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- [section:ellint_intro Elliptic Integral Overview]
- The main reference for the elliptic integrals is:
- [:M. Abramowitz and I. A. Stegun (Eds.) (1964)
- Handbook of Mathematical Functions with Formulas, Graphs, and
- Mathematical Tables,
- National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C.]
- and its recently revised version __DMLF, in particular
- [:[@https://dlmf.nist.gov/19 Elliptic Integrals, B. C. Carlson]]
- Mathworld also contain a lot of useful background information:
- [:[@http://mathworld.wolfram.com/EllipticIntegral.html Weisstein, Eric W.
- "Elliptic Integral." From MathWorld--A Wolfram Web Resource.]]
- As does [@http://en.wikipedia.org/wiki/Elliptic_integral Wikipedia Elliptic integral].
- [h4 Notation]
- All variables are real numbers unless otherwise noted.
- [h4 Definition]
- [equation ellint1]
- is called elliptic integral if ['R(t, s)] is a rational function
- of ['t] and ['s], and ['s[super 2]] is a cubic or quartic polynomial
- in ['t].
- Elliptic integrals generally cannot be expressed in terms of
- elementary functions. However, Legendre showed that all elliptic
- integrals can be reduced to the following three canonical forms:
- Elliptic Integral of the First Kind (Legendre form)
- [equation ellint2]
- Elliptic Integral of the Second Kind (Legendre form)
- [equation ellint3]
- Elliptic Integral of the Third Kind (Legendre form)
- [equation ellint4]
- where
- [equation ellint5]
- [note ['[phi]] is called the amplitude.
- ['k] is called the elliptic modulus or eccentricity.
- ['[alpha]] is called the modular angle.
- ['n] is called the characteristic.]
-
- [caution Perhaps more than any other special functions the elliptic
- integrals are expressed in a variety of different ways. In particular,
- the final parameter /k/ (the modulus) may be expressed using a modular
- angle [alpha], or a parameter /m/. These are related by:
- [expression k = sin[thin][alpha]]
- [expression m = k[super 2] = sin[super 2][alpha]]
- So that the integral of the third kind (for example) may be expressed as
- either:
- [expression [Pi](n, [phi], k)]
- [expression [Pi](n, [phi] \\ [alpha])]
- [expression [Pi](n, [phi] | m)]
- To further complicate matters, some texts refer to the ['complement
- of the parameter m], or 1 - m, where:
- [expression 1 - m = 1 - k[super 2] = cos[super 2][alpha]]
- This implementation uses /k/ throughout: this matches the requirements
- of the [@http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2005/n1836.pdf
- Technical Report on C++ Library Extensions].[br]
- So you should be extra careful when using these functions!]
- [warning Boost.Math order of arguments differs from other implementations: /k/ is always the *first* argument.]
- A simple example comparing use of __WolframAlpha with Boost.Math (including much higher precision using __multiprecision)
- is [@../../example/jacobi_zeta_example.cpp jacobi_zeta_example.cpp].
- When ['[phi]] = ['[pi]] / 2, the elliptic integrals are called ['complete].
- Complete Elliptic Integral of the First Kind (Legendre form)
- [equation ellint6]
- Complete Elliptic Integral of the Second Kind (Legendre form)
- [equation ellint7]
- Complete Elliptic Integral of the Third Kind (Legendre form)
- [equation ellint8]
- Legendre also defined a fourth integral /D([phi],k)/ which is a combination of the other three:
- [equation ellint_d]
- Like the other Legendre integrals this comes in both complete and incomplete forms.
- [h4 Carlson Elliptic Integrals]
- Carlson [[link ellint_ref_carlson77 Carlson77]] [[link ellint_ref_carlson78 Carlson78]] gives an alternative definition of
- elliptic integral's canonical forms:
- Carlson's Elliptic Integral of the First Kind
- [equation ellint9]
- where ['x], ['y], ['z] are nonnegative and at most one of them
- may be zero.
- Carlson's Elliptic Integral of the Second Kind
- [equation ellint10]
- where ['x], ['y] are nonnegative, at most one of them may be zero,
- and ['z] must be positive.
- Carlson's Elliptic Integral of the Third Kind
- [equation ellint11]
- where ['x], ['y], ['z] are nonnegative, at most one of them may be
- zero, and ['p] must be nonzero.
- Carlson's Degenerate Elliptic Integral
- [equation ellint12]
- where ['x] is nonnegative and ['y] is nonzero.
- [note ['R[sub C](x, y) = R[sub F](x, y, y)]
- ['R[sub D](x, y, z) = R[sub J](x, y, z, z)]]
- Carlson's Symmetric Integral
- [equation ellint27]
- [h4 Duplication Theorem]
- Carlson proved in [[link ellint_ref_carlson78 Carlson78]] that
- [equation ellint13]
- [h4 Carlson's Formulas]
- The Legendre form and Carlson form of elliptic integrals are related
- by equations:
- [equation ellint14]
- In particular,
- [equation ellint15]
- [h4 Miscellaneous Elliptic Integrals]
- There are two functions related to the elliptic integrals which otherwise
- defy categorisation, these are the Jacobi Zeta function:
- [equation jacobi_zeta]
- and the Heuman Lambda function:
- [equation heuman_lambda]
- Both of these functions are easily implemented in terms of Carlson's integrals, and are
- provided in this library as __jacobi_zeta and __heuman_lambda.
- [h4 Numerical Algorithms]
- The conventional methods for computing elliptic integrals are Gauss
- and Landen transformations, which converge quadratically and work
- well for elliptic integrals of the first and second kinds.
- Unfortunately they suffer from loss of significant digits for the
- third kind.
- Carlson's algorithm [[link ellint_ref_carlson79 Carlson79]] [[link ellint_ref_carlson78 Carlson78]], by contrast,
- provides a unified method for all three kinds of elliptic integrals with satisfactory precisions.
- [h4 References]
- Special mention goes to:
- [:A. M. Legendre, ['Trait[eacute] des Fonctions Elliptiques et des Integrales
- Euleriennes], Vol. 1. Paris (1825).]
- However the main references are:
- # [#ellint_ref_AS]M. Abramowitz and I. A. Stegun (Eds.) (1964)
- Handbook of Mathematical Functions with Formulas, Graphs, and
- Mathematical Tables,
- National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C.
- # [@https://dlmf.nist.gov/19 NIST Digital Library of Mathematical Functions, Elliptic Integrals, B. C. Carlson]
- # [#ellint_ref_carlson79]B.C. Carlson, ['Computing elliptic integrals by duplication],
- Numerische Mathematik, vol 33, 1 (1979).
- # [#ellint_ref_carlson77]B.C. Carlson, ['Elliptic Integrals of the First Kind],
- SIAM Journal on Mathematical Analysis, vol 8, 231 (1977).
- # [#ellint_ref_carlson78]B.C. Carlson, ['Short Proofs of Three Theorems on Elliptic Integrals],
- SIAM Journal on Mathematical Analysis, vol 9, 524 (1978).
- # [#ellint_ref_carlson81]B.C. Carlson and E.M. Notis, ['ALGORITHM 577: Algorithms for Incomplete
- Elliptic Integrals], ACM Transactions on Mathematmal Software,
- vol 7, 398 (1981).
- # B. C. Carlson, ['On computing elliptic integrals and functions]. J. Math. and
- Phys., 44 (1965), pp. 36-51.
- # B. C. Carlson, ['A table of elliptic integrals of the second kind]. Math. Comp., 49
- (1987), pp. 595-606. (Supplement, ibid., pp. S13-S17.)
- # B. C. Carlson, ['A table of elliptic integrals of the third kind]. Math. Comp., 51 (1988),
- pp. 267-280. (Supplement, ibid., pp. S1-S5.)
- # B. C. Carlson, ['A table of elliptic integrals: cubic cases]. Math. Comp., 53 (1989), pp.
- 327-333.
- # B. C. Carlson, ['A table of elliptic integrals: one quadratic factor]. Math. Comp., 56 (1991),
- pp. 267-280.
- # B. C. Carlson, ['A table of elliptic integrals: two quadratic factors]. Math. Comp., 59
- (1992), pp. 165-180.
- # B. C. Carlson, ['[@http://arxiv.org/abs/math.CA/9409227
- Numerical computation of real or complex elliptic integrals]]. Numerical Algorithms,
- Volume 10, Number 1 / March, 1995, p13-26.
- # B. C. Carlson and John L. Gustafson, ['[@http://arxiv.org/abs/math.CA/9310223
- Asymptotic Approximations for Symmetric Elliptic Integrals]],
- SIAM Journal on Mathematical Analysis, Volume 25, Issue 2 (March 1994), 288-303.
- The following references, while not directly relevent to our implementation,
- may also be of interest:
- # R. Burlisch, ['Numerical Compuation of Elliptic Integrals and Elliptic Functions.]
- Numerical Mathematik 7, 78-90.
- # R. Burlisch, ['An extension of the Bartky Transformation to Incomplete
- Elliptic Integrals of the Third Kind]. Numerical Mathematik 13, 266-284.
- # R. Burlisch, ['Numerical Compuation of Elliptic Integrals and Elliptic Functions. III].
- Numerical Mathematik 13, 305-315.
- # T. Fukushima and H. Ishizaki, ['[@http://adsabs.harvard.edu/abs/1994CeMDA..59..237F
- Numerical Computation of Incomplete Elliptic Integrals of a General Form.]]
- Celestial Mechanics and Dynamical Astronomy, Volume 59, Number 3 / July, 1994,
- 237-251.
- [endsect] [/section:ellint_intro Elliptic Integral Overview]
- [/
- Copyright (c) 2006 Xiaogang Zhang
- Use, modification and distribution are subject to the
- Boost Software License, Version 1.0. (See accompanying file
- LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
- ]
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