tests/rtest_sqrt.mac

   1 /*******************************************************************************
   2  *
   3  *          Sqrt function
   4  *
   5  * The examples show what a Maxima user can expect from the sqrt function.
   6  * Examples are taken from functions.wolfram.com.
   7  *
   8  ******************************************************************************/
   9
  10 /* ---------- Initialization ------------------------------------------------ */
  11
  12 kill(all);
  13 done;
  14
  15 (reset(domain), reset(radexpand), done);
  16 done;
  17
  18 /* ---------- Specific values ----------------------------------------------- */
  19
  20 sqrt([0, 0.0, 0.0b0]);
  21 [0, 0.0, 0.0b0];
  22
  23 sqrt([-1, -1.0, -1.0b0]);
  24 [%i, 1.0*%i, 1.0b0*%i];
  25
  26 /* Maxima does not simplify -%i -> -(-1)^(3/4) */
  27 sqrt([%i, -%i]);
  28 [(-1)^(1/4), sqrt(-%i)];
  29
  30 /* --- Polarform of specific values --- */
  31
  32 %emode:false;
  33 false;
  34
  35 polarform(sqrt([-1, %i, -%i]));
  36 [%e^(%i*%pi/2), %e^(%i*%pi/4), %e^(-%i*%pi/4)];
  37
  38 %emode:true;
  39 true;
  40
  41 /* ---------- Values at infinities ------------------------------------------ */
  42
  43 /* Only limit can handle infinities. We check this. */
  44
  45 limit([sqrt(inf), sqrt(-inf), sqrt(minf), sqrt(-minf), sqrt(infinity)]);
  46 [inf, infinity, infinity, inf, infinity];
  47
  48 limit(sqrt(x),x,inf);
  49 inf;
  50 limit(sqrt(x),x,-inf);
  51 infinity;
  52 limit(sqrt(x),x,minf);
  53 infinity;
  54 limit(sqrt(x),x,-minf);
  55 inf;
  56
  57 /* ---------- Numerical evaluation ------------------------------------------ */
  58
  59 /* Check only for some values to be sure that it works */
  60
  61 (closeto(value,compare,tol):=
  62   block(
  63     [abse],
  64     abse:abs(value-compare),if(abse<tol) then true else abse),
  65     done);
  66 done;
  67
  68 /* --- Float precision --- */
  69
  70 closeto(
  71   sqrt(0.5),
  72   0.707106781186547524400844362104849039284835937688474036588339868995366239,
  73   1.0e-15);
  74 true;
  75
  76 closeto(
  77   sqrt(0.5+%i),
  78   0.899453719973933636130613791812127819163772491812606935297702716148303720 +
  79   0.555892970251421171992048047897569250691057814997037061577650731037508243*%i,
  80   1.0e-15);
  81 true;
  82
  83 closeto(
  84   sqrt(0.5*%i),
  85   0.500000000000000000000000000000000000000000000000000000000000000000000000 +
  86   0.500000000000000000000000000000000000000000000000000000000000000000000000*%i,
  87   1.0e-15);
  88 true;
  89
  90 closeto(
  91   sqrt(-0.5+%i),
  92   0.555892970251421171992048047897569250691057814997037061577650731037508243 +
  93   0.899453719973933636130613791812127819163772491812606935297702716148303720*%i,
  94   1.0e-15);
  95 true;
  96
  97 closeto(
  98   sqrt(-0.5),
  99   0.707106781186547524400844362104849039284835937688474036588339868995366239*%i,
 100   1.0e-15);
 101 true;
 102
 103 closeto(
 104   sqrt(-0.5-%i),
 105   0.555892970251421171992048047897569250691057814997037061577650731037508243 -
 106   0.899453719973933636130613791812127819163772491812606935297702716148303720*%i,
 107   1.0e-15);
 108 true;
 109
 110 closeto(
 111   sqrt(-0.5*%i),
 112   0.500000000000000000000000000000000000000000000000000000000000000000000000 -
 113   0.500000000000000000000000000000000000000000000000000000000000000000000000*%i,
 114   1.0e-15);
 115 true;
 116
 117 closeto(
 118   sqrt(0.5-%i),
 119   0.899453719973933636130613791812127819163772491812606935297702716148303720 -
 120   0.555892970251421171992048047897569250691057814997037061577650731037508243*%i,
 121   1.0e-15);
 122 true;
 123
 124 /* --- Bigfloat precision with fpprec 72 --- */
 125
 126 (save_fpprec: fpprec, fpprec: 72);
 127 72;
 128
 129 closeto(
 130   sqrt(0.5b0),
 131   0.707106781186547524400844362104849039284835937688474036588339868995366239b0,
 132   1.0b-72);
 133 true;
 134
 135 closeto(
 136   sqrt(0.5b0+%i),
 137   0.899453719973933636130613791812127819163772491812606935297702716148303720b0 +
 138   0.555892970251421171992048047897569250691057814997037061577650731037508243b0*%i,
 139   1.0b-72);
 140 true;
 141
 142 closeto(
 143   sqrt(0.5b0*%i),
 144   0.500000000000000000000000000000000000000000000000000000000000000000000000b0 +
 145   0.500000000000000000000000000000000000000000000000000000000000000000000000b0*%i,
 146   1.0b-72);
 147 true;
 148
 149 closeto(
 150   sqrt(-0.5b0+%i),
 151   0.555892970251421171992048047897569250691057814997037061577650731037508243b0 +
 152   0.899453719973933636130613791812127819163772491812606935297702716148303720b0*%i,
 153   1.0b-72);
 154 true;
 155
 156 closeto(
 157   sqrt(-0.5b0),
 158   0.707106781186547524400844362104849039284835937688474036588339868995366239b0*%i,
 159   1.0b-72);
 160 true;
 161
 162 closeto(
 163   sqrt(-0.5b0-%i),
 164   0.555892970251421171992048047897569250691057814997037061577650731037508243b0 -
 165   0.899453719973933636130613791812127819163772491812606935297702716148303720b0*%i,
 166   1.0b-72);
 167 true;
 168
 169 closeto(
 170   sqrt(-0.5b0*%i),
 171   0.500000000000000000000000000000000000000000000000000000000000000000000000b0 -
 172   0.500000000000000000000000000000000000000000000000000000000000000000000000b0*%i,
 173   1.0b-72);
 174 true;
 175
 176 closeto(
 177   sqrt(0.5b0-%i),
 178   0.899453719973933636130613791812127819163772491812606935297702716148303720b0 -
 179   0.555892970251421171992048047897569250691057814997037061577650731037508243b0*%i,
 180   1.0b-72);
 181 true;
 182
 183 fpprec:save_fpprec;
 184 ''save_fpprec;
 185
 186 /* ---------- Mirror symmetry ----------------------------------------------- */
 187
 188 /* We have no Mirror symmetry on the real negative axis. Maxima knows this. */
 189
 190 conjugate(sqrt(2+%i*y));
 191 sqrt(2-%i*y);
 192
 193 conjugate(sqrt(-2+%i*y));
 194 conjugate(sqrt(-2+%i*y));
 195
 196 conjugate(sqrt(x+%i*y));
 197 conjugate(sqrt(x+%i*y));
 198
 199 assume(x>0);
 200 [x>0];
 201
 202 conjugate(sqrt(x+%i*y));
 203 sqrt(x-%i*y);
 204
 205 forget(x>0);
 206 [x>0];
 207
 208 /* ---------- Series representations ---------------------------------------- */
 209
 210 /* Check taylor expansion */
 211
 212 taylor(sqrt(x), x, x0, 2);
 213 sqrt(x0)+sqrt(x0)*(x-x0)/2/x0-sqrt(x0)*(x-x0)^2/8/x0^2;
 214
 215 /* Check taylor expansion as an argument to sqrt */
 216 sqrt(taylor(x,x,x0,2));
 217 sqrt(x0)+sqrt(x0)*(x-x0)/2/x0-sqrt(x0)*(x-x0)^2/8/x0^2;
 218
 219 taylor(sqrt(x), x, 1, 3);
 220 1 + (x-1)/2 - (x-1)^2/8 + (x-1)^3/16;
 221
 222 taylor(sqrt(x+1), x, 0, 3);
 223 1+x/2-x^2/8+x^3/16;
 224
 225 taylor(sqrt(x), x, inf, 2);
 226 sqrt(x);
 227
 228 /* ---------- Differential equations ---------------------------------------- */
 229
 230 depends(y,x);
 231 [y(x)];
 232 %e_to_numlog:true; /* Simplify %e^log(x) -> x */
 233 true;
 234
 235 ode2('diff(y,x)-y/(2*x), y, x);
 236 y=%c*sqrt(x);
 237
 238 ode2(4*'diff(y,x,2)*x^2+y, y, x);
 239 y=sqrt(x)*(%k1 + %k2*log(x)/2);
 240
 241 depends([g,h],x);
 242 [g(x), h(x)];
 243
 244 ode2('diff(y,x)-'diff(g(x),x)/(2*g(x))*y, y,x);
 245 y=%c*sqrt(g(x));
 246
 247 ode2('diff(y,x)-('diff(g(x),x)/(2*g(x))+'diff(h(x),x)/h(x))*y, y, x), expand;
 248 y=%c*h(x)*sqrt(g(x));
 249
 250 remove([y,g,h], depends);
 251 done;
 252 %e_to_numlog:false;
 253 false;
 254
 255 /* ---------- Transformations and argument simplifications ------------------ */
 256
 257 declare(z,complex);
 258 done;
 259
 260 assume(xp>0, xn<0);
 261 [xp>0, xn<0];
 262
 263 /* --- sqrt(-z) --- */
 264
 265 sqrt([-z,-x,-xp]);
 266 [sqrt(-z), sqrt(-x), %i*sqrt(xp)];
 267
 268 sqrt([-1/2, -3/4, -1, -3/2, -2]);
 269 [%i/sqrt(2), %i*sqrt(3)/2, %i, %i*sqrt(3)/sqrt(2), %i*sqrt(2)];
 270
 271 sqrt(-(xp+2));
 272 sqrt(-(xp+2));
 273
 274 /* --- sqrt(%i*z) --- */
 275
 276 sqrt([%i*z,%i*x,%i*xp]);
 277 [sqrt(%i*z), sqrt(%i*x), (-1)^(1/4)*sqrt(xp)];
 278
 279 sqrt([%i*1/2, %i*3/4, %i, %i*3/2, %i*2]);
 280 [(-1)^(1/4)/sqrt(2), (-1)^(1/4)*sqrt(3)/2, (-1)^(1/4),
 281  (-1)^(1/4)*sqrt(3)/sqrt(2), (-1)^(1/4)*sqrt(2)];
 282
 283 sqrt([%i*(xp+2), %i*xp+%i*2]);
 284 [(-1)^(1/4)*sqrt(xp+2),sqrt(%i*xp+%i*2)];
 285
 286 /* --- sqrt(-%i*z) --- */
 287
 288 sqrt([-%i*z,-%i*x,-%i*xp]);
 289 [sqrt(-%i*z), sqrt(-%i*x), sqrt(-%i)*sqrt(xp)];
 290
 291 sqrt([-%i*1/2, -%i*3/4, -%i, -%i*3/2, -%i*2]);
 292 [sqrt(-%i)/sqrt(2), sqrt(-%i)*sqrt(3)/2, sqrt(-%i),
 293  sqrt(-%i)*sqrt(3)/sqrt(2), sqrt(-%i)*sqrt(2)];
 294
 295 sqrt([-%i*(xp+2), -%i*xp-%i*2]);
 296 [sqrt(-%i)*sqrt(xp+2),sqrt(-%i*xp-%i*2)];
 297
 298 /* --- sqrt(1/z) --- */
 299
 300 sqrt(1/xp) - 1/sqrt(xp);
 301 0;
 302 sqrt(1/-xp) + 1/sqrt(-xp);
 303 0;
 304 sqrt(1/-xn) - 1/sqrt(-xn);
 305 0;
 306
 307 /* --- sqrt(-1/z) --- */
 308
 309 sqrt([-1/x, -1/xp, -1/xn, -1/(xn-2), -1/(xp+2)]);
 310 [%i/sqrt(x), %i/sqrt(xp), 1/sqrt(-xn), 1/sqrt(-xn+2), %i/sqrt(xp+2)];
 311
 312 /* --- sqrt(%i/z) --- */
 313
 314 sqrt([%i/x, %i/xp, %i/xn, %i/(xn-2), %i/(xp+2)]);
 315 [sqrt(%i/x), (-1)^(1/4)/sqrt(xp), sqrt(-%i)/sqrt(-xn), sqrt(-%i)/sqrt(-xn+2),
 316  (-1)^(1/4)/sqrt(xp+2)];
 317
 318  /* --- sqrt(-%i/z) --- */
 319
 320 sqrt([-%i/x, -%i/xp, -%i/xn, -%i/(xn-2), -%i/(xp+2)]);
 321 [sqrt(-%i/x), sqrt(-%i)/sqrt(xp), (-1)^(1/4)/sqrt(-xn), (-1)^(1/4)/sqrt(-xn+2),
 322  sqrt(-%i)/sqrt(xp+2)];
 323
 324 /* --- Half-angle formula --- */
 325
 326 sqrt(z/2);
 327 sqrt(z)/sqrt(2);
 328
 329 /* --- Multiple arguments for products --- */
 330
 331 assume(a>0);
 332 [a>0];
 333
 334 sqrt(a*z);
 335 sqrt(a)*sqrt(z);
 336
 337 sqrt(b*z); /* Do not simplify if the sign is not known to be positive */
 338 sqrt(b*z);
 339
 340 radcan(sqrt(z-z^2));
 341 sqrt(z)*sqrt(1-z);
 342
 343 radcan(sqrt(-z^2-z));
 344 sqrt(z)*sqrt(-z-1);
 345
 346 /* --- Multiple arguments for quotients --- */
 347
 348 radcan(sqrt(z)/sqrt(z+1));
 349 sqrt(z)/sqrt(z+1);
 350
 351 radcan(sqrt(z/(z-1)));
 352 sqrt(z)/sqrt(z-1);
 353
 354 /* radcan always simplifies sqrt(z1/z2) -> sqrt(z1)/sqrt(z2).
 355  * That is not correct.
 356  */
 357
 358 declare([z1,z2],complex);
 359 done;
 360
 361 radcan(sqrt(z1/z2));
 362 sqrt(z1)/sqrt(z2);
 363
 364 /* More examples of this type */
 365
 366 radcan(sqrt(z/(1-z)));
 367 sqrt(z)*sqrt(1/(1-z));
 368
 369 radcan(sqrt(-z/(1+z)));
 370 sqrt(z)*sqrt(1/(-1+-z));
 371
 372 /* --- Power of arguments --- */
 373
 374 sqrt([z^2, x^2, xp^2, xn^2]);
 375 [sqrt(z^2), abs(x), xp, -xn];
 376
 377 (domain:complex, done);
 378 done;
 379
 380 res:sqrt(-z^2); /* Wrong: Maxima simplifies to %i*sqrt(z) */
 381 sqrt(-z^2);
 382
 383 res,z=%i; /* We insert %i and get a wrong sign */
 384 1;
 385
 386 (reset(domain), done);
 387 done;
 388
 389 /* --- Exponent of arguments --- */
 390
 391 /* Maxima simplifes immediately sqrt(exp(z)) -> exp(z/2)
 392  * That is wrong for imagpart(z) > %pi or < -%pi
 393  * We give an example.
 394  */
 395
 396 sqrt(exp(2*%pi*%i)); /* That is the expected result */
 397 1;
 398
 399 sqrt(exp(%i*y)); /* This expression simplifies wrongly */
 400 sqrt(exp(%i*y));
 401
 402 %,y=2*%pi; /* We get a wrong sign */
 403 1;
 404
 405 /* ---------- Nested sqrt functions ----------------------------------------- */
 406
 407 /* The following 19 examples should simplify or evaluate to zero for all real
 408  * or complex numbers. We check this. This way we can detect simplification
 409  * errors for the sqrt function.
 410  */
 411
 412 (domain:complex, done);
 413 done;
 414
 415 /* --- Example 1 --- */
 416
 417 /* Wrong simplification because of sqrt(1/z) -> 1/sqrt(z) */
 418
 419 expr(z):=sqrt(-1-z)*sqrt(1/(1+z))*sqrt(1/z)*sqrt(-z)+1;
 420 expr(z):=sqrt(-1-z)*sqrt(1/(1+z))*sqrt(1/z)*sqrt(-z)+1;
 421
 422 /* Check correct simplification of expression */
 423 res:expr(z);
 424 sqrt(-1-z)*sqrt(1/(1+z))*sqrt(1/z)*sqrt(-z)+1;
 425
 426 closeto(float(rectform(res)),0.0,1e-15), z=1/2;
 427 true;
 428 closeto(float(rectform(res)),0.0,1e-15), z=-1/2; /* wrong -> 2 */
 429 true;
 430 closeto(float(rectform(res)),0.0,1e-15), z=%i/2;
 431 true;
 432 closeto(float(rectform(res)),0.0,1e-15), z=-%i/2;
 433 true;
 434 closeto(float(rectform(res)),0.0,1e-15), z=1/2+%i;
 435 true;
 436 closeto(float(rectform(res)),0.0,1e-15), z=-1/2+%i;
 437 true;
 438 closeto(float(rectform(res)),0.0,1e-15), z=-1/2-%i;
 439 true;
 440 closeto(float(rectform(res)),0.0,1e-15), z=1/2-%i;
 441 true;
 442
 443 /* --- Example 2 --- */
 444
 445 /* Wrong simplification because of sqrt(-1/z) -> 1/sqrt(-z) */
 446
 447 expr(z):=sqrt((z-1)/z)*sqrt(z/(1-z))-sqrt(-1/z)*sqrt(z);
 448 expr(z):=sqrt((z-1)/z)*sqrt(z/(1-z))-sqrt(-1/z)*sqrt(z);
 449
 450 /* Check correct simplification of equation */
 451 res:expr(z);
 452 sqrt((z-1)/z)*sqrt(z/(1-z))-sqrt(-1/z)*sqrt(z);
 453
 454 closeto(float(rectform(res)),0.0,1e-15), z=1/2; /* Wrong -> 2 */
 455 true;
 456 closeto(float(rectform(res)),0.0,1e-15), z=-1/2;
 457 true;
 458 closeto(float(rectform(res)),0.0,1e-15), z=%i/2;
 459 true;
 460 closeto(float(rectform(res)),0.0,1e-15), z=-%i/2;
 461 true;
 462 closeto(float(rectform(res)),0.0,1e-15), z=1/2+%i;
 463 true;
 464 closeto(float(rectform(res)),0.0,1e-15), z=-1/2+%i;
 465 true;
 466 closeto(float(rectform(res)),0.0,1e-15), z=-1/2-%i;
 467 true;
 468 closeto(float(rectform(res)),0.0,1e-15), z=1/2-%i;
 469 true;
 470
 471 /* --- Example 3 --- */
 472
 473 /* Wrong simplification because of sqrt(z^2)/sqrt(-z^2) -> -%i */
 474
 475 expr(z):=sqrt(sqrt(z^2+1)-1)/sqrt(1-sqrt(z^2+1))-sqrt(z^2)/sqrt(-z^2);
 476 expr(z):=sqrt(sqrt(z^2+1)-1)/sqrt(1-sqrt(z^2+1))-sqrt(z^2)/sqrt(-z^2);
 477
 478 /* Check correct simplification of equation */
 479 res:expr(z);
 480 sqrt(sqrt(z^2+1)-1)/sqrt(1-sqrt(z^2+1))-sqrt(z^2)/sqrt(-z^2);
 481
 482 closeto(float(rectform(res)),0.0,1e-15), z=1/2;
 483 true;
 484 closeto(float(rectform(res)),0.0,1e-15), z=-1/2;
 485 true;
 486 closeto(float(rectform(res)),0.0,1e-15), z=%i/2;      /* Wrong -> 2 */
 487 true;
 488 closeto(float(rectform(res)),0.0,1e-15), z=-%i/2;     /* Wrong -> 2 */
 489 true;
 490 closeto(float(rectform(res)),0.0,1e-15), z=1/2+%i;  /* Wrong -> 2 */
 491 true;
 492 closeto(float(rectform(res)),0.0,1e-15), z=-1/2+%i;
 493 true;
 494 closeto(float(rectform(res)),0.0,1e-15), z=-1/2-%i; /* Wrong -> 2 */
 495 true;
 496 closeto(float(rectform(res)),0.0,1e-15), z=1/2-%i;
 497 true;
 498
 499 /* --- Example 4 --- */
 500
 501 /* Wrong simplification sqrt(-z^2)/sqrt(z^2) -> %i */
 502
 503 expr(z):=sqrt(sqrt(1-z^2)-1)/sqrt(1-sqrt(1-z^2))-sqrt(-z^2)/sqrt(z^2);
 504 expr(z):=sqrt(sqrt(1-z^2)-1)/sqrt(1-sqrt(1-z^2))-sqrt(-z^2)/sqrt(z^2);
 505
 506 /* Check correct simplification of equation */
 507 res:expr(z);
 508 sqrt(sqrt(1-z^2)-1)/sqrt(1-sqrt(1-z^2))-sqrt(-z^2)/sqrt(z^2);
 509
 510 closeto(float(rectform(res)),0.0,1e-15), z=1/2;
 511 true;
 512 closeto(float(rectform(res)),0.0,1e-15), z=-1/2;
 513 true;
 514 closeto(float(rectform(res)),0.0,1e-15), z=%i/2;     /* Wrong -> 2 */
 515 true;
 516 closeto(float(rectform(res)),0.0,1e-15), z=-%i/2;    /* Wrong -> 2 */
 517 true;
 518 closeto(float(rectform(res)),0.0,1e-15), z=1/2+%i; /* Wrong -> 2 */
 519 true;
 520 closeto(float(rectform(res)),0.0,1e-15), z=-1/2+%i;
 521 true;
 522 closeto(float(rectform(res)),0.0,1e-15), z=-1/2-%i; /* Wrong -> 2 */
 523 true;
 524 closeto(float(rectform(res)),0.0,1e-15), z=1/2-%i;
 525 true;
 526
 527 /* --- Example 5 --- */
 528
 529 /* Wrong simplification sqrt(-1/z) -> 1/sqrt(-z) */
 530
 531 expr(z):=sqrt(sqrt(z^2+1)-z)/sqrt(z-sqrt(z^2+1))+sqrt(z)*sqrt(-1/z);
 532 expr(z):=sqrt(sqrt(z^2+1)-z)/sqrt(z-sqrt(z^2+1))+sqrt(z)*sqrt(-1/z);
 533
 534 /* Check correct simplification of equation */
 535 res:expr(z);
 536 sqrt(sqrt(z^2+1)-z)/sqrt(z-sqrt(z^2+1))+sqrt(z)*sqrt(-1/z);
 537
 538 closeto(float(rectform(res)),0.0,1e-15), z=1/2; /* Wrong -> 2 */
 539 true;
 540 closeto(float(rectform(res)),0.0,1e-15), z=-1/2;
 541 true;
 542 closeto(float(rectform(res)),0.0,1e-15), z=%i/2;
 543 true;
 544 closeto(float(rectform(res)),0.0,1e-15), z=-%i/2;
 545 true;
 546 closeto(float(rectform(res)),0.0,1e-15), z=1/2+%i;
 547 true;
 548 closeto(float(rectform(res)),0.0,1e-15), z=-1/2+%i;
 549 true;
 550 closeto(float(rectform(res)),0.0,1e-15), z=-1/2-%i;
 551 true;
 552 closeto(float(rectform(res)),0.0,1e-15), z=1/2-%i;
 553 true;
 554
 555 /* --- Example 6 --- */
 556
 557 /* Wrong simplification sqrt(1/z) -> 1/sqrt(z) */
 558
 559 expr(z):=sqrt(z+sqrt(z^2+1))/sqrt(-z-sqrt(z^2+1))+sqrt(-z)*sqrt(1/z);
 560 expr(z):=sqrt(z+sqrt(z^2+1))/sqrt(-z-sqrt(z^2+1))+sqrt(-z)*sqrt(1/z);
 561
 562 /* Check correct simplification of equation */
 563 res:expr(z);
 564 sqrt(z+sqrt(z^2+1))/sqrt(-z-sqrt(z^2+1))+sqrt(-z)*sqrt(1/z);
 565
 566 closeto(float(rectform(res)),0.0,1e-15), z=1/2;
 567 true;
 568 closeto(float(rectform(res)),0.0,1e-15), z=-1/2; /* Wrong -> 2 */
 569 true;
 570 closeto(float(rectform(res)),0.0,1e-15), z=%i/2;
 571 true;
 572 closeto(float(rectform(res)),0.0,1e-15), z=-%i/2;
 573 true;
 574 closeto(float(rectform(res)),0.0,1e-15), z=1/2+%i;
 575 true;
 576 closeto(float(rectform(res)),0.0,1e-15), z=-1/2+%i;
 577 true;
 578 closeto(float(rectform(res)),0.0,1e-15), z=-1/2-%i;
 579 true;
 580 closeto(float(rectform(res)),0.0,1e-15), z=1/2-%i;
 581 true;
 582
 583 /* --- Example 7 --- */
 584
 585 /* This expression does not simplify at all. All numeric values are correct.
 586  * rootscontract gives the result 2*%i. That is wrong. Because we have no
 587  * problem with the simplification of the expression, the error is in
 588  * rootscontract for this example */
 589
 590 expr(z):=sqrt(sqrt(1-z^2)-%i*z)/sqrt(%i*z-sqrt(1-z^2))+sqrt(%i*z)*sqrt(%i/z);
 591 expr(z):=sqrt(sqrt(1-z^2)-%i*z)/sqrt(%i*z-sqrt(1-z^2))+sqrt(%i*z)*sqrt(%i/z);
 592
 593 /* Check correct simplification of equation */
 594 res:expr(z);
 595 sqrt(sqrt(1-z^2)-%i*z)/sqrt(%i*z-sqrt(1-z^2))+sqrt(%i*z)*sqrt(%i/z);
 596
 597 closeto(float(rectform(res)),0.0,1e-15), z=1/2;
 598 true;
 599 closeto(float(rectform(res)),0.0,1e-15), z=-1/2;
 600 true;
 601 closeto(float(rectform(res)),0.0,1e-15), z=%i/2;
 602 true;
 603 closeto(float(rectform(res)),0.0,1e-15), z=-%i/2;
 604 true;
 605 closeto(float(rectform(res)),0.0,1e-15), z=1/2+%i;
 606 true;
 607 closeto(float(rectform(res)),0.0,1e-15), z=-1/2+%i;
 608 true;
 609 closeto(float(rectform(res)),0.0,1e-15), z=-1/2-%i;
 610 true;
 611 closeto(float(rectform(res)),0.0,1e-15), z=1/2-%i;
 612 true;
 613
 614 /* --- Example 8 --- */
 615
 616 /* This expression does not simplify at all. All numeric values are correct.
 617  * rootscontract gives the result 2*%i. That is wrong. Because we have no
 618  * problem with the simplification of the expression, the error is in
 619  * rootscontract for this example. See also example 7.
 620  */
 621
 622 expr(z):=sqrt(%i*z+sqrt(1-z^2))/sqrt(-%i*z-sqrt(1-z^2))+sqrt(-%i*z)*sqrt(-%i/z);
 623 expr(z):=sqrt(%i*z+sqrt(1-z^2))/sqrt(-%i*z-sqrt(1-z^2))+sqrt(-%i*z)*sqrt(-%i/z);
 624
 625 /* Check correct simplification of equation */
 626 res:expr(z);
 627 sqrt(%i*z+sqrt(1-z^2))/sqrt(-%i*z-sqrt(1-z^2))+sqrt(-%i*z)*sqrt(-%i/z);
 628
 629 closeto(float(rectform(res)),0.0,1e-15), z=1/2;
 630 true;
 631 closeto(float(rectform(res)),0.0,1e-15), z=-1/2;
 632 true;
 633 closeto(float(rectform(res)),0.0,1e-15), z=%i/2;
 634 true;
 635 closeto(float(rectform(res)),0.0,1e-15), z=-%i/2;
 636 true;
 637 closeto(float(rectform(res)),0.0,1e-15), z=1/2+%i;
 638 true;
 639 closeto(float(rectform(res)),0.0,1e-15), z=-1/2+%i;
 640 true;
 641 closeto(float(rectform(res)),0.0,1e-15), z=-1/2-%i;
 642 true;
 643 closeto(float(rectform(res)),0.0,1e-15), z=1/2-%i;
 644 true;
 645
 646 /* --- Example 9 --- */
 647
 648 /* This examples simplifies to zero immediately. This is by accident.
 649  * We know that the applied simplifications are not correct.
 650  */
 651
 652 expr(z):=sqrt(z^2/(1-sqrt(1-z^2)))*sqrt((1-sqrt(1-z^2))/z^2)-1;
 653 expr(z):=sqrt(z^2/(1-sqrt(1-z^2)))*sqrt((1-sqrt(1-z^2))/z^2)-1;
 654
 655 /* Check correct simplification of equation */
 656 res:expr(z);
 657 sqrt(z^2/(1-sqrt(1-z^2)))*sqrt((1-sqrt(1-z^2))/z^2)-1;
 658
 659 closeto(float(rectform(res)),0.0,1e-15), z=1/2;
 660 true;
 661 closeto(float(rectform(res)),0.0,1e-15), z=-1/2;
 662 true;
 663 closeto(float(rectform(res)),0.0,1e-15), z=%i/2;
 664 true;
 665 closeto(float(rectform(res)),0.0,1e-15), z=-%i/2;
 666 true;
 667 closeto(float(rectform(res)),0.0,1e-15), z=1/2+%i;
 668 true;
 669 closeto(float(rectform(res)),0.0,1e-15), z=-1/2+%i;
 670 true;
 671 closeto(float(rectform(res)),0.0,1e-15), z=-1/2-%i;
 672 true;
 673 closeto(float(rectform(res)),0.0,1e-15), z=1/2-%i;
 674 true;
 675
 676 /* --- Example 10 --- */
 677
 678 /* Wrong simplification sqrt(1/z) -> 1/sqrt(z) */
 679
 680 expr(z):=sqrt(z/(sqrt(z^2+1)-z))*sqrt((sqrt(z^2+1)-z)/z)-sqrt(1/(z^2+1))*sqrt(z^2+1)-sqrt(1/z)*sqrt(z)+1;
 681 expr(z):=sqrt(z/(sqrt(z^2+1)-z))*sqrt((sqrt(z^2+1)-z)/z)-sqrt(1/(z^2+1))*sqrt(z^2+1)-sqrt(1/z)*sqrt(z)+1;
 682
 683 /* Check correct simplification of equation */
 684 res:expr(z);
 685 sqrt(z/(sqrt(z^2+1)-z))*sqrt((sqrt(z^2+1)-z)/z)-sqrt(1/(z^2+1))*sqrt(z^2+1)-sqrt(1/z)*sqrt(z)+1;
 686
 687 closeto(float(rectform(res)),0.0,1e-15), z=1/2;
 688 true;
 689 closeto(float(rectform(res)),0.0,1e-15), z=-1/2;
 690 true;
 691 closeto(float(rectform(res)),0.0,1e-15), z=%i/2; /* Wrong -> 2 */
 692 true;
 693 closeto(float(rectform(res)),0.0,1e-15), z=-%i/2; /* Wrong -> 2 */
 694 true;
 695 closeto(float(rectform(res)),0.0,1e-15), z=1/2+%i;
 696 true;
 697 closeto(float(rectform(res)),0.0,1e-15), z=-1/2+%i;
 698 true;
 699 closeto(float(rectform(res)),0.0,1e-15), z=-1/2-%i;
 700 true;
 701 closeto(float(rectform(res)),0.0,1e-15), z=1/2-%i;
 702 true;
 703
 704 /* --- Example 11 --- */
 705
 706 /* This example is correct:
 707  * No simplification of this expression. All numeric results are correct.
 708  * radcan does not simplify too. That is not wrong.
 709  * rootscontract simplifies correctly to zero.
 710  */
 711
 712 expr(z):=sqrt(sqrt(z)-1)*sqrt(sqrt(z)+1)-sqrt(z-1);
 713 expr(z):=sqrt(sqrt(z)-1)*sqrt(sqrt(z)+1)-sqrt(z-1);
 714
 715 /* Check correct simplification of equation */
 716 res:expr(z);
 717 sqrt(sqrt(z)-1)*sqrt(sqrt(z)+1)-sqrt(z-1);
 718
 719 closeto(float(rectform(res)),0.0,1e-15), z=1/2;
 720 true;
 721 closeto(float(rectform(res)),0.0,1e-15), z=-1/2;
 722 true;
 723 closeto(float(rectform(res)),0.0,1e-15), z=%i/2;
 724 true;
 725 closeto(float(rectform(res)),0.0,1e-15), z=-%i/2;
 726 true;
 727 closeto(float(rectform(res)),0.0,1e-15), z=1/2+%i;
 728 true;
 729 closeto(float(rectform(res)),0.0,1e-15), z=-1/2+%i;
 730 true;
 731 closeto(float(rectform(res)),0.0,1e-15), z=-1/2-%i;
 732 true;
 733 closeto(float(rectform(res)),0.0,1e-15), z=1/2-%i;
 734 true;
 735
 736 /* --- Example 12 --- */
 737
 738 /* This example is correct:
 739  * No simplification of this expression. All numeric results are correct.
 740  * radcan does not simplify too. That is not wrong.
 741  * rootscontract simplifies correctly to zero. See also example 11.
 742  */
 743
 744 expr(z):=sqrt(sqrt(z+1)-1)*sqrt((sqrt(z+1)+1))-sqrt(z);
 745 expr(z):=sqrt(sqrt(z+1)-1)*sqrt((sqrt(z+1)+1))-sqrt(z);
 746
 747 /* Check correct simplification of equation */
 748 res:expr(z);
 749 sqrt(sqrt(z+1)-1)*sqrt((sqrt(z+1)+1))-sqrt(z);
 750
 751 closeto(float(rectform(res)),0.0,1e-15), z=1/2;
 752 true;
 753 closeto(float(rectform(res)),0.0,1e-15), z=-1/2;
 754 true;
 755 closeto(float(rectform(res)),0.0,1e-15), z=%i/2;
 756 true;
 757 closeto(float(rectform(res)),0.0,1e-15), z=-%i/2;
 758 true;
 759 closeto(float(rectform(res)),0.0,1e-15), z=1/2+%i;
 760 true;
 761 closeto(float(rectform(res)),0.0,1e-15), z=-1/2+%i;
 762 true;
 763 closeto(float(rectform(res)),0.0,1e-15), z=-1/2-%i;
 764 true;
 765 closeto(float(rectform(res)),0.0,1e-15), z=1/2-%i;
 766 true;
 767
 768 /* --- Example 13 --- */
 769
 770 /* This example does not simplify. All numeric results are correct.
 771  * radcan gives a modified result, which is correct too.
 772  * rootscontract gives a wrong expression.
 773  */
 774
 775 expr(z):=sqrt(2)*sqrt(1-sqrt(1-z))*sqrt(sqrt(1-z)+%i*sqrt(z))/(%i*(1-sqrt(1-z))+sqrt(z))-1;
 776 expr(z):=sqrt(2)*sqrt(1-sqrt(1-z))*sqrt(sqrt(1-z)+%i*sqrt(z))/(%i*(1-sqrt(1-z))+sqrt(z))-1;
 777
 778 /* Check correct simplification of equation */
 779 res:expr(z);
 780 sqrt(2)*sqrt(1-sqrt(1-z))*sqrt(sqrt(1-z)+%i*sqrt(z))/(%i*(1-sqrt(1-z))+sqrt(z))-1;
 781
 782 closeto(float(rectform(res)),0.0,1e-15), z=1/2;
 783 true;
 784 closeto(float(rectform(res)),0.0,1e-15), z=-1/2;
 785 true;
 786 closeto(float(rectform(res)),0.0,1e-15), z=%i/2;
 787 true;
 788 closeto(float(rectform(res)),0.0,1e-15), z=-%i/2;
 789 true;
 790 closeto(float(rectform(res)),0.0,1e-15), z=1/2+%i;
 791 true;
 792 closeto(float(rectform(res)),0.0,1e-15), z=-1/2+%i;
 793 true;
 794 closeto(float(rectform(res)),0.0,1e-15), z=-1/2-%i;
 795 true;
 796 closeto(float(rectform(res)),0.0,1e-15), z=1/2-%i;
 797 true;
 798
 799 /* --- Example 14 --- */
 800
 801 /* This example does not simplify. All numeric results are correct.
 802  * radcan gives the correct result zero.
 803  * rootscontract gives a wrong expression.
 804  */
 805
 806 expr(z):=sqrt(1-z)*(1+z-%i*sqrt(1-z^2))/(sqrt(z+1)*(1-z+%i*sqrt(1-z^2)))+%i;
 807 expr(z):=sqrt(1-z)*(1+z-%i*sqrt(1-z^2))/(sqrt(z+1)*(1-z+%i*sqrt(1-z^2)))+%i;
 808
 809 /* Check correct simplification of equation */
 810 res:expr(z);
 811 sqrt(1-z)*(1+z-%i*sqrt(1-z^2))/(sqrt(z+1)*(1-z+%i*sqrt(1-z^2)))+%i;
 812
 813 closeto(float(rectform(res)),0.0,1e-15), z=1/2;
 814 true;
 815 closeto(float(rectform(res)),0.0,1e-15), z=-1/2;
 816 true;
 817 closeto(float(rectform(res)),0.0,1e-15), z=%i/2;
 818 true;
 819 closeto(float(rectform(res)),0.0,1e-15), z=-%i/2;
 820 true;
 821 closeto(float(rectform(res)),0.0,1e-15), z=1/2+%i;
 822 true;
 823 closeto(float(rectform(res)),0.0,1e-15), z=-1/2+%i;
 824 true;
 825 closeto(float(rectform(res)),0.0,1e-15), z=-1/2-%i;
 826 true;
 827 closeto(float(rectform(res)),0.0,1e-15), z=1/2-%i;
 828 true;
 829
 830 /* --- Example 15 --- */
 831
 832 /* This example does not simplify. All numeric results are correct.
 833  * radcan gives the correct result zero.
 834  * rootscontract gives a wrong expression.
 835  */
 836
 837 expr(z):=sqrt(1-z)*(1+z+%i*sqrt(1-z^2))/(sqrt(z+1)*(1-z-%i*sqrt(1-z^2)))-%i;
 838 expr(z):=sqrt(1-z)*(1+z+%i*sqrt(1-z^2))/(sqrt(z+1)*(1-z-%i*sqrt(1-z^2)))-%i;
 839
 840 /* Check correct simplification of equation */
 841 res:expr(z);
 842 sqrt(1-z)*(1+z+%i*sqrt(1-z^2))/(sqrt(z+1)*(1-z-%i*sqrt(1-z^2)))-%i;
 843
 844 closeto(float(rectform(res)),0.0,1e-15), z=1/2;
 845 true;
 846 closeto(float(rectform(res)),0.0,1e-15), z=-1/2;
 847 true;
 848 closeto(float(rectform(res)),0.0,1e-15), z=%i/2;
 849 true;
 850 closeto(float(rectform(res)),0.0,1e-15), z=-%i/2;
 851 true;
 852 closeto(float(rectform(res)),0.0,1e-15), z=1/2+%i;
 853 true;
 854 closeto(float(rectform(res)),0.0,1e-15), z=-1/2+%i;
 855 true;
 856 closeto(float(rectform(res)),0.0,1e-15), z=-1/2-%i;
 857 true;
 858 closeto(float(rectform(res)),0.0,1e-15), z=1/2-%i;
 859 true;
 860
 861 /* --- Example 16 --- */
 862
 863 /* This example does not simplify. All numeric results are correct.
 864  * radcan gives a changed expression, which is correct too.
 865  * rootscontract gives only a small change.
 866  */
 867
 868 expr(z):=1/sqrt(1-sqrt(z))+1/sqrt(1+sqrt(z))-sqrt(2)*sqrt(sqrt(1-z)+1)/sqrt(1-z);
 869 expr(z):=1/sqrt(1-sqrt(z))+1/sqrt(1+sqrt(z))-sqrt(2)*sqrt(sqrt(1-z)+1)/sqrt(1-z);
 870
 871 /* Check correct simplification of equation */
 872 res:expr(z);
 873 1/sqrt(1-sqrt(z))+1/sqrt(1+sqrt(z))-sqrt(2)*sqrt(sqrt(1-z)+1)/sqrt(1-z);
 874
 875 closeto(float(rectform(res)),0.0,1e-15), z=1/2;
 876 true;
 877 closeto(float(rectform(res)),0.0,1e-15), z=-1/2;
 878 true;
 879 closeto(float(rectform(res)),0.0,1e-15), z=%i/2;
 880 true;
 881 closeto(float(rectform(res)),0.0,1e-15), z=-%i/2;
 882 true;
 883 closeto(float(rectform(res)),0.0,1e-15), z=1/2+%i;
 884 true;
 885 closeto(float(rectform(res)),0.0,1e-15), z=-1/2+%i;
 886 true;
 887 closeto(float(rectform(res)),0.0,1e-15), z=-1/2-%i;
 888 true;
 889 closeto(float(rectform(res)),0.0,1e-15), z=1/2-%i;
 890 true;
 891
 892 /* --- Example 17 --- */
 893
 894 /* This example does not simplify. All numeric results are correct.
 895  * radcan gives a changed expression, which is correct too.
 896  * rootscontract gives only a small change.
 897  */
 898
 899 expr(z):=1/sqrt(1-sqrt(z))-1/sqrt(1+sqrt(z))-sqrt(2)*sqrt(1-sqrt(1-z))/sqrt(1-z);
 900 expr(z):=1/sqrt(1-sqrt(z))-1/sqrt(1+sqrt(z))-sqrt(2)*sqrt(1-sqrt(1-z))/sqrt(1-z);
 901
 902 /* Check correct simplification of equation */
 903 res:expr(z);
 904 1/sqrt(1-sqrt(z))-1/sqrt(1+sqrt(z))-sqrt(2)*sqrt(1-sqrt(1-z))/sqrt(1-z);
 905
 906 closeto(float(rectform(res)),0.0,1e-15), z=1/2;
 907 true;
 908 closeto(float(rectform(res)),0.0,1e-15), z=-1/2;
 909 true;
 910 closeto(float(rectform(res)),0.0,1e-15), z=%i/2;
 911 true;
 912 closeto(float(rectform(res)),0.0,1e-15), z=-%i/2;
 913 true;
 914 closeto(float(rectform(res)),0.0,1e-15), z=1/2+%i;
 915 true;
 916 closeto(float(rectform(res)),0.0,1e-15), z=-1/2+%i;
 917 true;
 918 closeto(float(rectform(res)),0.0,1e-15), z=-1/2-%i;
 919 true;
 920 closeto(float(rectform(res)),0.0,1e-15), z=1/2-%i;
 921 true;
 922
 923 /* --- Example 18 --- */
 924
 925 /* All numeric values are correct. radcan and rootscontract does not
 926  * change the expression.
 927  */
 928
 929 expr(z):=sqrt(1+sqrt(z))+sqrt(1-sqrt(z))-sqrt(2)*sqrt(1+sqrt(1-z));
 930 expr(z):=sqrt(1+sqrt(z))+sqrt(1-sqrt(z))-sqrt(2)*sqrt(1+sqrt(1-z));
 931
 932 /* Check correct simplification of equation */
 933 res:expr(z);
 934 sqrt(1+sqrt(z))+sqrt(1-sqrt(z))-sqrt(2)*sqrt(1+sqrt(1-z));
 935
 936 closeto(float(rectform(res)),0.0,1e-15), z=1/2;
 937 true;
 938 closeto(float(rectform(res)),0.0,1e-15), z=-1/2;
 939 true;
 940 closeto(float(rectform(res)),0.0,1e-15), z=%i/2;
 941 true;
 942 closeto(float(rectform(res)),0.0,1e-15), z=-%i/2;
 943 true;
 944 closeto(float(rectform(res)),0.0,1e-15), z=1/2+%i;
 945 true;
 946 closeto(float(rectform(res)),0.0,1e-15), z=-1/2+%i;
 947 true;
 948 closeto(float(rectform(res)),0.0,1e-15), z=-1/2-%i;
 949 true;
 950 closeto(float(rectform(res)),0.0,1e-15), z=1/2-%i;
 951 true;
 952
 953 /* --- Example 19 --- */
 954
 955 /* All numeric values are correct. radcan and rootscontract does not
 956  * change the expression.
 957  */
 958
 959 expr(z):=sqrt(1+sqrt(z))-sqrt(1-sqrt(z))-sqrt(2)*sqrt(1-sqrt(1-z));
 960 expr(z):=sqrt(1+sqrt(z))-sqrt(1-sqrt(z))-sqrt(2)*sqrt(1-sqrt(1-z));
 961
 962 /* Check correct simplification of equation */
 963 res:expr(z);
 964 sqrt(1+sqrt(z))-sqrt(1-sqrt(z))-sqrt(2)*sqrt(1-sqrt(1-z));
 965
 966 closeto(float(rectform(res)),0.0,1e-15), z=1/2;
 967 true;
 968 closeto(float(rectform(res)),0.0,1e-15), z=-1/2;
 969 true;
 970 closeto(float(rectform(res)),0.0,1e-15), z=%i/2;
 971 true;
 972 closeto(float(rectform(res)),0.0,1e-15), z=-%i/2;
 973 true;
 974 closeto(float(rectform(res)),0.0,1e-15), z=1/2+%i;
 975 true;
 976 closeto(float(rectform(res)),0.0,1e-15), z=-1/2+%i;
 977 true;
 978 closeto(float(rectform(res)),0.0,1e-15), z=-1/2-%i;
 979 true;
 980 closeto(float(rectform(res)),0.0,1e-15), z=1/2-%i;
 981 true;
 982
 983 (reset(domain), done);
 984 done;
 985
 986 /* ---------- Complex characteristics --------------------------------------- */
 987
 988 /* --- Real part --- */
 989
 990 realpart(sqrt(x+%i*y));
 991 (x^2+y^2)^(1/4)*cos(1/2*atan2(y,x));
 992
 993 /* --- Imaginary part --- */
 994
 995 imagpart(sqrt(x+%i*y));
 996 (x^2+y^2)^(1/4)*sin(1/2*atan2(y,x));
 997
 998 /* --- Absolute value --- */
 999
1000 abs(sqrt(x+%i*y));
1001 abs(sqrt(%i*y+x))$
1002
1003 /* --- Argument --- */
1004
1005 carg(sqrt(x+%i*y));
1006 1/2*atan2(y,x);
1007
1008 /* --- Conjugate value --- */
1009
1010 conjugate(sqrt(xp+%i*y));
1011 sqrt(xp-%i*y);
1012
1013 rectform(conjugate(sqrt(xp+%i*y)));
1014 (xp^2+y^2)^(1/4)*cos(1/2*atan(y/xp))-%i*(xp^2+y^2)^(1/4)*sin(1/2*atan(y/xp));
1015
1016 /* --- Signum value --- */
1017
1018 /* Maxima has no signum function for complex expressions */
1019
1020 signum(sqrt(z));
1021 'signum(sqrt(z));
1022
1023 signum(sqrt(xp));
1024 1;
1025
1026 signum(sqrt(xn));
1027 'signum(sqrt(xn));
1028
1029 /* ---------- Differentiation ----------------------------------------------- */
1030
1031 diff(sqrt(z),z);
1032 1/(2*sqrt(z));
1033
1034 diff(sqrt(z),z,2);
1035 -1/(4*z^(3/2));
1036
1037 /* ---------- Integration --------------------------------------------------- */
1038
1039 /* --- Indefinite integration --- */
1040
1041 integrate(sqrt(z),z);
1042 2/3*z^(3/2);
1043
1044 /* --- Definite integration --- */
1045
1046 integrate(sqrt(t),t,0,1);
1047 2/3;
1048
1049 integrate(sqrt(t)*log(t),t,0,1);
1050 -4/9;
1051
1052 /* Maxima does not know the general result: -> 2/3*2F1(3/2,-a;5/2;-1) */
1053 integrate(sqrt(t)*(t+1)^a,t,0,1);
1054 'integrate(sqrt(t)*(t+1)^a,t,0,1);
1055
1056 /* Maxima can evaluate the integral for a an integer or a half integral value */
1057 integrate(sqrt(t)*(t+1)^2,t,0,1);
1058 184/105;
1059
1060 integrate(sqrt(t)*(t+1)^-2,t,0,1);
1061 %pi/4-1/2$
1062
1063 integrate(sqrt(t)*(t+1)^(1/2),t,0,1);
1064 (-log(sqrt(2)+1)+log(1-sqrt(2))+3*2^(3/2))/8-log(-1)/8$
1065
1066 integrate(sqrt(t)*(t+1)^(-1/2),t,0,1);
1067 (-log(sqrt(2)+1)/2)+log(sqrt(2)-1)/2+sqrt(2);
1068
1069 integrate(sqrt(t)*(t+1)^(3/2),t,0,1);
1070 (-3*log(sqrt(2)+1)+3*log(1-sqrt(2))+25*2^(3/2))/48-log(-1)/16$
1071
1072 integrate(sqrt(t)*(t+1)^(-3/2),t,0,1);
1073 log(sqrt(2)+1)-log(sqrt(2)-1)-sqrt(2);
1074
1075 /* ---------- Integral transforms ------------------------------------------- */
1076
1077 /* --- Laplace transform --- */
1078
1079 laplace(sqrt(t),t,s);
1080 sqrt(%pi)/(2*s^(3/2));
1081
1082 /* Maxima does not know the inverse laplace transform
1083  *    -> -1/(2*t^(3/2)*sqrt(%pi)
1084  */
1085 ilt(sqrt(s),s,t);
1086 'ilt(sqrt(s),s,t);
1087
1088 /* ---------- Limits -------------------------------------------------------- */
1089
1090 assume(b>1);
1091 [b>1];
1092
1093 limit(sqrt(z)/b^z,z,inf);
1094 0;
1095
1096 forget(b>1);
1097 [b>1];
1098
1099 limit(sqrt(z)*log(z),z,0);
1100 0;
1101
1102 limit(log(z)/sqrt(z),z,inf);
1103 0;
1104
1105 /* ---------- Representation through hypergeometric functions --------------- */
1106
1107 hgfred([-1/2],[],1-z);
1108 sqrt(z);
1109
1110 hgfred([],[], 1/2*log(z)), %e_to_numlog:true;
1111 sqrt(z);
1112
1113 hgfred([-1/2,b],[b],1-z);
1114 sqrt(z);
1115
1116 (reset(domain), done);
1117 done;
1118
1119 /* bug reported to mailing list 2018-11-01: "radcan(sqrt(-1/(sqrt(-3)+1))*sqrt(2))" */
1120
1121 radcan (sqrt(-1/(sqrt(-3)+1))*sqrt(2));
1122 sqrt(2)/sqrt(-sqrt(3)*%i - 1);
1123
1124 radcan (sqrt(-1/(sqrt(-3)+1)));
1125 1/sqrt(-sqrt(3)*%i - 1);
1126
1127 radcan (sqrt(1/(sqrt(-3)+1))*sqrt(2));
1128 sqrt(2)/sqrt((sqrt(3)*%i + 1));