doc/info/Elliptic.texi.m4

   1 @c -*- mode: texinfo -*-
   2
   3 @menu
   4 * Introduction to Elliptic Functions and Integrals::
   5 * Functions and Variables for Elliptic Functions::
   6 * Functions and Variables for Elliptic Integrals::
   7 @end menu
   8
   9
  10
  11 @node Introduction to Elliptic Functions and Integrals, Functions and Variables for Elliptic Functions, , Elliptic Functions
  12 @comment  node-name,  next,  previous,  up
  13 @section Introduction to Elliptic Functions and Integrals
  14
  15 Maxima includes support for Jacobian elliptic functions and for
  16 complete and incomplete elliptic integrals.  This includes symbolic
  17 manipulation of these functions and numerical evaluation as well.
  18 Definitions of these functions and many of their properties can by
  19 found in Abramowitz and Stegun, @urlaands{Chapter 16, 567} and
  20 @urlaands{Chapter 17., 587}  See also @urldlmf{22.2}.  As much as possible,
  21 we use the definitions and relationships given in Abramowitz and Stegun.
  22
  23 In particular, all elliptic functions and integrals use the parameter
  24 @math{m} instead of the modulus @math{k} or the modular angle
  25 @math{\alpha}.  The following relationships are true:
  26
  27 m4_displaymath(
  28 <<<\eqalign{
  29 m &= k^2 \cr
  30 k &= \sin\alpha
  31 }>>>
  32 ,
  33 <<<@math{m = k^2}
  34
  35 @math{k = sin(alpha)}>>>
  36 )
  37
  38 Note that Abramowitz and Stegun uses the notation
  39 m4_math(<<<{\rm
  40 sn}(u|m)>>>, <<<sn(u|m)>>>)
  41 where we use
  42 m4_math(<<<{\rm
  43 sn}(u,m)>>>, <<<sn(u,m)>>>)
  44 instead.  The DLMF uses modulus @math{k}
  45 instead of the parameter @math{m}.
  46
  47 The elliptic functions and integrals are primarily intended to support
  48 symbolic computation.  Therefore, most of derivatives of the functions
  49 and integrals are known.  However, if floating-point values are given,
  50 a floating-point result is returned.
  51
  52 Support for most of the other properties of elliptic functions and
  53 integrals other than derivatives has not yet been written.
  54
  55 Some examples of elliptic functions:
  56 @c ===beg===
  57 @c jacobi_sn (u, m);
  58 @c jacobi_sn (u, 1);
  59 @c jacobi_sn (u, 0);
  60 @c diff (jacobi_sn (u, m), u);
  61 @c diff (jacobi_sn (u, m), m);
  62 @c ===end===
  63 @example
  64 @group
  65 (%i1) jacobi_sn (u, m);
  66 (%o1)                    jacobi_sn(u, m)
  67 @end group
  68 @group
  69 (%i2) jacobi_sn (u, 1);
  70 (%o2)                        tanh(u)
  71 @end group
  72 @group
  73 (%i3) jacobi_sn (u, 0);
  74 (%o3)                        sin(u)
  75 @end group
  76 @group
  77 (%i4) diff (jacobi_sn (u, m), u);
  78 (%o4)            jacobi_cn(u, m) jacobi_dn(u, m)
  79 @end group
  80 @group
  81 (%i5) diff (jacobi_sn (u, m), m);
  82 (%o5) (jacobi_cn(u, m) jacobi_dn(u, m)
  83       elliptic_e(asin(jacobi_sn(u, m)), m)
  84  (u - ------------------------------------))/(2 m)
  85                      1 - m
  86             2
  87    jacobi_cn (u, m) jacobi_sn(u, m)
  88  + --------------------------------
  89               2 (1 - m)
  90 @end group
  91 @end example
  92
  93 Some examples of elliptic integrals:
  94 @c ===beg===
  95 @c elliptic_f (phi, m);
  96 @c elliptic_f (phi, 0);
  97 @c elliptic_f (phi, 1);
  98 @c elliptic_e (phi, 1);
  99 @c elliptic_e (phi, 0);
 100 @c elliptic_kc (1/2);
 101 @c makegamma (%);
 102 @c diff (elliptic_f (phi, m), phi);
 103 @c diff (elliptic_f (phi, m), m);
 104 @c ===end===
 105 @example
 106 @group
 107 (%i1) elliptic_f (phi, m);
 108 (%o1)                  elliptic_f(phi, m)
 109 @end group
 110 @group
 111 (%i2) elliptic_f (phi, 0);
 112 (%o2)                          phi
 113 @end group
 114 @group
 115 (%i3) elliptic_f (phi, 1);
 116                                phi   %pi
 117 (%o3)                  log(tan(--- + ---))
 118                                 2     4
 119 @end group
 120 @group
 121 (%i4) elliptic_e (phi, 1);
 122                     phi                  phi
 123 (%o4)       2 round(---) - sin(%pi round(---) - phi)
 124                     %pi                  %pi
 125 @end group
 126 @group
 127 (%i5) elliptic_e (phi, 0);
 128 (%o5)                          phi
 129 @end group
 130 @group
 131 (%i6) elliptic_kc (1/2);
 132                                 3/2
 133                              %pi
 134 (%o6)                      -----------
 135                                   2 3
 136                            2 gamma (-)
 137                                     4
 138 @end group
 139 @group
 140 (%i7) makegamma (%);
 141                                 3/2
 142                              %pi
 143 (%o7)                      -----------
 144                                   2 3
 145                            2 gamma (-)
 146                                     4
 147 @end group
 148 @group
 149 (%i8) diff (elliptic_f (phi, m), phi);
 150                                 1
 151 (%o8)                 ---------------------
 152                                     2
 153                       sqrt(1 - m sin (phi))
 154 @end group
 155 @group
 156 (%i9) diff (elliptic_f (phi, m), m);
 157        elliptic_e(phi, m) - (1 - m) elliptic_f(phi, m)
 158 (%o9) (-----------------------------------------------
 159                               m
 160                                  cos(phi) sin(phi)
 161                              - ---------------------)/(2 (1 - m))
 162                                              2
 163                                sqrt(1 - m sin (phi))
 164 @end group
 165 @end example
 166
 167 Support for elliptic functions and integrals was written by Raymond
 168 Toy.  It is placed under the terms of the General Public License (GPL)
 169 that governs the distribution of Maxima.
 170
 171 @opencatbox{Categories:}
 172 @category{Elliptic functions}
 173 @closecatbox
 174
 175 @node Functions and Variables for Elliptic Functions, Functions and Variables for Elliptic Integrals, Introduction to Elliptic Functions and Integrals, Elliptic Functions
 176 @comment  node-name,  next,  previous,  up
 177 @section Functions and Variables for Elliptic Functions
 178 See @urlaands{Section 6.12, 569} and @urldlmf{22.2} for more
 179 information.
 180
 181 @anchor{jacobi_sn}
 182 @deffn {Function} jacobi_sn (@var{u}, @var{m})
 183 The Jacobian elliptic function
 184 m4_mathdot(<<<{\rm sn}(u,m)>>>, <<<sn(u,m)>>>)
 185
 186
 187 @opencatbox{Categories:}
 188 @category{Elliptic functions}
 189 @closecatbox
 190 @end deffn
 191
 192 @anchor{jacobi_cn}
 193 @deffn {Function} jacobi_cn (@var{u}, @var{m})
 194 The Jacobian elliptic function
 195 m4_mathdot(<<<{\rm cn}(u,m)>>>, <<<cn(u,m)>>>)
 196
 197 @opencatbox{Categories:}
 198 @category{Elliptic functions}
 199 @closecatbox
 200 @end deffn
 201
 202 @anchor{jacobi_dn}
 203 @deffn {Function} jacobi_dn (@var{u}, @var{m})
 204 The Jacobian elliptic function
 205 m4_mathdot(<<<{\rm dn}(u,m)>>>, <<<dn(u,m)>>>)
 206
 207 @opencatbox{Categories:}
 208 @category{Elliptic functions}
 209 @closecatbox
 210 @end deffn
 211
 212 @anchor{jacobi_ns}
 213 @deffn {Function} jacobi_ns (@var{u}, @var{m})
 214 The Jacobian elliptic function
 215 m4_mathdot(<<<{\rm ns}(u,m) = 1/{\rm
 216 sn}(u,m)>>>, <<<ns(u,m) = 1/sn(u,m)>>>)
 217
 218 @opencatbox{Categories:}
 219 @category{Elliptic functions}
 220 @closecatbox
 221 @end deffn
 222
 223 @anchor{jacobi_sc}
 224 @deffn {Function} jacobi_sc (@var{u}, @var{m})
 225 The Jacobian elliptic function
 226 m4_mathdot(<<<{\rm sc}(u,m) = {\rm
 227 sn}(u,m)/{\rm cn}(u,m)>>>, <<<sc(u,m) = sn(u,m)/cn(u,m)>>>)
 228
 229 @opencatbox{Categories:}
 230 @category{Elliptic functions}
 231 @closecatbox
 232 @end deffn
 233
 234 @anchor{jacobi_sd}
 235 @deffn {Function} jacobi_sd (@var{u}, @var{m})
 236 The Jacobian elliptic function
 237 m4_mathdot({\rm sd}(u,m) = {\rm
 238 sn}(u,m)/{\rm dn}(u,m), <<<sd(u,m) = sn(u,m)/dn(u,m)>>>)
 239
 240 @opencatbox{Categories:}
 241 @category{Elliptic functions}
 242 @closecatbox
 243 @end deffn
 244
 245 @anchor{jacobi_nc}
 246 @deffn {Function} jacobi_nc (@var{u}, @var{m})
 247 The Jacobian elliptic function
 248 m4_mathdot(<<<{\rm nc}(u,m) = 1/{\rm cn}(u,m)>>>, <<<nc(u,m) = 1/cn(u,m)>>>)
 249
 250 @opencatbox{Categories:}
 251 @category{Elliptic functions}
 252 @closecatbox
 253 @end deffn
 254
 255 @anchor{jacobi_cs}
 256 @deffn {Function} jacobi_cs (@var{u}, @var{m})
 257 The Jacobian elliptic function
 258 m4_mathdot(<<<{\rm cs}(u,m) = {\rm
 259 cn}(u,m)/{\rm sn}(u,m)>>>, <<<cs(u,m) = cn(u,m)/sn(u,m)>>>)
 260
 261 @opencatbox{Categories:}
 262 @category{Elliptic functions}
 263 @closecatbox
 264 @end deffn
 265
 266 @anchor{jacobi_cd}
 267 @deffn {Function} jacobi_cd (@var{u}, @var{m})
 268 The Jacobian elliptic function
 269 m4_mathdot(<<<{\rm cd}(u,m) = {\rm cn}(u,m)/{\rm dn}(u,m)>>>, <<<cd(u,m) = cn(u,m)/dn(u,m)>>>)
 270
 271 @opencatbox{Categories:}
 272 @category{Elliptic functions}
 273 @closecatbox
 274 @end deffn
 275
 276 @anchor{jacobi_nd}
 277 @deffn {Function} jacobi_nd (@var{u}, @var{m})
 278 The Jacobian elliptic function
 279 m4_mathdot(<<<{\rm nd}(u,m) = 1/{\rm dn}(u,m)>>>,
 280 <<<nd(u,m) = 1/dn(u,m)>>>)
 281
 282 @opencatbox{Categories:}
 283 @category{Elliptic functions}
 284 @closecatbox
 285 @end deffn
 286
 287 @anchor{jacobi_ds}
 288 @deffn {Function} jacobi_ds (@var{u}, @var{m})
 289 The Jacobian elliptic function
 290 m4_mathdot(<<<{\rm ds}(u,m) =
 291 {\rm dn}(u,m)/{\rm sn}(u,m)>>>, <<<ds(u,m) = dn(u,m)/sn(u,m)>>>)
 292
 293 @opencatbox{Categories:}
 294 @category{Elliptic functions}
 295 @closecatbox
 296 @end deffn
 297
 298 @anchor{jacobi_dc}
 299 @deffn {Function} jacobi_dc (@var{u}, @var{m})
 300 The Jacobian elliptic function
 301 m4_mathdot(<<<{\rm dc}(u,m) =
 302 {\rm dn}(u,m)/{\rm cn}(u,m)>>>, <<<dc(u,m) = dn(u,m)/cn(u,m)>>>)
 303
 304 @opencatbox{Categories:}
 305 @category{Elliptic functions}
 306 @closecatbox
 307 @end deffn
 308
 309 @anchor{jacobi_am}
 310 @deffn {Function} jacobi_am (@var{u}, @var{m})
 311 The Jacobi amplitude function, @code{jacobi_am}, is defined implicitly by (see
 312 @url{http://functions.wolfram.com/09.24.02.0001.01})
 313 m4_math(<<<z = {\rm am}(w, m)>>>,<<<z = am(w, m)>>>)
 314 where @math{w = F(z,m)} where @math{F(z,m)} is the incomplete elliptic
 315 integral of the first kind (@pxref{elliptic_f}).  It is defined for
 316 all real and complex values of @math{z} and @math{m}.  In particular
 317 for real @math{z} and @math{m} with @math{|m|<1},
 318 m4_math(<<<{\rm am}(z,m)>>>,<<<am(z,m)>>>)
 319 maps the entire real line to the entire real line.  For other values
 320 of @math{z} and @math{m}, the following relationship is used:
 321 m4_mathdot(<<<{\rm am}(z,m) = \sin^{-1}({\rm jacobi\_sn}(z, m))>>>,
 322 <<<am(z,m) = asin(jacobi_sn(z,m))>>>)
 323
 324 Some examples:
 325 @c ===beg===
 326 @c jacobi_am(z,0);
 327 @c jacobi_am(z,1);
 328 @c jacobi_am(0,m);
 329 @c jacobi_am(100, .5);
 330 @c jacobi_am(0.5, 1.5);
 331 @c jacobi_am(1.5b0, 1.5b0+%i);
 332 @c ===end===
 333 @example
 334 @group
 335 (%i1) jacobi_am(z,0);
 336 (%o1)                           z
 337 @end group
 338 @group
 339 (%i2) jacobi_am(z,1);
 340                                  z    %pi
 341 (%o2)                   2 atan(%e ) - ---
 342                                        2
 343 @end group
 344 @group
 345 (%i3) jacobi_am(0,m);
 346 (%o3)                           0
 347 @end group
 348 @group
 349 (%i4) jacobi_am(100, .5);
 350 (%o4)                   84.70311272411382
 351 @end group
 352 @group
 353 (%i5) jacobi_am(0.5, 1.5);
 354 (%o5)                  0.4707197897046991
 355 @end group
 356 @group
 357 (%i6) jacobi_am(1.5b0, 1.5b0+%i);
 358 (%o6)    9.340542168700782b-1 - 3.723960452146071b-1 %i
 359 @end group
 360 @end example
 361
 362 @c ===beg===
 363 @c plot2d([jacobi_am(x,.4),jacobi_am(x,.7),jacobi_am(x,.99),jacobi_am(x,.999999)],[x,0,10*%pi]);
 364 @c ===end===
 365 @example
 366 @group
 367 (%i1) plot2d([jacobi_am(x,.4),jacobi_am(x,.7),jacobi_am(x,.99),jacobi_am(x,.999999)],[x,0,10*%pi]);
 368 (%o1)                         false
 369 @end group
 370 @end example
 371
 372 @ifnotinfo
 373 Compare this plot with the plot from @urldlmf{22.16.iv}:
 374
 375 @image{figures/jacobi_am}
 376 @end ifnotinfo
 377
 378 @opencatbox{Categories:}
 379 @category{Elliptic functions}
 380 @closecatbox
 381 @end deffn
 382
 383 @anchor{inverse_jacobi_dn}
 384 @deffn {Function} inverse_jacobi_dn (@var{u}, @var{m})
 385 The inverse of the Jacobian elliptic function
 386 m4_mathdot(<<<{\rm
 387 dn}(u,m)>>>, <<<dn(u,m)>>>)
 388 For
 389 m4_mathcomma(\sqrt{1-m}\le u \le 1,
 390 sqrt(1-m) <= u <= 1)
 391 it can also be written (@urldlmf{22.15.E14}):
 392 m4_displaymath(
 393 {\rm inverse\_jacobi\_dn}(u, m) = \int_u^1 {dt\over \sqrt{(1-t^2)(t^2-(1-m))}},
 394 @example
 395                                 1
 396                                /
 397                                [               1
 398      inverse_jacobi_dn(u, m) = I  --------------------------- dt
 399                                ]             2    2
 400                                /  sqrt((1 - t ) (t  + m - 1))
 401                                 u
 402 @end example
 403 )
 404
 405
 406 @opencatbox{Categories:}
 407 @category{Elliptic functions}
 408 @closecatbox
 409 @end deffn
 410
 411 @anchor{inverse_jacobi_ns}
 412 @deffn {Function} inverse_jacobi_ns (@var{u}, @var{m})
 413 The inverse of the Jacobian elliptic function
 414 m4_mathdot(<<<{\rm
 415 ns}(u,m)>>>, <<<ns(u,m)>>>)
 416 For
 417 m4_mathcomma(1 \le u, 1 <= u)
 418 it can also be written (@urldlmf{22.15.E121}):
 419 m4_displaymath(
 420 {\rm inverse\_jacobi\_ns}(u, m) = \int_u^{\infty} {dt\over \sqrt{(1-t^2)(t^2-m)}},
 421 @example
 422                                  inf
 423                                 /
 424                                 [               1
 425       inverse_jacobi_ns(u, m) = I    ----------------------- dt
 426                                 ]               2    2
 427                                 /    sqrt((1 - t ) (t  - m))
 428                                  u
 429 @end example
 430 )
 431
 432 @opencatbox{Categories:}
 433 @category{Elliptic functions}
 434 @closecatbox
 435 @end deffn
 436
 437 @anchor{inverse_jacobi_sc}
 438 @deffn {Function} inverse_jacobi_sc (@var{u}, @var{m})
 439 The inverse of the Jacobian elliptic function
 440 m4_mathdot(<<<{\rm
 441 sc}(u,m)>>>, <<<sc(u,m)>>>)
 442 For all @math{u} it can also be written (@urldlmf{22.15.E20}):
 443 m4_displaymath(
 444 {\rm inverse\_jacobi\_sc}(u, m) = \int_0^u {dt\over \sqrt{(1+t^2)(1+(1-m)t^2)}},
 445 @example
 446                               u
 447                              /
 448                              [                 1
 449    inverse_jacobi_sc(u, m) = I  ------------------------------- dt
 450                              ]             2                2
 451                              /  sqrt((1 - t ) (1 - (1 - m) t ))
 452                               0
 453 @end example
 454 )
 455
 456 @opencatbox{Categories:}
 457 @category{Elliptic functions}
 458 @closecatbox
 459 @end deffn
 460
 461 @anchor{inverse_jacobi_sd}
 462 @deffn {Function} inverse_jacobi_sd (@var{u}, @var{m})
 463 The inverse of the Jacobian elliptic function
 464 m4_mathdot(<<<{\rm
 465 sd}(u,m)>>>, <<<sd(u,m)>>>)
 466 For
 467 m4_mathcomma(-1/\sqrt{1-m}\le u \le
 468 1/\sqrt{1-m}, -1/sqrt(1-m) <= 1/sqrt(1-m))
 469 it can also be written (@urldlmf{22.15.E16}):
 470 m4_displaymath(
 471 {\rm inverse\_jacobi\_sd}(u, m) = \int_0^u {dt\over \sqrt{(1-(1-m)t^2)(1+mt^2)}},
 472 @example
 473                             u
 474                            /
 475                            [                  1
 476  inverse_jacobi_sd(u, m) = I  --------------------------------- dt
 477                            ]                     2      2
 478                            /  sqrt((1 - (1 - m) t ) (m t  + 1))
 479                             0
 480 @end example
 481 )
 482
 483 @opencatbox{Categories:}
 484 @category{Elliptic functions}
 485 @closecatbox
 486 @end deffn
 487
 488 @anchor{inverse_jacobi_nc}
 489 @deffn {Function} inverse_jacobi_nc (@var{u}, @var{m})
 490 The inverse of the Jacobian elliptic function
 491 m4_mathdot(<<<{\rm
 492 nc}(u,m)>>>, <<<nc(u,m)>>>)
 493 For
 494 m4_mathcomma(1\le u, 1 <= u)
 495 it can also be written (@urldlmf{22.15.E19}):
 496 m4_displaymath(
 497 {\rm inverse\_jacobi\_nc}(u, m) = \int_1^u {dt\over \sqrt{(t^2-1)(m+(1-m)t^2)}},
 498 @example
 499                               1
 500                              /
 501                              [                 1
 502  inverse_jacobi_nc(u, m) = - I  ------------------------------- dt
 503                              ]         2                2
 504                              /  sqrt((t  - 1) ((1 - m) t  + m))
 505                               u
 506 @end example
 507 )
 508
 509 @opencatbox{Categories:}
 510 @category{Elliptic functions}
 511 @closecatbox
 512 @end deffn
 513
 514 @anchor{inverse_jacobi_cs}
 515 @deffn {Function} inverse_jacobi_cs (@var{u}, @var{m})
 516 The inverse of the Jacobian elliptic function
 517 m4_mathdot(<<<{\rm
 518 cs}(u,m)>>>, <<<cs(u,m)>>>)
 519 For all @math{u} it can also be written (@urldlmf{22.15.E23}):
 520 m4_displaymath(
 521 {\rm inverse\_jacobi\_cs}(u, m) = \int_u^{\infty} {dt\over \sqrt{(1+t^2)(t^2+(1-m))}},
 522 @example
 523                              inf
 524                             /
 525                             [                  1
 526   inverse_jacobi_cs(u, m) = I    ----------------------------- dt
 527                             ]          2            2
 528                             /    sqrt(t  + 1) sqrt(t  - m + 1)
 529                              u
 530 @end example
 531 )
 532
 533
 534 @opencatbox{Categories:}
 535 @category{Elliptic functions}
 536 @closecatbox
 537 @end deffn
 538
 539 @anchor{inverse_jacobi_cd}
 540 @deffn {Function} inverse_jacobi_cd (@var{u}, @var{m})
 541 The inverse of the Jacobian elliptic function
 542 m4_mathdot(<<<{\rm
 543 cd}(u,m)>>>, <<<cd(u,m)>>>)
 544 For
 545 m4_mathcomma(-1\le u \le 1, -1 <= u <=
 546 1)
 547 it can also be written (@urldlmf{22.15.E15}):
 548 m4_displaymath(
 549 {\rm inverse\_jacobi\_cd}(u, m) = \int_u^1 {dt\over \sqrt{(1-t^2)(1-mt^2)}},
 550 @example
 551                                 1
 552                                /
 553                                [              1
 554      inverse_jacobi_cd(u, m) = I  ------------------------- dt
 555                                ]             2          2
 556                                /  sqrt((1 - t ) (1 - m t ))
 557                                 u
 558 @end example
 559 )
 560
 561
 562 @opencatbox{Categories:}
 563 @category{Elliptic functions}
 564 @closecatbox
 565 @end deffn
 566
 567 @anchor{inverse_jacobi_nd}
 568 @deffn {Function} inverse_jacobi_nd (@var{u}, @var{m})
 569 The inverse of the Jacobian elliptic function
 570 m4_mathdot(<<<{\rm
 571 nd}(u,m)>>>, <<<nd(u,m)>>>)
 572 For
 573 m4_mathcomma(1\le u \le 1/\sqrt{1-m},
 574 1 <= u <= 1/sqrt(1-m))
 575 it can also be written (@urldlmf{22.15.E17}):
 576 m4_displaymath(
 577 {\rm inverse\_jacobi\_nd}(u, m) = \int_1^u {dt\over \sqrt{(t^2-1)(1-(1-m)t^2)}},
 578 @example
 579                               1
 580                              /
 581                              [                 1
 582  inverse_jacobi_nd(u, m) = - I  ------------------------------- dt
 583                              ]         2                    2
 584                              /  sqrt((t  - 1) (1 - (1 - m) t ))
 585                               u
 586 @end example
 587 )
 588
 589
 590 @opencatbox{Categories:}
 591 @category{Elliptic functions}
 592 @closecatbox
 593 @end deffn
 594
 595 @anchor{inverse_jacobi_ds}
 596 @deffn {Function} inverse_jacobi_ds (@var{u}, @var{m})
 597 The inverse of the Jacobian elliptic function
 598 m4_mathdot(<<<{\rm
 599 ds}(u,m)>>>, <<<ds(u,m)>>>)
 600 For
 601 m4_mathcomma(\sqrt{1-m}\le u,
 602 sqrt(1-m) <= u)
 603 it can also be written (@urldlmf{22.15.E22}):
 604 m4_displaymath(
 605 {\rm inverse\_jacobi\_ds}(u, m) = \int_u^{\infty} {dt\over \sqrt{(t^2+m)(t^2-(1-m))}},
 606 @example
 607                               inf
 608                              /
 609                              [                 1
 610    inverse_jacobi_ds(u, m) = I    --------------------------- dt
 611                              ]           2            2
 612                              /    sqrt((t  + m - 1) (t  + m))
 613                               u
 614 @end example
 615 )
 616
 617
 618 @opencatbox{Categories:}
 619 @category{Elliptic functions}
 620 @closecatbox
 621 @end deffn
 622
 623 @anchor{inverse_jacobi_dc}
 624 @deffn {Function} inverse_jacobi_dc (@var{u}, @var{m})
 625 The inverse of the Jacobian elliptic function
 626 m4_mathdot(<<<{\rm
 627 dc}(u,m)>>>, <<<dc(u,m)>>>)
 628 For
 629 m4_mathcomma(1 \le u, 1 <= u)
 630 it can also be written (@urldlmf{22.15.E18}):
 631 m4_displaymath(
 632 {\rm inverse\_jacobi\_dc}(u, m) = \int_1^u {dt\over \sqrt{(t^2-1)(t^2-m)}},
 633 @example
 634                                   u
 635                                  /
 636                                  [             1
 637      inverse_jacobi_dc(u, m) =   I  ----------------------- dt
 638                                  ]         2        2
 639                                  /  sqrt((t  - 1) (t  - m))
 640                                   1
 641 @end example
 642 )
 643
 644
 645 @opencatbox{Categories:}
 646 @category{Elliptic functions}
 647 @closecatbox
 648 @end deffn
 649
 650
 651 @node Functions and Variables for Elliptic Integrals,  , Functions and Variables for Elliptic Functions, Elliptic Functions
 652 @comment  node-name,  next,  previous,  up
 653 @section Functions and Variables for Elliptic Integrals
 654
 655 @anchor{elliptic_f}
 656 @deffn {Function} elliptic_f (@var{phi}, @var{m})
 657 The incomplete elliptic integral of the first kind, defined as
 658
 659 m4_displaymath(
 660 <<<\int_0^{\phi} {\frac{d\theta}{\sqrt{1-m\sin^2\theta}}}>>>
 661 ,
 662 <<<@math{integrate(1/sqrt(1 - m*sin(x)^2), x, 0, phi)}>>>
 663 )
 664
 665 See also @ref{elliptic_e} and @ref{elliptic_kc}.
 666
 667 @opencatbox{Categories:}
 668 @category{Elliptic integrals}
 669 @closecatbox
 670 @end deffn
 671
 672 @anchor{elliptic_e}
 673 @deffn {Function} elliptic_e (@var{phi}, @var{m})
 674 The incomplete elliptic integral of the second kind, defined as
 675
 676 m4_displaymath(
 677 <<<\int_0^\phi {\sqrt{1 - m\sin^2\theta}}\, d\theta>>>
 678 ,
 679 <<<@math{elliptic_e(phi, m) = integrate(sqrt(1 - m*sin(x)^2), x, 0, phi)}>>>
 680 )
 681
 682 See also @ref{elliptic_f} and @ref{elliptic_ec}.
 683
 684 @opencatbox{Categories:}
 685 @category{Elliptic integrals}
 686 @closecatbox
 687 @end deffn
 688
 689 @anchor{elliptic_eu}
 690 @deffn {Function} elliptic_eu (@var{u}, @var{m})
 691 The incomplete elliptic integral of the second kind, defined as
 692
 693 m4_displaymath(
 694 <<<E(u, m) = \int_0^u {\rm dn}(v, m)\, dv  = \int_0^\tau \sqrt{\frac{1-m t^2}{1-t^2}}\, dt>>>
 695 ,
 696 <<<@math{elliptic_eu(u, m) = integrate(dn(v,m)^2,v,0,u) = integrate(sqrt(1-m*t^2)/sqrt(1-t^2), t, 0, tau)}>>>
 697 )
 698
 699 where
 700 m4_mathdot(
 701 <<<\tau = {\rm sn}(u,m)>>>
 702 ,
 703 <<<tau = sn(u,m)>>>
 704 )
 705
 706
 707 This is related to @code{elliptic_e} by
 708
 709 m4_displaymath(
 710 <<<E(u,m) = E(\sin^{-1} {\rm sn}(u, m), m)>>>
 711 ,
 712 <<<@math{elliptic_eu(u, m) = elliptic_e(asin(sn(u,m)),m)}>>>
 713 )
 714
 715 See also @ref{elliptic_e}.
 716 @opencatbox{Categories:}
 717 @category{Elliptic integrals}
 718 @closecatbox
 719 @end deffn
 720
 721 @anchor{elliptic_pi}
 722 @deffn {Function} elliptic_pi (@var{n}, @var{phi}, @var{m})
 723 The incomplete elliptic integral of the third kind, defined as
 724
 725 m4_displaymath(
 726 <<<\int_0^\phi {{d\theta}\over{(1-n\sin^2 \theta)\sqrt{1 - m\sin^2\theta}}}>>>
 727 ,
 728 <<<@math{integrate(1/(1-n*sin(x)^2)/sqrt(1 - m*sin(x)^2), x, 0, phi)}>>>
 729 )
 730
 731 @opencatbox{Categories:}
 732 @category{Elliptic integrals}
 733 @closecatbox
 734 @end deffn
 735
 736 @anchor{elliptic_kc}
 737 @deffn {Function} elliptic_kc (@var{m})
 738 The complete elliptic integral of the first kind, defined as
 739
 740 m4_displaymath(
 741 <<<\int_0^{\frac{\pi}{2}} {{d\theta}\over{\sqrt{1 - m\sin^2\theta}}}>>>
 742 ,
 743 <<<@math{integrate(1/sqrt(1 - m*sin(x)^2), x, 0, pi/2)}>>>
 744 )
 745
 746 For certain values of @math{m}, the value of the integral is known in
 747 terms of @math{Gamma} functions.  Use @mref{makegamma} to evaluate them.
 748
 749 @opencatbox{Categories:}
 750 @category{Elliptic integrals}
 751 @closecatbox
 752 @end deffn
 753
 754 @anchor{elliptic_ec}
 755 @deffn {Function} elliptic_ec (@var{m})
 756 The complete elliptic integral of the second kind, defined as
 757
 758 m4_displaymath(
 759 <<<\int_0^{\frac{\pi}{2}} \sqrt{1 - m\sin^2\theta}\, d\theta>>>
 760 ,
 761 <<<@math{integrate(sqrt(1 - m*sin(x)^2), x, 0, pi/2)}>>>
 762 )
 763
 764 For certain values of @math{m}, the value of the integral is known in
 765 terms of @math{Gamma} functions.  Use @mref{makegamma} to evaluate them.
 766
 767 @opencatbox{Categories:}
 768 @category{Elliptic integrals}
 769 @closecatbox
 770 @end deffn
 771
 772 @anchor{carlson_rc}
 773 @deffn {Function} carlson_rc (@var{x}, @var{y})
 774 Carlson's RC integral is defined by
 775
 776 m4_displaymath(
 777 <<<R_C(x, y) = \frac{1}{2} \int_0^{\infty} \frac{1}{\sqrt{t+x}(t+y)}\, dt>>>
 778 ,
 779 <<<@math{integrate(1/2*(t+x)^(-1/2)/(t+y), t, 0, inf)}>>>
 780 )
 781
 782 This integral is related to many elementary functions in the following
 783 way:
 784
 785 m4_displaymath(
 786 <<<\eqalign{
 787 \log x &= (x-1) R_C\left(\left({\frac{1+x}{2}}\right)^2, x\right), \quad x > 0 \cr
 788 \sin^{-1} x &= x R_C(1-x^2, 1), \quad |x| \le 1 \cr
 789 \cos^{-1} x &= \sqrt{1-x^2} R_C(x^2,1), \quad 0 \le x \le 1  \cr
 790 \tan^{-1} x &= x  R_C(1,1+x^2)  \cr
 791 \sinh^{-1} x &= x  R_C(1+x^2,1)  \cr
 792 \cosh^{-1} x &= \sqrt{x^2-1}  R_C(x^2,1), \quad x \ge 1  \cr
 793 \tanh^{-1}(x) &= x  R_C(1,1-x^2), \quad |x| \le 1
 794 }>>>
 795 ,
 796 <<<@math{log(x)  = (x-1)*rc(((1+x)/2)^2, x), x > 0}
 797
 798 @math{asin(x) = x * rc(1-x^2, 1), |x|<= 1}
 799
 800 @math{acos(x) = sqrt(1-x^2)*rc(x^2,1), 0 <= x <=1}
 801
 802 @math{atan(x) = x * rc(1,1+x^2)}
 803
 804 @math{asinh(x) = x * rc(1+x^2,1)}
 805
 806 @math{acosh(x) = sqrt(x^2-1) * rc(x^2,1), x >= 1}
 807
 808 @math{atanh(x) = x * rc(1,1-x^2), |x|<=1}>>>
 809 )
 810
 811 Also, we have the relationship
 812
 813 m4_displaymath(
 814 <<<R_C(x,y) = R_F(x,y,y)>>>
 815 ,
 816 @math{R_C(x,y) = R_F(x,y,y)}
 817 )
 818
 819 Some special values:
 820 m4_displaymath(
 821 <<<\eqalign{R_C(0, 1) &= \frac{\pi}{2} \cr
 822 R_C(0, 1/4) &= \pi \cr
 823 R_C(2,1) &= \log(\sqrt{2} + 1) \cr
 824 R_C(i,i+1) &= \frac{\pi}{4} + \frac{i}{2} \log(\sqrt{2}-1) \cr
 825 R_C(0,i) &= (1-i)\frac{\pi}{2\sqrt{2}} \cr
 826 }>>>
 827 ,
 828 @math{R_C(0,1) = pi/2}
 829
 830 @math{R_C(0,1/4) = pi}
 831
 832 @math{R_C(2,1) = log(sqrt(2)+1)}
 833
 834 @math{R_C(i, i+1) = pi/4 + i/2*log(sqrt(2)+1)}
 835
 836 @math{R_C(0, i) = (1-i)*pi/(2*sqrt(2))}
 837 )
 838
 839
 840 @opencatbox{Categories:}
 841 @category{Elliptic integrals}
 842 @closecatbox
 843 @end deffn
 844
 845 @anchor{carlson_rd}
 846 @deffn {Function} carlson_rd (@var{x}, @var{y}, @var{z})
 847 Carlson's RD integral is defined by
 848
 849 m4_displaymath(
 850 <<<R_D(x,y,z) = \frac{3}{2} \int_0^{\infty} \frac{1}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}\,(t+z)}\, dt>>>
 851 ,
 852 <<<@math{R_D(x,y,z) = 3/2*integrate(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+z)*(t+z)), t, 0, inf)}>>>
 853 )
 854
 855 We also have the special values
 856
 857 m4_displaymath(
 858 <<<\eqalign{
 859 R_D(x,x,x) &= x^{-\frac{3}{2}} \cr
 860 R_D(0,y,y) &= \frac{3}{4} \pi y^{-\frac{3}{2}} \cr
 861 R_D(0,2,1) &= 3 \sqrt{\pi} \frac{\Gamma(\frac{3}{4})}{\Gamma(\frac{1}{4})}
 862 }>>>
 863 ,
 864 <<<@math{R_D(x,x,x) = x^(-3/2)}
 865
 866 @math{R_D(0,y,y) = 3/4*pi*y^(-3/2)}
 867
 868 @math{R_D(0,2,1) = 3 sqrt(pi) gamma(3/4)/gamma(1/4)}>>>
 869 )
 870
 871
 872 It is also related to the complete elliptic E function as follows
 873
 874 m4_displaymath(
 875 <<<E(m) = R_F(0, 1 - m, 1) - \frac{m}{3} R_D(0, 1 - m, 1)>>>
 876 ,
 877 <<<@math{E(m) = R_F(0, 1 - m, 1) - (m/3)* R_D(0, 1 - m, 1)}>>>
 878 )
 879
 880 @opencatbox{Categories:}
 881 @category{Elliptic integrals}
 882 @closecatbox
 883 @end deffn
 884
 885 @anchor{carlson_rf}
 886 @deffn {Function} carlson_rf (@var{x}, @var{y}, @var{z})
 887 Carlson's RF integral is defined by
 888
 889 m4_displaymath(
 890 <<<R_F(x,y,z) = \frac{1}{2} \int_0^{\infty} \frac{1}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}}\, dt>>>
 891 ,
 892 <<<@math{R_F(x,y,z) = 1/2*integrate(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+z)), t, 0, inf)}>>>
 893 )
 894
 895 We also have the special values
 896
 897 m4_displaymath(
 898 <<<\eqalign{
 899 R_F(0,1,2)  &= \frac{\Gamma({\frac{1}{4}})^2}{4\sqrt{2\pi}} \cr
 900 R_F(i,-i,0) &= \frac{\Gamma({\frac{1}{4}})^2}{4\sqrt{\pi}}
 901 }>>>
 902 ,
 903 <<<
 904 @math{R_F(0,1,2) = gamma(1/4)^2/(4*sqrt(2*pi))}
 905
 906 @math{R_F(i,-i,0) = gamma(1/4)^2/(4*sqrt(pi))}
 907 >>>
 908 )
 909
 910 It is also related to the complete elliptic E function as follows
 911
 912 m4_displaymath(
 913 <<<E(m) = R_F(0, 1 - m, 1) - \frac{m}{3} R_D(0, 1 - m, 1)>>>
 914 ,
 915 <<<@math{E(m) = R_F(0, 1 - m, 1) - (m/3)* R_D(0, 1 - m, 1)}>>>
 916 )
 917
 918 @opencatbox{Categories:}
 919 @category{Elliptic integrals}
 920 @closecatbox
 921 @end deffn
 922
 923 @anchor{carlson_rj}
 924 @deffn {Function} carlson_rj (@var{x}, @var{y}, @var{z}, @var{p})
 925 Carlson's RJ integral is defined by
 926
 927 m4_displaymath(
 928 <<<R_J(x,y,z) = \frac{1}{2} \int_0^{\infty} \frac{1}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}\,(t+p)}\, dt>>>
 929 ,
 930 <<<
 931 @math{R_J(x,y,z) = 1/2*integrate(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+z)*(t+p)), t, 0, inf)}
 932 >>>)
 933
 934 @opencatbox{Categories:}
 935 @category{Elliptic integrals}
 936 @closecatbox
 937 @end deffn
 938