tests/rtest_log.mac

   1 /*******************************************************************************
   2  *
   3  *          Log function
   4  *
   5  * The examples show what a Maxima user can expect from the log function.
   6  * Examples are taken from functions.wolfram.com.
   7  *
   8  ******************************************************************************/
   9
  10 /* ---------- Initialization ------------------------------------------------ */
  11
  12 kill(all);
  13 done;
  14
  15 declare(z,complex, n,integer);
  16 done;
  17
  18 assume(xp>0, notequal(v,0));
  19 [xp>0, notequal(v,0)];
  20
  21 (load(functs),done);
  22 done;
  23
  24 (domain:complex, logexpand:false, done);
  25 done;
  26
  27 /* ----- Specific values ---------------------------------------------------- */
  28
  29 log([exp(1), exp(2), exp(-2), exp(n), exp(n+2)]);
  30 [1, 2, -2, n, n+2];
  31
  32 limit(log(x),x,0,plus);
  33 minf;
  34
  35 log([1, 1.0, 1.0b0]);
  36 [0, 0.0, 0.0b0];
  37
  38 log(-1),lognegint;
  39 %i*%pi;
  40 log(-1.0)-float(%i*%pi);
  41 0.0;
  42 log(-1.0b0)-bfloat(%i*%pi);
  43 0.0b0;
  44
  45 log(%e);
  46 1;
  47
  48 /* For the following simplifications Maxima needs help */
  49
  50 log(-%e),lognegint, logexpand:all;
  51 1+%i*%pi;
  52 log(-%e),rectform;
  53 1+%i*%pi;
  54
  55 log(%i),rectform;
  56 %i*%pi/2;
  57
  58 log(-%i),rectform;
  59 -%i*%pi/2;
  60
  61 /* ----- Values at infinities ----------------------------------------------- */
  62
  63 /* functions.wolfram.com gives the result infinity for limit(log(minf)) and
  64    limit(log(infinity)).
  65  */
  66
  67 limit(log([inf,minf,infinity]));
  68 [inf, infinity, infinity];
  69
  70 /* ----- Mirror symmetry ---------------------------------------------------- */
  71
  72 conjugate(log(z));
  73 '(conjugate(log(z)));
  74
  75 conjugate(log(x+%i*y));
  76 '(conjugate(log(x+%i*y)));
  77
  78 conjugate(log(xp+%i*y));
  79 log(xp-%i*y);
  80
  81 conjugate(log(10+%i*y));
  82 log(10-%i*y);
  83
  84 conjugate(log(-10+%i*y));
  85 '(conjugate(log(-10+%i*y)));
  86
  87 /* ---- Series representations ---------------------------------------------- */
  88
  89 taylor(log(x),x,x0,2);
  90 log(x0)+(x-x0)/x0-(x-x0)^2/(2*x0^2);
  91
  92 taylor(log(f(x)),x,x0,2);
  93 log(f(x0))+('at('diff(f(x),x,1),x = x0))*(x-x0)/f(x0)
  94           +(f(x0)*('at('diff(f(x),x,2),x = x0))
  95            -('at('diff(f(x),x,1),x = x0))^2)
  96            *(x-x0)^2
  97            /(2*f(x0)^2);
  98
  99 taylor(log(x),x,1,3) - (x-1-(x-1)^2/2+(x-1)^3/3), ratsimp;
 100 0;
 101
 102 taylor(log(1+x),x,0,3);
 103 x-x^2/2+x^3/3;
 104
 105 taylor(log(1+x),x,inf,3);
 106 +log(x)+1/x-1/(2*x^2)+1/(3*x^3);
 107
 108 taylor(log(x),x,0,2);
 109 +log(x);
 110
 111 /* ----- Differential equations --------------------------------------------- */
 112
 113 depends([w,g,h],z);
 114 [w(z), g(z), h(z)];
 115
 116 ode2(z*'diff(w,z)-1,w,z);
 117 w=log(z)+%c;
 118
 119 ic1(%,z=1,w=0);
 120 w=log(z);
 121
 122 ode2(z*'diff(w,z,2)+'diff(w,z),w,z);
 123 w = %k1*log(z) + %k2;
 124
 125 determinant(wronskian([1,log(z)],z));
 126 1/z;
 127
 128 ode2('diff(w,z,2)
 129      +('diff(g(z),z)/g(z)-'diff(g(z),z,2)/'diff(g(z),z))*'diff(w,z),w,z);
 130 w = %k1*log(g(z)) + %k2;
 131
 132 determinant(wronskian([log(g(z)),1],z));
 133 -'diff(g(z),z,1)/g(z)$
 134
 135 /* Maxima can not solve this ode
 136 ode2( 'diff(w,z,2)
 137      + ('diff(g(z),z)/g(z) - 2*'diff(h(z),z)/h(z)
 138                            - 'diff(g(z),z,2)/'diff(g(z),z)) * 'diff(w,z)
 139      +(2*'diff(h(z),z)^2/h(z)^2 + 'diff(g(z),z,2)*'diff(h(z),z)/h(z)/'diff(g(z),z)
 140                                 - 'diff(g(z),z)*'diff(h(z),z)/g(z)/h(z)
 141                                 - 'diff(h(z),z,2)/h(z)) * w,
 142      w,z);
 143 w=%k1*h(z)*log(g(z))+%k2*h(z)$
 144 */
 145
 146 determinant(wronskian([h(z)*log(g(z)), h(z)],z)),ratsimp;
 147 -h(z)^2*'diff(g(z),z,1)/g(z)$
 148
 149 ode2 ('diff(w,z,2)*z^2+(1-2*s)*'diff(w,z,1)*z+s^2*w, w,z)$
 150 w = z^s*(%k2*log(z)+%k1)$
 151
 152 determinant(wronskian([z^s*log(z),z^s],z)), ratsimp;
 153 -z^(2*s-1)$
 154
 155 assume(not(equal(log(s),0)));
 156 [notequal(log(s),0)];
 157 ode2(diff(w,z,2)-2*log(s)*diff(w,z,1)+log(s)^2*w, w,z), radcan;
 158 w = %k2*s^z*z+%k1*s^z$
 159 forget(notequal(log(s),0));
 160 [notequal(log(s),0)];
 161
 162 determinant(wronskian([s^z*z, s^z], z)), ratsimp;
 163 -s^(2*z)$
 164
 165 /* ----- Transformations and argument simplifications ----------------------- */
 166
 167 (reset(domain, logexpand), done);
 168 done$
 169
 170 domain;
 171 real$
 172 logexpand;
 173 true$
 174
 175 /* Simplifications with the standard settings of domain and logexpand */
 176
 177 (expr1: (x-1)^3, expr2: (2*y-3), done);
 178 done$
 179
 180 log(a^b);
 181 b*log(a)$
 182
 183 log(x^a^b);
 184 a^b*log(x)$
 185
 186 log((x^a)^b);
 187 a*b*log(x)$
 188
 189 log(1/x);
 190 -log(x)$
 191
 192 log(1/2);
 193 -log(2)$
 194
 195 log(1/(x-1)^3);
 196 -3*log(x-1)$
 197
 198 log(((x-1)^3)^(2*y-3));
 199 3*(2*y-3)*log(x-1)$
 200
 201 /* Simplification of products with logexpand:all */
 202
 203 log(-z), logexpand:all;
 204 log(-1)+log(z);
 205
 206 log(2/3*a*b), logexpand:all;
 207 log(a)+log(b)+log(2/3)$
 208
 209 log(2/3*a/b), logexpand:all;
 210 log(a)-log(b)+log(2/3)$
 211
 212 log(2/3*(x-1)^3*(2*y-3)), logexpand:all;
 213 3*log(x-1)+log(2*y-3)+log(2/3)$
 214
 215 log(2/3*(x-1)^3/(2*y-3)), logexpand:all;
 216 3*log(x-1)-log(2*y-3)+log(2/3)$
 217
 218 log(2/3*((x-1)^3)^(a/b)*(2*y-3)^(b/a)), logexpand:all;
 219 3*a*log(x-1)/b + b*log(2*y-3)/a+log(2/3)$
 220
 221 /* Simplificaton of products with logexpand:super
 222    In addition log(n/d) -> log(n)-log(d), where n/d is a rational number
 223  */
 224
 225 log(2/3*a*b), logexpand:super;
 226 log(a)+log(b)+log(2)-log(3)$
 227
 228 log(2/3*a/b), logexpand:super;
 229 log(a)-log(b)+log(2)-log(3)$
 230
 231 log(2/3*(x-1)^3*(2*y-3)), logexpand:super;
 232 3*log(x-1)+log(2*y-3)+log(2)-log(3)$
 233
 234 log(2/3*(x-1)^3/(2*y-3)), logexpand:super;
 235 3*log(x-1)-log(2*y-3)+log(2)-log(3)$
 236
 237 log(2/3*((x-1)^3)^(a/b)*(2*y-3)^(b/a)), logexpand:super;
 238 3*a*log(x-1)/b + b*log(2*y-3)/a+log(2)-log(3)$
 239
 240 /* Simplifications with domain:complex */
 241
 242 /* The following examples show that the expressions do not simplify
 243    with domain:complex or logexand:false
 244  */
 245
 246 subst(a^b=1,log(a^b)),domain:complex;
 247 0$
 248 subst(a^b=1,log(a^b)),logexpand:false;
 249 0$
 250
 251 subst(xp^z=1,log(xp^z)),domain:complex;
 252 0;
 253 subst(xp^z=1,log(xp^z)),logexpand:false;
 254 0;
 255
 256 subst(exp(x+%i*y)=1,log(exp(x+%i*y))),domain:complex;
 257 0$
 258 subst(exp(x+%i*y)=1,log(exp(x+%i*y))),logexpand:false;
 259 0$
 260
 261 /* These examples simplifies with domain:complex and logexpand:false */
 262
 263 log(xp^a), domain:complex, logexpand:false;
 264 a*log(xp);
 265
 266 log(exp(x)), domain:complex, logexpand:false;
 267 x;
 268 log(exp(%i*x)), domain:complex, logexpand:false;
 269 %i*x;
 270
 271 log(1/2), domain:complex, logexpand:false;
 272 -log(2);
 273
 274 log(sqrt(z)), domain:complex, logexpand:false;
 275 log(z)/2;
 276
 277 log(z^(1/3)), domain:complex, logexpand:false;
 278 log(z)/3;
 279
 280 log(z^(1/(n^2+1))), domain:complex, logexpand:false;
 281 log(z)/(n^2+1);
 282
 283 /* logexpand:all and logexpand:true overwrites the setting domain:complex */
 284
 285 log(2/3*((x-1)^3)^(a/b)*(2*y-3)^(b/a)), domain:complex, logexpand:all;
 286 3*a*log(x-1)/b + b*log(2*y-3)/a+log(2/3)$
 287
 288 log(2/3*((x-1)^3)^(a/b)*(2*y-3)^(b/a)), domain:complex, logexpand:super;
 289 3*a*log(x-1)/b + b*log(2*y-3)/a+log(2)-log(3)$
 290
 291 /* Bug ID: 3377347 - log(1/(1+%i)) gives error
 292  * Check this case.
 293  */
 294 log(1/(1+%i)),logexpand:false;
 295 -log(1+%i);
 296
 297 /* Bug 2597: logcontract(42+log[x]) was returning 42+log(x) */
 298 is(logcontract(42+log[x]) = 42+log[x]);
 299 true$
 300
 301 /* ----- Complex characteristics -------------------------------------------- */
 302
 303 realpart(log(x+%i*y));
 304 log(y^2+x^2)/2;
 305
 306 realpart(log(z));
 307 log(abs(z));
 308
 309 realpart(log(x));
 310 log(abs(x));
 311
 312 imagpart(log(x+%i*y));
 313 atan2(y,x);
 314
 315 imagpart(log(z));
 316 carg(z);
 317
 318 imagpart(log(x));
 319 atan2(0,x);
 320
 321 cabs(log(x+%i*y));
 322 sqrt(log(y^2+x^2)^2/4+atan2(y,x)^2);
 323
 324 cabs(log(z));
 325 sqrt(log(abs(z))^2+carg(z)^2);
 326
 327 carg(log(x+%i*y));
 328 atan2(atan2(y,x),log(y^2+x^2)/2);
 329
 330 carg(log(z));
 331 atan2(carg(z),log(abs(z)));
 332
 333 /* ----- Differentiation ---------------------------------------------------- */
 334
 335 diff(log(z),z);
 336 1/z;
 337
 338 diff(log(z),z,2);
 339 -1/z^2;
 340
 341 /* ----- Indefinite integration --------------------------------------------- */
 342
 343 integrate(log(z),z);
 344 z*log(z)-z;
 345
 346 integrate(z^(v-1)*log(z),z);
 347 z^v*log(z)/v-z^v/v^2;
 348
 349 integrate(1/z*log((b+a*z)/(d+c*z)),z);
 350 log(z)*log((a*z+b)/(c*z+d))-log(c*z+d)*log(1-(c*z+d)/d)
 351                            +log(a*z+b)*log(1-(a*z+b)/b)-li[2]((c*z+d)/d)
 352                            -log(z)*(log(a*z+b)-log(c*z+d))+li[2]((a*z+b)/b);
 353
 354 assume(c>0,d>0);
 355 [c>0,d>0];
 356 integrate(log(z)/sqrt(c*z^2+d),z);
 357 'integrate(log(z)/sqrt(c*z^2+d),z);
 358 forget(c>0,d>0);
 359 [c>0,d>0];
 360
 361 integrate(log(b+a*z)/(d+c*z),z);
 362 (log(a*z+b)*log((a*c*z+b*c)/(a*d-b*c)+1)+li[2]((a*c*z+b*c)/(a*d-b*c)/-1))/c;
 363
 364 integrate(log(a*z^2+b*z+e)/(d+c*z),z);
 365 'integrate(log(a*z^2+b*z+e)/(d+c*z),z);
 366
 367 integrate(log(z)^2/(1-z),z);
 368 2*(-log(1-z)*log(z)^2/2-li[2](z)*log(z)+li[3](z));
 369
 370 integrate(log(1+z)^2/z,z);
 371 -2*(-log(-z)*log(z+1)^2/2-li[2](z+1)*log(z+1)+li[3](z+1));
 372
 373 integrate(log(a+b*z)^2/(e+f*z),z);
 374 -2*(-log(b*z+a)^2*log(1-f*(b*z+a)/(a*f-b*e))/2
 375    +li[3](f*(b*z+a)/(a*f-b*e))-log(b*z+a)*li[2](f*(b*z+a)/(a*f-b*e)))/f;
 376
 377 integrate(1/z*log((a+b*z)/(c+d*z))^2,z);
 378 'integrate(1/z*log((a+b*z)/(c+d*z))^2,z);
 379
 380 integrate(log(z)*log(1-z)/z,z);
 381 li[3](z)-li[2](z)*log(z);
 382
 383 integrate(log(a+b*z)*log(c+d*z)/(e+f*z),z);
 384 'integrate(log(a+b*z)*log(c+d*z)/(e+f*z),z);
 385
 386 integrate(log(z)*log(z-1)/z,z);
 387 log(z-1)*log(z)^2/2-log(1-z)*log(z)^2/2-li[2](z)*log(z)+li[3](z);
 388
 389 integrate(log(z)^3/(1-z),z);
 390 -6*(log(1-z)*log(z)^3/6+li[2](z)*log(z)^2/2-li[3](z)*log(z)+li[4](z));
 391
 392 integrate(log(z)^2*log(1-z)/z,z);
 393 log(1-z)*log(z)^3/3-2*(log(1-z)*log(z)^3/6+li[2](z)*log(z)^2/2-li[3](z)*log(z)
 394                                           +li[4](z))$
 395
 396 integrate(log(z)^2*log(1-z)/(1-z),z);
 397 'integrate(log(z)^2*log(1-z)/(1-z),z);
 398 /*  'integrate(log(1-z)^2*log(z)/z,z)-log(1-z)^2*log(z)^2/2;  */
 399
 400 /* Hier fehlen noch Integrale */
 401
 402 /* ----- Definite integration ----------------------------------------------- */
 403
 404 integrate(log(t),t,0,1);
 405 -1;
 406
 407 integrate(log(t^2-2*cos(z)*t+1)/t,t,0,1);
 408 'integrate(log(t^2-2*cos(z)*t+1)/t,t,0,1);
 409
 410 /* Lisp error in Maxima 5.24
 411 integrate(log(t)*log(t+1),t,0,1),expand,lognegint;
 412 -2*log(2)-%pi^2/12+2;
 413 */
 414
 415 integrate(log(t+1)*log(1+1/t^2)/t,t,0,inf);
 416 'integrate(log(t+1)*log(1+1/t^2)/t,t,0,inf);
 417
 418 integrate(log(t+1)*log(1+1/t^2),t,0,inf);
 419 'integrate(log(t+1)*log(1+1/t^2),t,0,inf);
 420
 421 integrate(log(a*t+1)*log(z/t^2+1)/t,t,0,inf);
 422 'integrate(log(a*t+1)*log(z/t^2+1)/t,t,0,inf);
 423
 424 integrate(log(t)*log(1-t)^2/t,t,0,1);
 425 'integrate(log(t)*log(1-t)^2/t,t,0,1);
 426
 427 integrate(log(t*(1-t))^4/(1-t),t,0,1/2);
 428 'integrate(log(t*(1-t))^4/(1-t),t,0,1/2);
 429
 430 integrate(log(t*(1-t))^5/(1-t),t,0,1/2);
 431 'integrate(log(t*(1-t))^5/(1-t),t,0,1/2);
 432
 433 integrate(exp(-t)*log(t),t,0,1);
 434 'integrate(exp(-t)*log(t),t,0,1);
 435
 436 /* Hier fehlen noch Integrale */
 437
 438 /* ----- Integral transforms ------------------------------------------------ */
 439
 440 laplace(log(t),t,s);
 441 (-log(s)-%gamma)/s$
 442
 443 /* ----- Representations through more general functions --------------------- */
 444
 445 (z-1)*hgfred([1,1],[2],1-z);
 446 log(z)$
 447
 448 -li[1](1-z);
 449 log(z)$
 450
 451 /* SF bug #3105: "li[s](1) simplifies to zeta(s), but li[s](1.0) doesn't simplify" */
 452
 453 makelist (li[k](1), k, 2, 8);
 454 [%pi^2/6,zeta(3),%pi^4/90,zeta(5),%pi^6/945,zeta(7),%pi^8/9450]$
 455
 456 makelist (li[k](1.0), k, 2, 8);
 457 [1.644934066848226, 1.202056903159594, 1.082323233711138,
 458  1.03692775514337, 1.017343061984449, 1.008349277381923, 1.004077356197944] $
 459
 460 makelist (li[k](1b0), k, 2, 8);
 461 [1.644934066848226b0, 1.202056903159594b0, 1.082323233711138b0,
 462  1.03692775514337b0, 1.017343061984449b0, 1.008349277381923b0, 1.004077356197944b0] $
 463
 464 makelist (li[k](-1), k, 2, 8);
 465 [-%pi^2/12,-(3*zeta(3))/4,-(7*%pi^4)/720,-(15*zeta(5))/16,-(31*%pi^6)/30240,
 466  -(63*zeta(7))/64,-(127*%pi^8)/1209600]$
 467
 468 makelist (li[k](-1.0), k, 2, 8);
 469 [- 0.8224670334241131, - 0.9015426773696957, - 0.9470328294972459,
 470 - 0.9721197704469093, - 0.9855510912974348, - 0.9925938199228304,
 471 - 0.9962330018526477] $
 472
 473 makelist (li[k](-1b0), k, 2, 8);
 474 [- 0.8224670334241131b0, - 0.9015426773696957b0, - 0.9470328294972459b0,
 475 - 0.9721197704469093b0, - 0.9855510912974351b0, - 0.9925938199228304b0,
 476 - 0.9962330018526479b0] $
 477
 478 /* SF bug #3422: "li[2] and li[3] numerical evaluation fails for complex not in rectangular form" */
 479
 480 (reset (fpprec), 0);
 481 0;
 482
 483 li[2]((1 + %i)/2e0);
 484 0.45398526915029558 + 0.64376733288926875*%i;
 485
 486 li[2]((1 + %i)/2b0);
 487 0.45398526915029558b0 + 0.64376733288926875b0*%i;
 488
 489 li[3]((1 + %i)/2e0);
 490 0.48615953708556008 + 0.57007740708876898*%i;
 491
 492 li[3]((1 + %i)/2b0);
 493 0.48615953708556008b0 + 0.57007740708876898b0*%i;