u)
3811 ;; dc(0) = 1
3812 (if (zerop1 const)
3813 1
3814 (ftake '%jacobi_dc const m)))
3815 (1
3816 ;; dc(4*m*K + K + u) = dc(K+u) = -ns(u)
3817 ;; dc(K) = pole
3818 (if (zerop1 const)
3819 (dbz-err1 'jacobi_dc)
3820 (neg (ftake '%jacobi_ns const m))))
3821 (2
3822 ;; dc(4*m*K + 2*K + u) = dc(2*K+u) = -dc(u)
3823 ;; dc(2K) = -1
3824 (if (zerop1 const)
3825 -1
3826 (neg (ftake '%jacobi_dc const m))))
3827 (3
3828 ;; dc(4*m*K + 3*K + u) = dc(2*K + K + u) =
3829 ;; -dc(K+u) = ns(u)
3830 ;; dc(3*K) = ns(0) = inf
3831 (if (zerop1 const)
3832 (dbz-err1 'jacobi_dc)
3833 (ftake '%jacobi_dc const m)))))
3834 ((and (alike1 lin 1//2)
3835 (zerop1 const))
3836 ;; jacobi_dn/jacobi_cn
3837 (div
3838 (ftake '%jacobi_dn
3839 (mul 1//2 (ftake '%elliptic_kc m))
3840 m)
3841 (ftake '%jacobi_cn
3842 (mul 1//2 (ftake '%elliptic_kc m))
3843 m)))
3844 (t
3845 ;; Nothing to do
3846 (give-up)))))
3847 (t
3848 ;; Nothing to do
3849 (give-up)))))
3850
3851 ;;; Other inverse Jacobian functions
3852
3853 ;; inverse_jacobi_ns(x)
3854 ;;
3855 ;; Let u = inverse_jacobi_ns(x). Then jacobi_ns(u) = x or
3856 ;; 1/jacobi_sn(u) = x or
3857 ;;
3858 ;; jacobi_sn(u) = 1/x
3859 ;;
3860 ;; so u = inverse_jacobi_sn(1/x)
3861 (defprop %inverse_jacobi_ns
3862 ((x m)
3863 ;; Whittaker and Watson, example in 22.122
3864 ;; inverse_jacobi_ns(u,m) = integrate(1/sqrt(t^2-1)/sqrt(t^2-m), t, u, inf)
3865 ;; -> -1/sqrt(x^2-1)/sqrt(x^2-m)
3866 ((mtimes) -1
3867 ((mexpt) ((mplus) -1 ((mexpt) x 2)) ((rat) -1 2))
3868 ((mexpt)
3869 ((mplus) ((mtimes simp ratsimp) -1 m) ((mexpt) x 2))
3870 ((rat) -1 2)))
3871 ;; wrt m
3872 ; ((%derivative) ((%inverse_jacobi_ns) x m) m 1)
3873 nil)
3874 grad)
3875
3876 (def-simplifier inverse_jacobi_ns (u m)
3877 (let (args)
3878 (cond
3879 ((float-numerical-eval-p u m)
3880 ;; Numerically evaluate asn
3881 ;;
3882 ;; ans(x,m) = asn(1/x,m) = F(asin(1/x),m)
3883 (to (elliptic-f (cl:asin (/ ($float u))) ($float m))))
3884 ((complex-float-numerical-eval-p u m)
3885 (to (elliptic-f (cl:asin (/ (complex ($realpart ($float u)) ($imagpart ($float u)))))
3886 (complex ($realpart ($float m)) ($imagpart ($float m))))))
3887 ((bigfloat-numerical-eval-p u m)
3888 (to (bigfloat::bf-elliptic-f (bigfloat:asin (bigfloat:/ (bigfloat:to ($bfloat u))))
3889 (bigfloat:to ($bfloat m)))))
3890 ((setf args (complex-bigfloat-numerical-eval-p u m))
3891 (destructuring-bind (u m)
3892 args
3893 (to (bigfloat::bf-elliptic-f (bigfloat:asin (bigfloat:/ (bigfloat:to ($bfloat u))))
3894 (bigfloat:to ($bfloat m))))))
3895 ((zerop1 m)
3896 ;; ans(x,0) = F(asin(1/x),0) = asin(1/x)
3897 (ftake '%elliptic_f (ftake '%asin (div 1 u)) 0))
3898 ((onep1 m)
3899 ;; ans(x,1) = F(asin(1/x),1) = log(tan(pi/2+asin(1/x)/2))
3900 (ftake '%elliptic_f (ftake '%asin (div 1 u)) 1))
3901 ((onep1 u)
3902 (ftake '%elliptic_kc m))
3903 ((alike1 u -1)
3904 (neg (ftake '%elliptic_kc m)))
3905 ((and (eq $triginverses '$all)
3906 (listp u)
3907 (eq (caar u) '%jacobi_ns)
3908 (alike1 (third u) m))
3909 ;; inverse_jacobi_ns(ns(u)) = u
3910 (second u))
3911 (t
3912 ;; Nothing to do
3913 (give-up)))))
3914
3915 ;; inverse_jacobi_nc(x)
3916 ;;
3917 ;; Let u = inverse_jacobi_nc(x). Then jacobi_nc(u) = x or
3918 ;; 1/jacobi_cn(u) = x or
3919 ;;
3920 ;; jacobi_cn(u) = 1/x
3921 ;;
3922 ;; so u = inverse_jacobi_cn(1/x)
3923 (defprop %inverse_jacobi_nc
3924 ((x m)
3925 ;; Whittaker and Watson, example in 22.122
3926 ;; inverse_jacobi_nc(u,m) = integrate(1/sqrt(t^2-1)/sqrt((1-m)*t^2+m), t, 1, u)
3927 ;; -> 1/sqrt(x^2-1)/sqrt((1-m)*x^2+m)
3928 ((mtimes)
3929 ((mexpt) ((mplus) -1 ((mexpt) x 2)) ((rat) -1 2))
3930 ((mexpt)
3931 ((mplus) m
3932 ((mtimes) -1 ((mplus) -1 m) ((mexpt) x 2)))
3933 ((rat) -1 2)))
3934 ;; wrt m
3935 ; ((%derivative) ((%inverse_jacobi_nc) x m) m 1)
3936 nil)
3937 grad)
3938
3939 (def-simplifier inverse_jacobi_nc (u m)
3940 (cond ((or (float-numerical-eval-p u m)
3941 (complex-float-numerical-eval-p u m)
3942 (bigfloat-numerical-eval-p u m)
3943 (complex-bigfloat-numerical-eval-p u m))
3944 ;;
3945 (ftake '%inverse_jacobi_cn ($rectform (div 1 u)) m))
3946 ((onep1 u)
3947 0)
3948 ((alike1 u -1)
3949 (mul 2 (ftake '%elliptic_kc m)))
3950 ((and (eq $triginverses '$all)
3951 (listp u)
3952 (eq (caar u) '%jacobi_nc)
3953 (alike1 (third u) m))
3954 ;; inverse_jacobi_nc(nc(u)) = u
3955 (second u))
3956 (t
3957 ;; Nothing to do
3958 (give-up))))
3959
3960 ;; inverse_jacobi_nd(x)
3961 ;;
3962 ;; Let u = inverse_jacobi_nd(x). Then jacobi_nd(u) = x or
3963 ;; 1/jacobi_dn(u) = x or
3964 ;;
3965 ;; jacobi_dn(u) = 1/x
3966 ;;
3967 ;; so u = inverse_jacobi_dn(1/x)
3968 (defprop %inverse_jacobi_nd
3969 ((x m)
3970 ;; Whittaker and Watson, example in 22.122
3971 ;; inverse_jacobi_nd(u,m) = integrate(1/sqrt(t^2-1)/sqrt(1-(1-m)*t^2), t, 1, u)
3972 ;; -> 1/sqrt(u^2-1)/sqrt(1-(1-m)*t^2)
3973 ((mtimes)
3974 ((mexpt) ((mplus) -1 ((mexpt simp ratsimp) x 2))
3975 ((rat) -1 2))
3976 ((mexpt)
3977 ((mplus) 1
3978 ((mtimes) ((mplus) -1 m) ((mexpt simp ratsimp) x 2)))
3979 ((rat) -1 2)))
3980 ;; wrt m
3981 ; ((%derivative) ((%inverse_jacobi_nd) x m) m 1)
3982 nil)
3983 grad)
3984
3985 (def-simplifier inverse_jacobi_nd (u m)
3986 (cond ((or (float-numerical-eval-p u m)
3987 (complex-float-numerical-eval-p u m)
3988 (bigfloat-numerical-eval-p u m)
3989 (complex-bigfloat-numerical-eval-p u m))
3990 (ftake '%inverse_jacobi_dn ($rectform (div 1 u)) m))
3991 ((onep1 u)
3992 0)
3993 ((onep1 ($ratsimp (mul (power (sub 1 m) 1//2) u)))
3994 ;; jacobi_nd(1/sqrt(1-m),m) = K(m). This follows from
3995 ;; jacobi_dn(sqrt(1-m),m) = K(m).
3996 (ftake '%elliptic_kc m))
3997 ((and (eq $triginverses '$all)
3998 (listp u)
3999 (eq (caar u) '%jacobi_nd)
4000 (alike1 (third u) m))
4001 ;; inverse_jacobi_nd(nd(u)) = u
4002 (second u))
4003 (t
4004 ;; Nothing to do
4005 (give-up))))
4006
4007 ;; inverse_jacobi_sc(x)
4008 ;;
4009 ;; Let u = inverse_jacobi_sc(x). Then jacobi_sc(u) = x or
4010 ;; x = jacobi_sn(u)/jacobi_cn(u)
4011 ;;
4012 ;; x^2 = sn^2/cn^2
4013 ;; = sn^2/(1-sn^2)
4014 ;;
4015 ;; so
4016 ;;
4017 ;; sn^2 = x^2/(1+x^2)
4018 ;;
4019 ;; sn(u) = x/sqrt(1+x^2)
4020 ;;
4021 ;; u = inverse_sn(x/sqrt(1+x^2))
4022 ;;
4023 (defprop %inverse_jacobi_sc
4024 ((x m)
4025 ;; Whittaker and Watson, example in 22.122
4026 ;; inverse_jacobi_sc(u,m) = integrate(1/sqrt(1+t^2)/sqrt(1+(1-m)*t^2), t, 0, u)
4027 ;; -> 1/sqrt(1+x^2)/sqrt(1+(1-m)*x^2)
4028 ((mtimes)
4029 ((mexpt) ((mplus) 1 ((mexpt) x 2))
4030 ((rat) -1 2))
4031 ((mexpt)
4032 ((mplus) 1
4033 ((mtimes) -1 ((mplus) -1 m) ((mexpt) x 2)))
4034 ((rat) -1 2)))
4035 ;; wrt m
4036 ; ((%derivative) ((%inverse_jacobi_sc) x m) m 1)
4037 nil)
4038 grad)
4039
4040 (def-simplifier inverse_jacobi_sc (u m)
4041 (cond ((or (float-numerical-eval-p u m)
4042 (complex-float-numerical-eval-p u m)
4043 (bigfloat-numerical-eval-p u m)
4044 (complex-bigfloat-numerical-eval-p u m))
4045 (ftake '%inverse_jacobi_sn
4046 ($rectform (div u (power (add 1 (mul u u)) 1//2)))
4047 m))
4048 ((zerop1 u)
4049 ;; jacobi_sc(0,m) = 0
4050 0)
4051 ((and (eq $triginverses '$all)
4052 (listp u)
4053 (eq (caar u) '%jacobi_sc)
4054 (alike1 (third u) m))
4055 ;; inverse_jacobi_sc(sc(u)) = u
4056 (second u))
4057 (t
4058 ;; Nothing to do
4059 (give-up))))
4060
4061 ;; inverse_jacobi_sd(x)
4062 ;;
4063 ;; Let u = inverse_jacobi_sd(x). Then jacobi_sd(u) = x or
4064 ;; x = jacobi_sn(u)/jacobi_dn(u)
4065 ;;
4066 ;; x^2 = sn^2/dn^2
4067 ;; = sn^2/(1-m*sn^2)
4068 ;;
4069 ;; so
4070 ;;
4071 ;; sn^2 = x^2/(1+m*x^2)
4072 ;;
4073 ;; sn(u) = x/sqrt(1+m*x^2)
4074 ;;
4075 ;; u = inverse_sn(x/sqrt(1+m*x^2))
4076 ;;
4077 (defprop %inverse_jacobi_sd
4078 ((x m)
4079 ;; Whittaker and Watson, example in 22.122
4080 ;; inverse_jacobi_sd(u,m) = integrate(1/sqrt(1-(1-m)*t^2)/sqrt(1+m*t^2), t, 0, u)
4081 ;; -> 1/sqrt(1-(1-m)*x^2)/sqrt(1+m*x^2)
4082 ((mtimes)
4083 ((mexpt)
4084 ((mplus) 1 ((mtimes) ((mplus) -1 m) ((mexpt) x 2)))
4085 ((rat) -1 2))
4086 ((mexpt) ((mplus) 1 ((mtimes) m ((mexpt) x 2)))
4087 ((rat) -1 2)))
4088 ;; wrt m
4089 ; ((%derivative) ((%inverse_jacobi_sd) x m) m 1)
4090 nil)
4091 grad)
4092
4093 (def-simplifier inverse_jacobi_sd (u m)
4094 (cond ((or (float-numerical-eval-p u m)
4095 (complex-float-numerical-eval-p u m)
4096 (bigfloat-numerical-eval-p u m)
4097 (complex-bigfloat-numerical-eval-p u m))
4098 (ftake '%inverse_jacobi_sn
4099 ($rectform (div u (power (add 1 (mul m (mul u u))) 1//2)))
4100 m))
4101 ((zerop1 u)
4102 0)
4103 ((eql 0 ($ratsimp (sub u (div 1 (power (sub 1 m) 1//2)))))
4104 ;; inverse_jacobi_sd(1/sqrt(1-m), m) = elliptic_kc(m)
4105 ;;
4106 ;; We can see this from inverse_jacobi_sd(x,m) =
4107 ;; inverse_jacobi_sn(x/sqrt(1+m*x^2), m). So
4108 ;; inverse_jacobi_sd(1/sqrt(1-m),m) = inverse_jacobi_sn(1,m)
4109 (ftake '%elliptic_kc m))
4110 ((and (eq $triginverses '$all)
4111 (listp u)
4112 (eq (caar u) '%jacobi_sd)
4113 (alike1 (third u) m))
4114 ;; inverse_jacobi_sd(sd(u)) = u
4115 (second u))
4116 (t
4117 ;; Nothing to do
4118 (give-up))))
4119
4120 ;; inverse_jacobi_cs(x)
4121 ;;
4122 ;; Let u = inverse_jacobi_cs(x). Then jacobi_cs(u) = x or
4123 ;; 1/x = 1/jacobi_cs(u) = jacobi_sc(u)
4124 ;;
4125 ;; u = inverse_sc(1/x)
4126 ;;
4127 (defprop %inverse_jacobi_cs
4128 ((x m)
4129 ;; Whittaker and Watson, example in 22.122
4130 ;; inverse_jacobi_cs(u,m) = integrate(1/sqrt(t^2+1)/sqrt(t^2+(1-m)), t, u, inf)
4131 ;; -> -1/sqrt(x^2+1)/sqrt(x^2+(1-m))
4132 ((mtimes) -1
4133 ((mexpt) ((mplus) 1 ((mexpt simp ratsimp) x 2))
4134 ((rat) -1 2))
4135 ((mexpt) ((mplus) 1
4136 ((mtimes simp ratsimp) -1 m)
4137 ((mexpt simp ratsimp) x 2))
4138 ((rat) -1 2)))
4139 ;; wrt m
4140 ; ((%derivative) ((%inverse_jacobi_cs) x m) m 1)
4141 nil)
4142 grad)
4143
4144 (def-simplifier inverse_jacobi_cs (u m)
4145 (cond ((or (float-numerical-eval-p u m)
4146 (complex-float-numerical-eval-p u m)
4147 (bigfloat-numerical-eval-p u m)
4148 (complex-bigfloat-numerical-eval-p u m))
4149 (ftake '%inverse_jacobi_sc ($rectform (div 1 u)) m))
4150 ((zerop1 u)
4151 (ftake '%elliptic_kc m))
4152 (t
4153 ;; Nothing to do
4154 (give-up))))
4155
4156 ;; inverse_jacobi_cd(x)
4157 ;;
4158 ;; Let u = inverse_jacobi_cd(x). Then jacobi_cd(u) = x or
4159 ;; x = jacobi_cn(u)/jacobi_dn(u)
4160 ;;
4161 ;; x^2 = cn^2/dn^2
4162 ;; = (1-sn^2)/(1-m*sn^2)
4163 ;;
4164 ;; or
4165 ;;
4166 ;; sn^2 = (1-x^2)/(1-m*x^2)
4167 ;;
4168 ;; sn(u) = sqrt(1-x^2)/sqrt(1-m*x^2)
4169 ;;
4170 ;; u = inverse_sn(sqrt(1-x^2)/sqrt(1-m*x^2))
4171 ;;
4172 (defprop %inverse_jacobi_cd
4173 ((x m)
4174 ;; Whittaker and Watson, example in 22.122
4175 ;; inverse_jacobi_cd(u,m) = integrate(1/sqrt(1-t^2)/sqrt(1-m*t^2), t, u, 1)
4176 ;; -> -1/sqrt(1-x^2)/sqrt(1-m*x^2)
4177 ((mtimes) -1
4178 ((mexpt)
4179 ((mplus) 1 ((mtimes) -1 ((mexpt) x 2)))
4180 ((rat) -1 2))
4181 ((mexpt)
4182 ((mplus) 1 ((mtimes) -1 m ((mexpt) x 2)))
4183 ((rat) -1 2)))
4184 ;; wrt m
4185 ; ((%derivative) ((%inverse_jacobi_cd) x m) m 1)
4186 nil)
4187 grad)
4188
4189 (def-simplifier inverse_jacobi_cd (u m)
4190 (cond ((or (complex-float-numerical-eval-p u m)
4191 (complex-bigfloat-numerical-eval-p u m))
4192 (let (($numer t))
4193 (ftake '%inverse_jacobi_sn
4194 ($rectform (div (power (mul (sub 1 u) (add 1 u)) 1//2)
4195 (power (sub 1 (mul m (mul u u))) 1//2)))
4196 m)))
4197 ((onep1 u)
4198 0)
4199 ((zerop1 u)
4200 (ftake '%elliptic_kc m))
4201 ((and (eq $triginverses '$all)
4202 (listp u)
4203 (eq (caar u) '%jacobi_cd)
4204 (alike1 (third u) m))
4205 ;; inverse_jacobi_cd(cd(u)) = u
4206 (second u))
4207 (t
4208 ;; Nothing to do
4209 (give-up))))
4210
4211 ;; inverse_jacobi_ds(x)
4212 ;;
4213 ;; Let u = inverse_jacobi_ds(x). Then jacobi_ds(u) = x or
4214 ;; 1/x = 1/jacobi_ds(u) = jacobi_sd(u)
4215 ;;
4216 ;; u = inverse_sd(1/x)
4217 ;;
4218 (defprop %inverse_jacobi_ds
4219 ((x m)
4220 ;; Whittaker and Watson, example in 22.122
4221 ;; inverse_jacobi_ds(u,m) = integrate(1/sqrt(t^2-(1-m))/sqrt(t^2+m), t, u, inf)
4222 ;; -> -1/sqrt(x^2-(1-m))/sqrt(x^2+m)
4223 ((mtimes) -1
4224 ((mexpt)
4225 ((mplus) -1 m ((mexpt simp ratsimp) x 2))
4226 ((rat) -1 2))
4227 ((mexpt)
4228 ((mplus) m ((mexpt simp ratsimp) x 2))
4229 ((rat) -1 2)))
4230 ;; wrt m
4231 ; ((%derivative) ((%inverse_jacobi_ds) x m) m 1)
4232 nil)
4233 grad)
4234
4235 (def-simplifier inverse_jacobi_ds (u m)
4236 (cond ((or (float-numerical-eval-p u m)
4237 (complex-float-numerical-eval-p u m)
4238 (bigfloat-numerical-eval-p u m)
4239 (complex-bigfloat-numerical-eval-p u m))
4240 (ftake '%inverse_jacobi_sd ($rectform (div 1 u)) m))
4241 ((and $trigsign (mminusp* u))
4242 (neg (ftake* '%inverse_jacobi_ds (neg u) m)))
4243 ((eql 0 ($ratsimp (sub u (power (sub 1 m) 1//2))))
4244 ;; inverse_jacobi_ds(sqrt(1-m),m) = elliptic_kc(m)
4245 ;;
4246 ;; Since inverse_jacobi_ds(sqrt(1-m), m) =
4247 ;; inverse_jacobi_sd(1/sqrt(1-m),m). And we know from
4248 ;; above that this is elliptic_kc(m)
4249 (ftake '%elliptic_kc m))
4250 ((and (eq $triginverses '$all)
4251 (listp u)
4252 (eq (caar u) '%jacobi_ds)
4253 (alike1 (third u) m))
4254 ;; inverse_jacobi_ds(ds(u)) = u
4255 (second u))
4256 (t
4257 ;; Nothing to do
4258 (give-up))))
4259
4260
4261 ;; inverse_jacobi_dc(x)
4262 ;;
4263 ;; Let u = inverse_jacobi_dc(x). Then jacobi_dc(u) = x or
4264 ;; 1/x = 1/jacobi_dc(u) = jacobi_cd(u)
4265 ;;
4266 ;; u = inverse_cd(1/x)
4267 ;;
4268 (defprop %inverse_jacobi_dc
4269 ((x m)
4270 ;; Note: Whittaker and Watson, example in 22.122 says
4271 ;; inverse_jacobi_dc(u,m) = integrate(1/sqrt(t^2-1)/sqrt(t^2-m),
4272 ;; t, u, 1) but that seems wrong. A&S 17.4.47 says
4273 ;; integrate(1/sqrt(t^2-1)/sqrt(t^2-m), t, a, u) =
4274 ;; inverse_jacobi_cd(x,m). Lawden 3.2.8 says the same.
4275 ;; functions.wolfram.com says the derivative is
4276 ;; 1/sqrt(t^2-1)/sqrt(t^2-m).
4277 ((mtimes)
4278 ((mexpt)
4279 ((mplus) -1 ((mexpt simp ratsimp) x 2))
4280 ((rat) -1 2))
4281 ((mexpt)
4282 ((mplus)
4283 ((mtimes simp ratsimp) -1 m)
4284 ((mexpt simp ratsimp) x 2))
4285 ((rat) -1 2)))
4286 ;; wrt m
4287 ; ((%derivative) ((%inverse_jacobi_dc) x m) m 1)
4288 nil)
4289 grad)
4290
4291 (def-simplifier inverse_jacobi_dc (u m)
4292 (cond ((or (complex-float-numerical-eval-p u m)
4293 (complex-bigfloat-numerical-eval-p u m))
4294 (ftake '%inverse_jacobi_cd ($rectform (div 1 u)) m))
4295 ((onep1 u)
4296 0)
4297 ((and (eq $triginverses '$all)
4298 (listp u)
4299 (eq (caar u) '%jacobi_dc)
4300 (alike1 (third u) m))
4301 ;; inverse_jacobi_dc(dc(u)) = u
4302 (second u))
4303 (t
4304 ;; Nothing to do
4305 (give-up))))
4306
4307 ;; Convert an inverse Jacobian function into the equivalent elliptic
4308 ;; integral F.
4309 ;;
4310 ;; See A&S 17.4.41-17.4.52.
4311 (defun make-elliptic-f (e)
4312 (cond ((atom e)
4313 e)
4314 ((member (caar e) '(%inverse_jacobi_sc %inverse_jacobi_cs
4315 %inverse_jacobi_nd %inverse_jacobi_dn
4316 %inverse_jacobi_sn %inverse_jacobi_cd
4317 %inverse_jacobi_dc %inverse_jacobi_ns
4318 %inverse_jacobi_nc %inverse_jacobi_ds
4319 %inverse_jacobi_sd %inverse_jacobi_cn))
4320 ;; We have some inverse Jacobi function. Convert it to the F form.
4321 (destructuring-bind ((fn &rest ops) u m)
4322 e
4323 (declare (ignore ops))
4324 (ecase fn
4325 (%inverse_jacobi_sc
4326 ;; A&S 17.4.41
4327 (ftake '%elliptic_f (ftake '%atan u) m))
4328 (%inverse_jacobi_cs
4329 ;; A&S 17.4.42
4330 (ftake '%elliptic_f (ftake '%atan (div 1 u)) m))
4331 (%inverse_jacobi_nd
4332 ;; A&S 17.4.43
4333 (ftake '%elliptic_f
4334 (ftake '%asin
4335 (mul (power m -1//2)
4336 (div 1 u)
4337 (power (add -1 (mul u u))
4338 1//2)))
4339 m))
4340 (%inverse_jacobi_dn
4341 ;; A&S 17.4.44
4342 (ftake '%elliptic_f
4343 (ftake '%asin (mul
4344 (power m -1//2)
4345 (power (sub 1 (power u 2)) 1//2)))
4346 m))
4347 (%inverse_jacobi_sn
4348 ;; A&S 17.4.45
4349 (ftake '%elliptic_f (ftake '%asin u) m))
4350 (%inverse_jacobi_cd
4351 ;; A&S 17.4.46
4352 (ftake '%elliptic_f
4353 (ftake '%asin
4354 (power (mul (sub 1 (mul u u))
4355 (sub 1 (mul m u u)))
4356 1//2))
4357 m))
4358 (%inverse_jacobi_dc
4359 ;; A&S 17.4.47
4360 (ftake '%elliptic_f
4361 (ftake '%asin
4362 (power (mul (sub (mul u u) 1)
4363 (sub (mul u u) m))
4364 1//2))
4365 m))
4366 (%inverse_jacobi_ns
4367 ;; A&S 17.4.48
4368 (ftake '%elliptic_f (ftake '%asin (div 1 u)) m))
4369 (%inverse_jacobi_nc
4370 ;; A&S 17.4.49
4371 (ftake '%elliptic_f (ftake '%acos (div 1 u)) m))
4372 (%inverse_jacobi_ds
4373 ;; A&S 17.4.50
4374 (ftake '%elliptic_f
4375 (ftake '%asin
4376 (power (add m (mul u u))
4377 1//2)
4378 m)))
4379 (%inverse_jacobi_sd
4380 ;; A&S 17.4.51
4381 (ftake '%elliptic_f
4382 (ftake '%asin
4383 (div u
4384 (power (add 1 (mul m u u))
4385 1//2)))
4386 m))
4387 (%inverse_jacobi_cn
4388 ;; A&S 17.4.52
4389 (ftake '%elliptic_f (ftake '%acos u) m)))))
4390 (t
4391 (recur-apply #'make-elliptic-f e))))
4392
4393 (defmfun $make_elliptic_f (e)
4394 (if (atom e)
4395 e
4396 (simplify (make-elliptic-f e))))
4397
4398 (defun make-elliptic-e (e)
4399 (cond ((atom e) e)
4400 ((eq (caar e) '$elliptic_eu)
4401 (destructuring-bind ((ffun &rest ops) u m) e
4402 (declare (ignore ffun ops))
4403 (ftake '%elliptic_e (ftake '%asin (ftake '%jacobi_sn u m)) m)))
4404 (t
4405 (recur-apply #'make-elliptic-e e))))
4406
4407 (defmfun $make_elliptic_e (e)
4408 (if (atom e)
4409 e
4410 (simplify (make-elliptic-e e))))
4411
4412
4413 ;; Eu(u,m) = integrate(jacobi_dn(v,m)^2,v,0,u)
4414 ;; = integrate(sqrt((1-m*t^2)/(1-t^2)),t,0,jacobi_sn(u,m))
4415 ;;
4416 ;; Eu(u,m) = E(am(u),m)
4417 ;;
4418 ;; where E(u,m) is elliptic-e above.
4419 ;;
4420 ;; Checks.
4421 ;; Lawden gives the following relationships
4422 ;;
4423 ;; E(u+v) = E(u) + E(v) - m*sn(u)*sn(v)*sn(u+v)
4424 ;; E(u,0) = u, E(u,1) = tanh u
4425 ;;
4426 ;; E(i*u,k) = i*sc(u,k')*dn(u,k') - i*E(u,k') + i*u
4427 ;;
4428 ;; E(2*i*K') = 2*i*(K'-E')
4429 ;;
4430 ;; E(u + 2*i*K') = E(u) + 2*i*(K' - E')
4431 ;;
4432 ;; E(u+K) = E(u) + E - k^2*sn(u)*cd(u)
4433 (defun elliptic-eu (u m)
4434 (cond ((realp u)
4435 ;; E(u + 2*n*K) = E(u) + 2*n*E
4436 (let ((ell-k (to (elliptic-k m)))
4437 (ell-e (elliptic-ec m)))
4438 (multiple-value-bind (n u-rem)
4439 (floor u (* 2 ell-k))
4440 ;; 0 <= u-rem < 2*K
4441 (+ (* 2 n ell-e)
4442 (cond ((>= u-rem ell-k)
4443 ;; 0 <= u-rem < K so
4444 ;; E(u + K) = E(u) + E - m*sn(u)*cd(u)
4445 (let ((u-k (- u ell-k)))
4446 (- (+ (elliptic-e (cl:asin (bigfloat::sn u-k m)) m)
4447 ell-e)
4448 (/ (* m (bigfloat::sn u-k m) (bigfloat::cn u-k m))
4449 (bigfloat::dn u-k m)))))
4450 (t
4451 (elliptic-e (cl:asin (bigfloat::sn u m)) m)))))))
4452 ((complexp u)
4453 ;; From Lawden:
4454 ;;
4455 ;; E(u+i*v, m) = E(u,m) -i*E(v,m') + i*v + i*sc(v,m')*dn(v,m')
4456 ;; -i*m*sn(u,m)*sc(v,m')*sn(u+i*v,m)
4457 ;;
4458 (let ((u-r (realpart u))
4459 (u-i (imagpart u))
4460 (m1 (- 1 m)))
4461 (+ (elliptic-eu u-r m)
4462 (* #c(0 1)
4463 (- (+ u-i
4464 (/ (* (bigfloat::sn u-i m1) (bigfloat::dn u-i m1))
4465 (bigfloat::cn u-i m1)))
4466 (+ (elliptic-eu u-i m1)
4467 (/ (* m (bigfloat::sn u-r m) (bigfloat::sn u-i m1) (bigfloat::sn u m))
4468 (bigfloat::cn u-i m1))))))))))
4469
4470 (defprop $elliptic_eu
4471 ((u m)
4472 ((mexpt) ((%jacobi_dn) u m) 2)
4473 ;; wrt m
4474 )
4475 grad)
4476
4477 (def-simplifier elliptic_eu (u m)
4478 (cond
4479 ;; as it stands, ELLIPTIC-EU can't handle bigfloats or complex bigfloats,
4480 ;; so handle only floats and complex floats here.
4481 ((float-numerical-eval-p u m)
4482 (elliptic-eu ($float u) ($float m)))
4483 ((complex-float-numerical-eval-p u m)
4484 (let ((u-r ($realpart u))
4485 (u-i ($imagpart u))
4486 (m ($float m)))
4487 (complexify (elliptic-eu (complex u-r u-i) m))))
4488 (t
4489 (give-up))))
4490
4491 (def-simplifier jacobi_am (u m)
4492 (cond
4493 ;; as it stands, BIGFLOAT::SN can't handle bigfloats or complex bigfloats,
4494 ;; so handle only floats and complex floats here.
4495 ((float-numerical-eval-p u m)
4496 (cl:asin (bigfloat::sn ($float u) ($float m))))
4497 ((complex-float-numerical-eval-p u m)
4498 (let ((u-r ($realpart ($float u)))
4499 (u-i ($imagpart ($float u)))
4500 (m ($float m)))
4501 (complexify (cl:asin (bigfloat::sn (complex u-r u-i) m)))))
4502 (t
4503 ;; Nothing to do
4504 (give-up))))
4505
4506 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
4507 ;; Integrals. At present with respect to first argument only.
4508 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
4509
4510 ;; A&S 16.24.1: integrate(jacobi_sn(u,m),u)
4511 ;; = log(jacobi_dn(u,m)-sqrt(m)*jacobi_cn(u,m))/sqrt(m)
4512 (defprop %jacobi_sn
4513 ((u m)
4514 ((mtimes simp) ((mexpt simp) m ((rat simp) -1 2))
4515 ((%log simp)
4516 ((mplus simp)
4517 ((mtimes simp) -1 ((mexpt simp) m ((rat simp) 1 2))
4518 ((%jacobi_cn simp) u m))
4519 ((%jacobi_dn simp) u m))))
4520 nil)
4521 integral)
4522
4523 ;; A&S 16.24.2: integrate(jacobi_cn(u,m),u) = acos(jacobi_dn(u,m))/sqrt(m)
4524 (defprop %jacobi_cn
4525 ((u m)
4526 ((mtimes simp) ((mexpt simp) m ((rat simp) -1 2))
4527 ((%acos simp) ((%jacobi_dn simp) u m)))
4528 nil)
4529 integral)
4530
4531 ;; A&S 16.24.3: integrate(jacobi_dn(u,m),u) = asin(jacobi_sn(u,m))
4532 (defprop %jacobi_dn
4533 ((u m)
4534 ((%asin simp) ((%jacobi_sn simp) u m))
4535 nil)
4536 integral)
4537
4538 ;; A&S 16.24.4: integrate(jacobi_cd(u,m),u)
4539 ;; = log(jacobi_nd(u,m)+sqrt(m)*jacobi_sd(u,m))/sqrt(m)
4540 (defprop %jacobi_cd
4541 ((u m)
4542 ((mtimes simp) ((mexpt simp) m ((rat simp) -1 2))
4543 ((%log simp)
4544 ((mplus simp) ((%jacobi_nd simp) u m)
4545 ((mtimes simp) ((mexpt simp) m ((rat simp) 1 2))
4546 ((%jacobi_sd simp) u m)))))
4547 nil)
4548 integral)
4549
4550 ;; integrate(jacobi_sd(u,m),u)
4551 ;;
4552 ;; A&S 16.24.5 gives
4553 ;; asin(-sqrt(m)*jacobi_cd(u,m))/sqrt(m*m_1), where m + m_1 = 1
4554 ;; but this does not pass some simple tests.
4555 ;;
4556 ;; functions.wolfram.com 09.35.21.001.01 gives
4557 ;; -asin(sqrt(m)*jacobi_cd(u,m))*sqrt(1-m*jacobi_cd(u,m)^2)*jacobi_dn(u,m)/((1-m)*sqrt(m))
4558 ;; and this does pass.
4559 (defprop %jacobi_sd
4560 ((u m)
4561 ((mtimes simp) -1
4562 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m)) -1)
4563 ((mexpt simp) m ((rat simp) -1 2))
4564 ((mexpt simp)
4565 ((mplus simp) 1
4566 ((mtimes simp) -1 $m ((mexpt simp) ((%jacobi_cd simp) u m) 2)))
4567 ((rat simp) 1 2))
4568 ((%jacobi_dn simp) u m)
4569 ((%asin simp)
4570 ((mtimes simp) ((mexpt simp) m ((rat simp) 1 2))
4571 ((%jacobi_cd simp) u m))))
4572 nil)
4573 integral)
4574
4575 ;; integrate(jacobi_nd(u,m),u)
4576 ;;
4577 ;; A&S 16.24.6 gives
4578 ;; acos(jacobi_cd(u,m))/sqrt(m_1), where m + m_1 = 1
4579 ;; but this does not pass some simple tests.
4580 ;;
4581 ;; functions.wolfram.com 09.32.21.0001.01 gives
4582 ;; sqrt(1-jacobi_cd(u,m)^2)*acos(jacobi_cd(u,m))/((1-m)*jacobi_sd(u,m))
4583 ;; and this does pass.
4584 (defprop %jacobi_nd
4585 ((u m)
4586 ((mtimes simp) ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m)) -1)
4587 ((mexpt simp)
4588 ((mplus simp) 1
4589 ((mtimes simp) -1 ((mexpt simp) ((%jacobi_cd simp) u m) 2)))
4590 ((rat simp) 1 2))
4591 ((mexpt simp) ((%jacobi_sd simp) u m) -1)
4592 ((%acos simp) ((%jacobi_cd simp) u m)))
4593 nil)
4594 integral)
4595
4596 ;; A&S 16.24.7: integrate(jacobi_dc(u,m),u) = log(jacobi_nc(u,m)+jacobi_sc(u,m))
4597 (defprop %jacobi_dc
4598 ((u m)
4599 ((%log simp) ((mplus simp) ((%jacobi_nc simp) u m) ((%jacobi_sc simp) u m)))
4600 nil)
4601 integral)
4602
4603 ;; A&S 16.24.8: integrate(jacobi_nc(u,m),u)
4604 ;; = log(jacobi_dc(u,m)+sqrt(m_1)*jacobi_sc(u,m))/sqrt(m_1), where m + m_1 = 1
4605 (defprop %jacobi_nc
4606 ((u m)
4607 ((mtimes simp)
4608 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m))
4609 ((rat simp) -1 2))
4610 ((%log simp)
4611 ((mplus simp) ((%jacobi_dc simp) u m)
4612 ((mtimes simp)
4613 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m))
4614 ((rat simp) 1 2))
4615 ((%jacobi_sc simp) u m)))))
4616 nil)
4617 integral)
4618
4619 ;; A&S 16.24.9: integrate(jacobi_sc(u,m),u)
4620 ;; = log(jacobi_dc(u,m)+sqrt(m_1)*jacobi_nc(u,m))/sqrt(m_1), where m + m_1 = 1
4621 (defprop %jacobi_sc
4622 ((u m)
4623 ((mtimes simp)
4624 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m))
4625 ((rat simp) -1 2))
4626 ((%log simp)
4627 ((mplus simp) ((%jacobi_dc simp) u m)
4628 ((mtimes simp)
4629 ((mexpt simp) ((mplus simp) 1 ((mtimes simp) -1 m))
4630 ((rat simp) 1 2))
4631 ((%jacobi_nc simp) u m)))))
4632 nil)
4633 integral)
4634
4635 ;; A&S 16.24.10: integrate(jacobi_ns(u,m),u)
4636 ;; = log(jacobi_ds(u,m)-jacobi_cs(u,m))
4637 (defprop %jacobi_ns
4638 ((u m)
4639 ((%log simp)
4640 ((mplus simp) ((mtimes simp) -1 ((%jacobi_cs simp) u m))
4641 ((%jacobi_ds simp) u m)))
4642 nil)
4643 integral)
4644
4645 ;; integrate(jacobi_ds(u,m),u)
4646 ;;
4647 ;; A&S 16.24.11 gives
4648 ;; log(jacobi_ds(u,m)-jacobi_cs(u,m))
4649 ;; but this does not pass some simple tests.
4650 ;;
4651 ;; functions.wolfram.com 09.30.21.0001.01 gives
4652 ;; log((1-jacobi_cn(u,m))/jacobi_sn(u,m))
4653 ;;
4654 (defprop %jacobi_ds
4655 ((u m)
4656 ((%log simp)
4657 ((mtimes simp)
4658 ((mplus simp) 1 ((mtimes simp) -1 ((%jacobi_cn simp) u m)))
4659 ((mexpt simp) ((%jacobi_sn simp) u m) -1)))
4660 nil)
4661 integral)
4662
4663 ;; A&S 16.24.12: integrate(jacobi_cs(u,m),u) = log(jacobi_ns(u,m)-jacobi_ds(u,m))
4664 (defprop %jacobi_cs
4665 ((u m)
4666 ((%log simp)
4667 ((mplus simp) ((mtimes simp) -1 ((%jacobi_ds simp) u m))
4668 ((%jacobi_ns simp) u m)))
4669 nil)
4670 integral)
4671
4672 ;; functions.wolfram.com 09.48.21.0001.01
4673 ;; integrate(inverse_jacobi_sn(u,m),u) =
4674 ;; inverse_jacobi_sn(u,m)*u
4675 ;; - log( jacobi_dn(inverse_jacobi_sn(u,m),m)
4676 ;; -sqrt(m)*jacobi_cn(inverse_jacobi_sn(u,m),m)) / sqrt(m)
4677 (defprop %inverse_jacobi_sn
4678 ((u m)
4679 ((mplus simp) ((mtimes simp) u ((%inverse_jacobi_sn simp) u m))
4680 ((mtimes simp) -1 ((mexpt simp) m ((rat simp) -1 2))
4681 ((%log simp)
4682 ((mplus simp)
4683 ((mtimes simp) -1 ((mexpt simp) m ((rat simp) 1 2))
4684 ((%jacobi_cn simp) ((%inverse_jacobi_sn simp) u m) m))
4685 ((%jacobi_dn simp) ((%inverse_jacobi_sn simp) u m) m)))))
4686 nil)
4687 integral)
4688
4689 ;; functions.wolfram.com 09.38.21.0001.01
4690 ;; integrate(inverse_jacobi_cn(u,m),u) =
4691 ;; u*inverse_jacobi_cn(u,m)
4692 ;; -%i*log(%i*jacobi_dn(inverse_jacobi_cn(u,m),m)/sqrt(m)
4693 ;; -jacobi_sn(inverse_jacobi_cn(u,m),m))
4694 ;; /sqrt(m)
4695 (defprop %inverse_jacobi_cn
4696 ((u m)
4697 ((mplus simp) ((mtimes simp) u ((%inverse_jacobi_cn simp) u m))
4698 ((mtimes simp) -1 $%i ((mexpt simp) m ((rat simp) -1 2))
4699 ((%log simp)
4700 ((mplus simp)
4701 ((mtimes simp) $%i ((mexpt simp) m ((rat simp) -1 2))
4702 ((%jacobi_dn simp) ((%inverse_jacobi_cn simp) u m) m))
4703 ((mtimes simp) -1
4704 ((%jacobi_sn simp) ((%inverse_jacobi_cn simp) u m) m))))))
4705 nil)
4706 integral)
4707
4708 ;; functions.wolfram.com 09.41.21.0001.01
4709 ;; integrate(inverse_jacobi_dn(u,m),u) =
4710 ;; u*inverse_jacobi_dn(u,m)
4711 ;; - %i*log(%i*jacobi_cn(inverse_jacobi_dn(u,m),m)
4712 ;; +jacobi_sn(inverse_jacobi_dn(u,m),m))
4713 (defprop %inverse_jacobi_dn
4714 ((u m)
4715 ((mplus simp) ((mtimes simp) u ((%inverse_jacobi_dn simp) u m))
4716 ((mtimes simp) -1 $%i
4717 ((%log simp)
4718 ((mplus simp)
4719 ((mtimes simp) $%i
4720 ((%jacobi_cn simp) ((%inverse_jacobi_dn simp) u m) m))
4721 ((%jacobi_sn simp) ((%inverse_jacobi_dn simp) u m) m)))))
4722 nil)
4723 integral)
4724
4725 \f
4726 ;; Real and imaginary part for Jacobi elliptic functions.
4727 (defprop %jacobi_sn risplit-sn-cn-dn risplit-function)
4728 (defprop %jacobi_cn risplit-sn-cn-dn risplit-function)
4729 (defprop %jacobi_dn risplit-sn-cn-dn risplit-function)
4730
4731 (defun risplit-sn-cn-dn (expr)
4732 (let* ((arg (second expr))
4733 (param (third expr)))
4734 ;; We only split on the argument, not the order
4735 (destructuring-bind (arg-r . arg-i)
4736 (risplit arg)
4737 (cond ((=0 arg-i)
4738 ;; Pure real
4739 (cons (take (first expr) arg-r param)
4740 0))
4741 (t
4742 (let* ((s (ftake '%jacobi_sn arg-r param))
4743 (c (ftake '%jacobi_cn arg-r param))
4744 (d (ftake '%jacobi_dn arg-r param))
4745 (s1 (ftake '%jacobi_sn arg-i (sub 1 param)))
4746 (c1 (ftake '%jacobi_cn arg-i (sub 1 param)))
4747 (d1 (ftake '%jacobi_dn arg-i (sub 1 param)))
4748 (den (add (mul c1 c1)
4749 (mul param
4750 (mul (mul s s)
4751 (mul s1 s1))))))
4752 ;; Let s = jacobi_sn(x,m)
4753 ;; c = jacobi_cn(x,m)
4754 ;; d = jacobi_dn(x,m)
4755 ;; s1 = jacobi_sn(y,1-m)
4756 ;; c1 = jacobi_cn(y,1-m)
4757 ;; d1 = jacobi_dn(y,1-m)
4758 (case (caar expr)
4759 (%jacobi_sn
4760 ;; A&S 16.21.1
4761 ;; jacobi_sn(x+%i*y,m) =
4762 ;;
4763 ;; s*d1 + %i*c*d*s1*c1
4764 ;; -------------------
4765 ;; c1^2+m*s^2*s1^2
4766 ;;
4767 (cons (div (mul s d1) den)
4768 (div (mul c (mul d (mul s1 c1)))
4769 den)))
4770 (%jacobi_cn
4771 ;; A&S 16.21.2
4772 ;;
4773 ;; cn(x+%i_y, m) =
4774 ;;
4775 ;; c*c1 - %i*s*d*s1*d1
4776 ;; -------------------
4777 ;; c1^2+m*s^2*s1^2
4778 (cons (div (mul c c1) den)
4779 (div (mul -1
4780 (mul s (mul d (mul s1 d1))))
4781 den)))
4782 (%jacobi_dn
4783 ;; A&S 16.21.3
4784 ;;
4785 ;; dn(x+%i_y, m) =
4786 ;;
4787 ;; d*c1*d1 - %i*m*s*c*s1
4788 ;; ---------------------
4789 ;; c1^2+m*s^2*s1^2
4790 (cons (div (mul d (mul c1 d1))
4791 den)
4792 (div (mul -1 (mul param (mul s (mul c s1))))
4793 den))))))))))
4794
Maxima fork w/ changes for Cygwin