1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Elliptic Integral Overview</title> 5<link rel="stylesheet" href="../../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../ellint.html" title="Elliptic Integrals"> 9<link rel="prev" href="../ellint.html" title="Elliptic Integrals"> 10<link rel="next" href="ellint_carlson.html" title="Elliptic Integrals - Carlson Form"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="../ellint.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../ellint.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="ellint_carlson.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h3 class="title"> 27<a name="math_toolkit.ellint.ellint_intro"></a><a class="link" href="ellint_intro.html" title="Elliptic Integral Overview">Elliptic Integral Overview</a> 28</h3></div></div></div> 29<p> 30 The main reference for the elliptic integrals is: 31 </p> 32<div class="blockquote"><blockquote class="blockquote"><p> 33 M. Abramowitz and I. A. Stegun (Eds.) (1964) Handbook of Mathematical Functions 34 with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards 35 Applied Mathematics Series, U.S. Government Printing Office, Washington, 36 D.C. 37 </p></blockquote></div> 38<p> 39 and its recently revised version <a href="http://dlmf.nist.gov/" target="_top">NIST 40 Digital Library of Mathematical Functions (DMLF)</a>, in particular 41 </p> 42<div class="blockquote"><blockquote class="blockquote"><p> 43 <a href="https://dlmf.nist.gov/19" target="_top">Elliptic Integrals, B. C. Carlson</a> 44 </p></blockquote></div> 45<p> 46 Mathworld also contain a lot of useful background information: 47 </p> 48<div class="blockquote"><blockquote class="blockquote"><p> 49 <a href="http://mathworld.wolfram.com/EllipticIntegral.html" target="_top">Weisstein, 50 Eric W. "Elliptic Integral." From MathWorld--A Wolfram Web Resource.</a> 51 </p></blockquote></div> 52<p> 53 As does <a href="http://en.wikipedia.org/wiki/Elliptic_integral" target="_top">Wikipedia 54 Elliptic integral</a>. 55 </p> 56<h5> 57<a name="math_toolkit.ellint.ellint_intro.h0"></a> 58 <span class="phrase"><a name="math_toolkit.ellint.ellint_intro.notation"></a></span><a class="link" href="ellint_intro.html#math_toolkit.ellint.ellint_intro.notation">Notation</a> 59 </h5> 60<p> 61 All variables are real numbers unless otherwise noted. 62 </p> 63<h5> 64<a name="math_toolkit.ellint.ellint_intro.h1"></a> 65 <span class="phrase"><a name="math_toolkit.ellint.ellint_intro.definition"></a></span><a class="link" href="ellint_intro.html#math_toolkit.ellint.ellint_intro.definition">Definition</a> 66 </h5> 67<div class="blockquote"><blockquote class="blockquote"><p> 68 <span class="inlinemediaobject"><img src="../../../equations/ellint1.svg"></span> 69 70 </p></blockquote></div> 71<p> 72 is called elliptic integral if <span class="emphasis"><em>R(t, s)</em></span> is a rational 73 function of <span class="emphasis"><em>t</em></span> and <span class="emphasis"><em>s</em></span>, and <span class="emphasis"><em>s<sup>2</sup></em></span> 74 is a cubic or quartic polynomial in <span class="emphasis"><em>t</em></span>. 75 </p> 76<p> 77 Elliptic integrals generally cannot be expressed in terms of elementary functions. 78 However, Legendre showed that all elliptic integrals can be reduced to the 79 following three canonical forms: 80 </p> 81<p> 82 Elliptic Integral of the First Kind (Legendre form) 83 </p> 84<div class="blockquote"><blockquote class="blockquote"><p> 85 <span class="inlinemediaobject"><img src="../../../equations/ellint2.svg"></span> 86 87 </p></blockquote></div> 88<p> 89 Elliptic Integral of the Second Kind (Legendre form) 90 </p> 91<div class="blockquote"><blockquote class="blockquote"><p> 92 <span class="inlinemediaobject"><img src="../../../equations/ellint3.svg"></span> 93 94 </p></blockquote></div> 95<p> 96 Elliptic Integral of the Third Kind (Legendre form) 97 </p> 98<div class="blockquote"><blockquote class="blockquote"><p> 99 <span class="inlinemediaobject"><img src="../../../equations/ellint4.svg"></span> 100 101 </p></blockquote></div> 102<p> 103 where 104 </p> 105<div class="blockquote"><blockquote class="blockquote"><p> 106 <span class="inlinemediaobject"><img src="../../../equations/ellint5.svg"></span> 107 108 </p></blockquote></div> 109<div class="note"><table border="0" summary="Note"> 110<tr> 111<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../doc/src/images/note.png"></td> 112<th align="left">Note</th> 113</tr> 114<tr><td align="left" valign="top"> 115<p> 116 <span class="emphasis"><em>φ</em></span> is called the amplitude. 117 </p> 118<p> 119 <span class="emphasis"><em>k</em></span> is called the elliptic modulus or eccentricity. 120 </p> 121<p> 122 <span class="emphasis"><em>α</em></span> is called the modular angle. 123 </p> 124<p> 125 <span class="emphasis"><em>n</em></span> is called the characteristic. 126 </p> 127</td></tr> 128</table></div> 129<div class="caution"><table border="0" summary="Caution"> 130<tr> 131<td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../doc/src/images/caution.png"></td> 132<th align="left">Caution</th> 133</tr> 134<tr><td align="left" valign="top"> 135<p> 136 Perhaps more than any other special functions the elliptic integrals are 137 expressed in a variety of different ways. In particular, the final parameter 138 <span class="emphasis"><em>k</em></span> (the modulus) may be expressed using a modular angle 139 α, or a parameter <span class="emphasis"><em>m</em></span>. These are related by: 140 </p> 141<div class="blockquote"><blockquote class="blockquote"><p> 142 <span class="serif_italic">k = sin α</span> 143 </p></blockquote></div> 144<div class="blockquote"><blockquote class="blockquote"><p> 145 <span class="serif_italic">m = k<sup>2</sup> = sin<sup>2</sup>α</span> 146 </p></blockquote></div> 147<p> 148 So that the integral of the third kind (for example) may be expressed as 149 either: 150 </p> 151<div class="blockquote"><blockquote class="blockquote"><p> 152 <span class="serif_italic">Π(n, φ, k)</span> 153 </p></blockquote></div> 154<div class="blockquote"><blockquote class="blockquote"><p> 155 <span class="serif_italic">Π(n, φ \ α)</span> 156 </p></blockquote></div> 157<div class="blockquote"><blockquote class="blockquote"><p> 158 <span class="serif_italic">Π(n, φ | m)</span> 159 </p></blockquote></div> 160<p> 161 To further complicate matters, some texts refer to the <span class="emphasis"><em>complement 162 of the parameter m</em></span>, or 1 - m, where: 163 </p> 164<div class="blockquote"><blockquote class="blockquote"><p> 165 <span class="serif_italic">1 - m = 1 - k<sup>2</sup> = cos<sup>2</sup>α</span> 166 </p></blockquote></div> 167<p> 168 This implementation uses <span class="emphasis"><em>k</em></span> throughout: this matches 169 the requirements of the <a href="http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2005/n1836.pdf" target="_top">Technical 170 Report on C++ Library Extensions</a>.<br> 171 </p> 172<p> 173 So you should be extra careful when using these functions! 174 </p> 175</td></tr> 176</table></div> 177<div class="warning"><table border="0" summary="Warning"> 178<tr> 179<td rowspan="2" align="center" valign="top" width="25"><img alt="[Warning]" src="../../../../../../doc/src/images/warning.png"></td> 180<th align="left">Warning</th> 181</tr> 182<tr><td align="left" valign="top"><p> 183 Boost.Math order of arguments differs from other implementations: <span class="emphasis"><em>k</em></span> 184 is always the <span class="bold"><strong>first</strong></span> argument. 185 </p></td></tr> 186</table></div> 187<p> 188 A simple example comparing use of <a href="http://www.wolframalpha.com/" target="_top">Wolfram 189 Alpha</a> with Boost.Math (including much higher precision using <a href="../../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>) 190 is <a href="../../../../example/jacobi_zeta_example.cpp" target="_top">jacobi_zeta_example.cpp</a>. 191 </p> 192<p> 193 When <span class="emphasis"><em>φ</em></span> = <span class="emphasis"><em>π</em></span> / 2, the elliptic integrals 194 are called <span class="emphasis"><em>complete</em></span>. 195 </p> 196<p> 197 Complete Elliptic Integral of the First Kind (Legendre form) 198 </p> 199<div class="blockquote"><blockquote class="blockquote"><p> 200 <span class="inlinemediaobject"><img src="../../../equations/ellint6.svg"></span> 201 202 </p></blockquote></div> 203<p> 204 Complete Elliptic Integral of the Second Kind (Legendre form) 205 </p> 206<div class="blockquote"><blockquote class="blockquote"><p> 207 <span class="inlinemediaobject"><img src="../../../equations/ellint7.svg"></span> 208 209 </p></blockquote></div> 210<p> 211 Complete Elliptic Integral of the Third Kind (Legendre form) 212 </p> 213<div class="blockquote"><blockquote class="blockquote"><p> 214 <span class="inlinemediaobject"><img src="../../../equations/ellint8.svg"></span> 215 216 </p></blockquote></div> 217<p> 218 Legendre also defined a fourth integral /D(φ,k)/ which is a combination of 219 the other three: 220 </p> 221<div class="blockquote"><blockquote class="blockquote"><p> 222 <span class="inlinemediaobject"><img src="../../../equations/ellint_d.svg"></span> 223 224 </p></blockquote></div> 225<p> 226 Like the other Legendre integrals this comes in both complete and incomplete 227 forms. 228 </p> 229<h5> 230<a name="math_toolkit.ellint.ellint_intro.h2"></a> 231 <span class="phrase"><a name="math_toolkit.ellint.ellint_intro.carlson_elliptic_integrals"></a></span><a class="link" href="ellint_intro.html#math_toolkit.ellint.ellint_intro.carlson_elliptic_integrals">Carlson 232 Elliptic Integrals</a> 233 </h5> 234<p> 235 Carlson [<a class="link" href="ellint_intro.html#ellint_ref_carlson77">Carlson77</a>] [<a class="link" href="ellint_intro.html#ellint_ref_carlson78">Carlson78</a>] 236 gives an alternative definition of elliptic integral's canonical forms: 237 </p> 238<p> 239 Carlson's Elliptic Integral of the First Kind 240 </p> 241<div class="blockquote"><blockquote class="blockquote"><p> 242 <span class="inlinemediaobject"><img src="../../../equations/ellint9.svg"></span> 243 244 </p></blockquote></div> 245<p> 246 where <span class="emphasis"><em>x</em></span>, <span class="emphasis"><em>y</em></span>, <span class="emphasis"><em>z</em></span> 247 are nonnegative and at most one of them may be zero. 248 </p> 249<p> 250 Carlson's Elliptic Integral of the Second Kind 251 </p> 252<div class="blockquote"><blockquote class="blockquote"><p> 253 <span class="inlinemediaobject"><img src="../../../equations/ellint10.svg"></span> 254 255 </p></blockquote></div> 256<p> 257 where <span class="emphasis"><em>x</em></span>, <span class="emphasis"><em>y</em></span> are nonnegative, at 258 most one of them may be zero, and <span class="emphasis"><em>z</em></span> must be positive. 259 </p> 260<p> 261 Carlson's Elliptic Integral of the Third Kind 262 </p> 263<div class="blockquote"><blockquote class="blockquote"><p> 264 <span class="inlinemediaobject"><img src="../../../equations/ellint11.svg"></span> 265 266 </p></blockquote></div> 267<p> 268 where <span class="emphasis"><em>x</em></span>, <span class="emphasis"><em>y</em></span>, <span class="emphasis"><em>z</em></span> 269 are nonnegative, at most one of them may be zero, and <span class="emphasis"><em>p</em></span> 270 must be nonzero. 271 </p> 272<p> 273 Carlson's Degenerate Elliptic Integral 274 </p> 275<div class="blockquote"><blockquote class="blockquote"><p> 276 <span class="inlinemediaobject"><img src="../../../equations/ellint12.svg"></span> 277 278 </p></blockquote></div> 279<p> 280 where <span class="emphasis"><em>x</em></span> is nonnegative and <span class="emphasis"><em>y</em></span> is 281 nonzero. 282 </p> 283<div class="note"><table border="0" summary="Note"> 284<tr> 285<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../doc/src/images/note.png"></td> 286<th align="left">Note</th> 287</tr> 288<tr><td align="left" valign="top"> 289<p> 290 <span class="emphasis"><em>R<sub>C</sub>(x, y) = R<sub>F</sub>(x, y, y)</em></span> 291 </p> 292<p> 293 <span class="emphasis"><em>R<sub>D</sub>(x, y, z) = R<sub>J</sub>(x, y, z, z)</em></span> 294 </p> 295</td></tr> 296</table></div> 297<p> 298 Carlson's Symmetric Integral 299 </p> 300<div class="blockquote"><blockquote class="blockquote"><p> 301 <span class="inlinemediaobject"><img src="../../../equations/ellint27.svg"></span> 302 303 </p></blockquote></div> 304<h5> 305<a name="math_toolkit.ellint.ellint_intro.h3"></a> 306 <span class="phrase"><a name="math_toolkit.ellint.ellint_intro.duplication_theorem"></a></span><a class="link" href="ellint_intro.html#math_toolkit.ellint.ellint_intro.duplication_theorem">Duplication 307 Theorem</a> 308 </h5> 309<p> 310 Carlson proved in [<a class="link" href="ellint_intro.html#ellint_ref_carlson78">Carlson78</a>] 311 that 312 </p> 313<div class="blockquote"><blockquote class="blockquote"><p> 314 <span class="inlinemediaobject"><img src="../../../equations/ellint13.svg"></span> 315 316 </p></blockquote></div> 317<h5> 318<a name="math_toolkit.ellint.ellint_intro.h4"></a> 319 <span class="phrase"><a name="math_toolkit.ellint.ellint_intro.carlson_s_formulas"></a></span><a class="link" href="ellint_intro.html#math_toolkit.ellint.ellint_intro.carlson_s_formulas">Carlson's 320 Formulas</a> 321 </h5> 322<p> 323 The Legendre form and Carlson form of elliptic integrals are related by equations: 324 </p> 325<div class="blockquote"><blockquote class="blockquote"><p> 326 <span class="inlinemediaobject"><img src="../../../equations/ellint14.svg"></span> 327 328 </p></blockquote></div> 329<p> 330 In particular, 331 </p> 332<div class="blockquote"><blockquote class="blockquote"><p> 333 <span class="inlinemediaobject"><img src="../../../equations/ellint15.svg"></span> 334 335 </p></blockquote></div> 336<h5> 337<a name="math_toolkit.ellint.ellint_intro.h5"></a> 338 <span class="phrase"><a name="math_toolkit.ellint.ellint_intro.miscellaneous_elliptic_integrals"></a></span><a class="link" href="ellint_intro.html#math_toolkit.ellint.ellint_intro.miscellaneous_elliptic_integrals">Miscellaneous 339 Elliptic Integrals</a> 340 </h5> 341<p> 342 There are two functions related to the elliptic integrals which otherwise 343 defy categorisation, these are the Jacobi Zeta function: 344 </p> 345<div class="blockquote"><blockquote class="blockquote"><p> 346 <span class="inlinemediaobject"><img src="../../../equations/jacobi_zeta.svg"></span> 347 348 </p></blockquote></div> 349<p> 350 and the Heuman Lambda function: 351 </p> 352<div class="blockquote"><blockquote class="blockquote"><p> 353 <span class="inlinemediaobject"><img src="../../../equations/heuman_lambda.svg"></span> 354 355 </p></blockquote></div> 356<p> 357 Both of these functions are easily implemented in terms of Carlson's integrals, 358 and are provided in this library as <a class="link" href="jacobi_zeta.html" title="Jacobi Zeta Function">jacobi_zeta</a> 359 and <a class="link" href="heuman_lambda.html" title="Heuman Lambda Function">heuman_lambda</a>. 360 </p> 361<h5> 362<a name="math_toolkit.ellint.ellint_intro.h6"></a> 363 <span class="phrase"><a name="math_toolkit.ellint.ellint_intro.numerical_algorithms"></a></span><a class="link" href="ellint_intro.html#math_toolkit.ellint.ellint_intro.numerical_algorithms">Numerical 364 Algorithms</a> 365 </h5> 366<p> 367 The conventional methods for computing elliptic integrals are Gauss and Landen 368 transformations, which converge quadratically and work well for elliptic 369 integrals of the first and second kinds. Unfortunately they suffer from loss 370 of significant digits for the third kind. 371 </p> 372<p> 373 Carlson's algorithm [<a class="link" href="ellint_intro.html#ellint_ref_carlson79">Carlson79</a>] 374 [<a class="link" href="ellint_intro.html#ellint_ref_carlson78">Carlson78</a>], by contrast, provides 375 a unified method for all three kinds of elliptic integrals with satisfactory 376 precisions. 377 </p> 378<h5> 379<a name="math_toolkit.ellint.ellint_intro.h7"></a> 380 <span class="phrase"><a name="math_toolkit.ellint.ellint_intro.references"></a></span><a class="link" href="ellint_intro.html#math_toolkit.ellint.ellint_intro.references">References</a> 381 </h5> 382<p> 383 Special mention goes to: 384 </p> 385<div class="blockquote"><blockquote class="blockquote"><p> 386 A. M. Legendre, <span class="emphasis"><em>Traité des Fonctions Elliptiques et des Integrales 387 Euleriennes</em></span>, Vol. 1. Paris (1825). 388 </p></blockquote></div> 389<p> 390 However the main references are: 391 </p> 392<div class="orderedlist"><ol class="orderedlist" type="1"> 393<li class="listitem"> 394 <a name="ellint_ref_AS"></a>M. Abramowitz and I. A. Stegun (Eds.) (1964) 395 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical 396 Tables, National Bureau of Standards Applied Mathematics Series, U.S. 397 Government Printing Office, Washington, D.C. 398 </li> 399<li class="listitem"> 400 <a href="https://dlmf.nist.gov/19" target="_top">NIST Digital Library of Mathematical 401 Functions, Elliptic Integrals, B. C. Carlson</a> 402 </li> 403<li class="listitem"> 404 <a name="ellint_ref_carlson79"></a>B.C. Carlson, <span class="emphasis"><em>Computing 405 elliptic integrals by duplication</em></span>, Numerische Mathematik, 406 vol 33, 1 (1979). 407 </li> 408<li class="listitem"> 409 <a name="ellint_ref_carlson77"></a>B.C. Carlson, <span class="emphasis"><em>Elliptic Integrals 410 of the First Kind</em></span>, SIAM Journal on Mathematical Analysis, 411 vol 8, 231 (1977). 412 </li> 413<li class="listitem"> 414 <a name="ellint_ref_carlson78"></a>B.C. Carlson, <span class="emphasis"><em>Short Proofs 415 of Three Theorems on Elliptic Integrals</em></span>, SIAM Journal on Mathematical 416 Analysis, vol 9, 524 (1978). 417 </li> 418<li class="listitem"> 419 <a name="ellint_ref_carlson81"></a>B.C. Carlson and E.M. Notis, <span class="emphasis"><em>ALGORITHM 420 577: Algorithms for Incomplete Elliptic Integrals</em></span>, ACM Transactions 421 on Mathematical Software, vol 7, 398 (1981). 422 </li> 423<li class="listitem"> 424 B. C. Carlson, <span class="emphasis"><em>On computing elliptic integrals and functions</em></span>. 425 J. Math. and Phys., 44 (1965), pp. 36-51. 426 </li> 427<li class="listitem"> 428 B. C. Carlson, <span class="emphasis"><em>A table of elliptic integrals of the second 429 kind</em></span>. Math. Comp., 49 (1987), pp. 595-606. (Supplement, ibid., 430 pp. S13-S17.) 431 </li> 432<li class="listitem"> 433 B. C. Carlson, <span class="emphasis"><em>A table of elliptic integrals of the third kind</em></span>. 434 Math. Comp., 51 (1988), pp. 267-280. (Supplement, ibid., pp. S1-S5.) 435 </li> 436<li class="listitem"> 437 B. C. Carlson, <span class="emphasis"><em>A table of elliptic integrals: cubic cases</em></span>. 438 Math. Comp., 53 (1989), pp. 327-333. 439 </li> 440<li class="listitem"> 441 B. C. Carlson, <span class="emphasis"><em>A table of elliptic integrals: one quadratic 442 factor</em></span>. Math. Comp., 56 (1991), pp. 267-280. 443 </li> 444<li class="listitem"> 445 B. C. Carlson, <span class="emphasis"><em>A table of elliptic integrals: two quadratic 446 factors</em></span>. Math. Comp., 59 (1992), pp. 165-180. 447 </li> 448<li class="listitem"> 449 B. C. Carlson, <span class="emphasis"><em><a href="http://arxiv.org/abs/math.CA/9409227" target="_top">Numerical 450 computation of real or complex elliptic integrals</a></em></span>. 451 Numerical Algorithms, Volume 10, Number 1 / March, 1995, p13-26. 452 </li> 453<li class="listitem"> 454 B. C. Carlson and John L. Gustafson, <span class="emphasis"><em><a href="http://arxiv.org/abs/math.CA/9310223" target="_top">Asymptotic 455 Approximations for Symmetric Elliptic Integrals</a></em></span>, SIAM 456 Journal on Mathematical Analysis, Volume 25, Issue 2 (March 1994), 288-303. 457 </li> 458</ol></div> 459<p> 460 The following references, while not directly relevant to our implementation, 461 may also be of interest: 462 </p> 463<div class="orderedlist"><ol class="orderedlist" type="1"> 464<li class="listitem"> 465 R. Burlisch, <span class="emphasis"><em>Numerical Computation of Elliptic Integrals and 466 Elliptic Functions.</em></span> Numerical Mathematik 7, 78-90. 467 </li> 468<li class="listitem"> 469 R. Burlisch, <span class="emphasis"><em>An extension of the Bartky Transformation to Incomplete 470 Elliptic Integrals of the Third Kind</em></span>. Numerical Mathematik 471 13, 266-284. 472 </li> 473<li class="listitem"> 474 R. Burlisch, <span class="emphasis"><em>Numerical Computation of Elliptic Integrals and 475 Elliptic Functions. III</em></span>. Numerical Mathematik 13, 305-315. 476 </li> 477<li class="listitem"> 478 T. Fukushima and H. Ishizaki, <span class="emphasis"><em><a href="http://adsabs.harvard.edu/abs/1994CeMDA..59..237F" target="_top">Numerical 479 Computation of Incomplete Elliptic Integrals of a General Form.</a></em></span> 480 Celestial Mechanics and Dynamical Astronomy, Volume 59, Number 3 / July, 481 1994, 237-251. 482 </li> 483</ol></div> 484</div> 485<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 486<td align="left"></td> 487<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 488 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 489 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 490 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 491 Daryle Walker and Xiaogang Zhang<p> 492 Distributed under the Boost Software License, Version 1.0. (See accompanying 493 file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) 494 </p> 495</div></td> 496</tr></table> 497<hr> 498<div class="spirit-nav"> 499<a accesskey="p" href="../ellint.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../ellint.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="ellint_carlson.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 500</div> 501</body> 502</html> 503
