1[section:ellint_intro Elliptic Integral Overview] 2 3The main reference for the elliptic integrals is: 4 5[:M. Abramowitz and I. A. Stegun (Eds.) (1964) 6Handbook of Mathematical Functions with Formulas, Graphs, and 7Mathematical Tables, 8National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C.] 9 10and its recently revised version __DMLF, in particular 11[:[@https://dlmf.nist.gov/19 Elliptic Integrals, B. C. Carlson]] 12 13Mathworld also contain a lot of useful background information: 14 15[:[@http://mathworld.wolfram.com/EllipticIntegral.html Weisstein, Eric W. 16"Elliptic Integral." From MathWorld--A Wolfram Web Resource.]] 17 18As does [@http://en.wikipedia.org/wiki/Elliptic_integral Wikipedia Elliptic integral]. 19 20[h4 Notation] 21 22All variables are real numbers unless otherwise noted. 23 24[h4 Definition] 25 26[equation ellint1] 27 28is called elliptic integral if ['R(t, s)] is a rational function 29of ['t] and ['s], and ['s[super 2]] is a cubic or quartic polynomial 30in ['t]. 31 32Elliptic integrals generally cannot be expressed in terms of 33elementary functions. However, Legendre showed that all elliptic 34integrals can be reduced to the following three canonical forms: 35 36Elliptic Integral of the First Kind (Legendre form) 37 38[equation ellint2] 39 40Elliptic Integral of the Second Kind (Legendre form) 41 42[equation ellint3] 43 44Elliptic Integral of the Third Kind (Legendre form) 45 46[equation ellint4] 47 48where 49 50[equation ellint5] 51 52[note ['[phi]] is called the amplitude. 53 54['k] is called the elliptic modulus or eccentricity. 55 56['[alpha]] is called the modular angle. 57 58['n] is called the characteristic.] 59 60[caution Perhaps more than any other special functions the elliptic 61integrals are expressed in a variety of different ways. In particular, 62the final parameter /k/ (the modulus) may be expressed using a modular 63angle [alpha], or a parameter /m/. These are related by: 64 65[expression k = sin[thin][alpha]] 66 67[expression m = k[super 2] = sin[super 2][alpha]] 68 69So that the integral of the third kind (for example) may be expressed as 70either: 71 72[expression [Pi](n, [phi], k)] 73 74[expression [Pi](n, [phi] \\ [alpha])] 75 76[expression [Pi](n, [phi] | m)] 77 78To further complicate matters, some texts refer to the ['complement 79of the parameter m], or 1 - m, where: 80 81[expression 1 - m = 1 - k[super 2] = cos[super 2][alpha]] 82 83This implementation uses /k/ throughout: this matches the requirements 84of the [@http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2005/n1836.pdf 85Technical Report on C++ Library Extensions].[br] 86 87So you should be extra careful when using these functions!] 88 89[warning Boost.Math order of arguments differs from other implementations: /k/ is always the *first* argument.] 90 91A simple example comparing use of __WolframAlpha with Boost.Math (including much higher precision using __multiprecision) 92is [@../../example/jacobi_zeta_example.cpp jacobi_zeta_example.cpp]. 93 94When ['[phi]] = ['[pi]] / 2, the elliptic integrals are called ['complete]. 95 96Complete Elliptic Integral of the First Kind (Legendre form) 97 98[equation ellint6] 99 100Complete Elliptic Integral of the Second Kind (Legendre form) 101 102[equation ellint7] 103 104Complete Elliptic Integral of the Third Kind (Legendre form) 105 106[equation ellint8] 107 108Legendre also defined a fourth integral /D([phi],k)/ which is a combination of the other three: 109 110[equation ellint_d] 111 112Like the other Legendre integrals this comes in both complete and incomplete forms. 113 114[h4 Carlson Elliptic Integrals] 115 116Carlson [[link ellint_ref_carlson77 Carlson77]] [[link ellint_ref_carlson78 Carlson78]] gives an alternative definition of 117elliptic integral's canonical forms: 118 119Carlson's Elliptic Integral of the First Kind 120 121[equation ellint9] 122 123where ['x], ['y], ['z] are nonnegative and at most one of them 124may be zero. 125 126Carlson's Elliptic Integral of the Second Kind 127 128[equation ellint10] 129 130where ['x], ['y] are nonnegative, at most one of them may be zero, 131and ['z] must be positive. 132 133Carlson's Elliptic Integral of the Third Kind 134 135[equation ellint11] 136 137where ['x], ['y], ['z] are nonnegative, at most one of them may be 138zero, and ['p] must be nonzero. 139 140Carlson's Degenerate Elliptic Integral 141 142[equation ellint12] 143 144where ['x] is nonnegative and ['y] is nonzero. 145 146[note ['R[sub C](x, y) = R[sub F](x, y, y)] 147 148['R[sub D](x, y, z) = R[sub J](x, y, z, z)]] 149 150Carlson's Symmetric Integral 151 152[equation ellint27] 153 154[h4 Duplication Theorem] 155 156Carlson proved in [[link ellint_ref_carlson78 Carlson78]] that 157 158[equation ellint13] 159 160[h4 Carlson's Formulas] 161 162The Legendre form and Carlson form of elliptic integrals are related 163by equations: 164 165[equation ellint14] 166 167In particular, 168 169[equation ellint15] 170 171[h4 Miscellaneous Elliptic Integrals] 172 173There are two functions related to the elliptic integrals which otherwise 174defy categorisation, these are the Jacobi Zeta function: 175 176[equation jacobi_zeta] 177 178and the Heuman Lambda function: 179 180[equation heuman_lambda] 181 182Both of these functions are easily implemented in terms of Carlson's integrals, and are 183provided in this library as __jacobi_zeta and __heuman_lambda. 184 185[h4 Numerical Algorithms] 186 187The conventional methods for computing elliptic integrals are Gauss 188and Landen transformations, which converge quadratically and work 189well for elliptic integrals of the first and second kinds. 190Unfortunately they suffer from loss of significant digits for the 191third kind. 192 193Carlson's algorithm [[link ellint_ref_carlson79 Carlson79]] [[link ellint_ref_carlson78 Carlson78]], by contrast, 194provides a unified method for all three kinds of elliptic integrals with satisfactory precisions. 195 196[h4 References] 197 198Special mention goes to: 199 200[:A. M. Legendre, ['Trait[eacute] des Fonctions Elliptiques et des Integrales 201Euleriennes], Vol. 1. Paris (1825).] 202 203However the main references are: 204 205# [#ellint_ref_AS]M. Abramowitz and I. A. Stegun (Eds.) (1964) 206Handbook of Mathematical Functions with Formulas, Graphs, and 207Mathematical Tables, 208National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C. 209# [@https://dlmf.nist.gov/19 NIST Digital Library of Mathematical Functions, Elliptic Integrals, B. C. Carlson] 210# [#ellint_ref_carlson79]B.C. Carlson, ['Computing elliptic integrals by duplication], 211 Numerische Mathematik, vol 33, 1 (1979). 212# [#ellint_ref_carlson77]B.C. Carlson, ['Elliptic Integrals of the First Kind], 213 SIAM Journal on Mathematical Analysis, vol 8, 231 (1977). 214# [#ellint_ref_carlson78]B.C. Carlson, ['Short Proofs of Three Theorems on Elliptic Integrals], 215 SIAM Journal on Mathematical Analysis, vol 9, 524 (1978). 216# [#ellint_ref_carlson81]B.C. Carlson and E.M. Notis, ['ALGORITHM 577: Algorithms for Incomplete 217 Elliptic Integrals], ACM Transactions on Mathematical Software, 218 vol 7, 398 (1981). 219# B. C. Carlson, ['On computing elliptic integrals and functions]. J. Math. and 220Phys., 44 (1965), pp. 36-51. 221# B. C. Carlson, ['A table of elliptic integrals of the second kind]. Math. Comp., 49 222(1987), pp. 595-606. (Supplement, ibid., pp. S13-S17.) 223# B. C. Carlson, ['A table of elliptic integrals of the third kind]. Math. Comp., 51 (1988), 224pp. 267-280. (Supplement, ibid., pp. S1-S5.) 225# B. C. Carlson, ['A table of elliptic integrals: cubic cases]. Math. Comp., 53 (1989), pp. 226327-333. 227# B. C. Carlson, ['A table of elliptic integrals: one quadratic factor]. Math. Comp., 56 (1991), 228pp. 267-280. 229# B. C. Carlson, ['A table of elliptic integrals: two quadratic factors]. Math. Comp., 59 230(1992), pp. 165-180. 231# B. C. Carlson, ['[@http://arxiv.org/abs/math.CA/9409227 232Numerical computation of real or complex elliptic integrals]]. Numerical Algorithms, 233Volume 10, Number 1 / March, 1995, p13-26. 234# B. C. Carlson and John L. Gustafson, ['[@http://arxiv.org/abs/math.CA/9310223 235Asymptotic Approximations for Symmetric Elliptic Integrals]], 236SIAM Journal on Mathematical Analysis, Volume 25, Issue 2 (March 1994), 288-303. 237 238 239The following references, while not directly relevant to our implementation, 240may also be of interest: 241 242# R. Burlisch, ['Numerical Computation of Elliptic Integrals and Elliptic Functions.] 243Numerical Mathematik 7, 78-90. 244# R. Burlisch, ['An extension of the Bartky Transformation to Incomplete 245Elliptic Integrals of the Third Kind]. Numerical Mathematik 13, 266-284. 246# R. Burlisch, ['Numerical Computation of Elliptic Integrals and Elliptic Functions. III]. 247Numerical Mathematik 13, 305-315. 248# T. Fukushima and H. Ishizaki, ['[@http://adsabs.harvard.edu/abs/1994CeMDA..59..237F 249Numerical Computation of Incomplete Elliptic Integrals of a General Form.]] 250Celestial Mechanics and Dynamical Astronomy, Volume 59, Number 3 / July, 1994, 251237-251. 252 253[endsect] [/section:ellint_intro Elliptic Integral Overview] 254 255[/ 256Copyright (c) 2006 Xiaogang Zhang 257Use, modification and distribution are subject to the 258Boost Software License, Version 1.0. (See accompanying file 259LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) 260] 261 262