In mathematics, the Carlson symmetric forms of elliptic integrals are a small canonical set of elliptic integrals to which all others may be reduced. They are a modern alternative to the Legendre forms. The Legendre forms may be expressed in terms of the Carlson forms and vice versa.
The Carlson elliptic integrals are:
Since and are special cases of and , all elliptic integrals can ultimately be evaluated in terms of just and .
The term symmetric refers to the fact that in contrast to the Legendre forms, these functions are unchanged by the exchange of certain subsets of their arguments. The value of is the same for any permutation of its arguments, and the value of is the same for any permutation of its first three arguments.
The Carlson elliptic integrals are named after Bille C. Carlson (1924-2013).
Relation to the Legendre forms
Incomplete elliptic integrals
Incomplete elliptic integrals can be calculated easily using Carlson symmetric forms:
(Note: the above are only valid for and )
Complete elliptic integrals
Complete elliptic integrals can be calculated by substituting φ = 1⁄2π:
When any two, or all three of the arguments of are the same, then a substitution of renders the integrand rational. The integral can then be expressed in terms of elementary transcendental functions.
Similarly, when at least two of the first three arguments of are the same,
By substituting in the integral definitions for any constant , it is found that
In obtaining a Taylor series expansion for or it proves convenient to expand about the mean value of the several arguments. So for , letting the mean value of the arguments be , and using homogeneity, define , and by
that is etc. The differences , and are defined with this sign (such that they are subtracted), in order to be in agreement with Carlson's papers. Since is symmetric under permutation of , and , it is also symmetric in the quantities , and . It follows that both the integrand of and its integral can be expressed as functions of the elementary symmetric polynomials in , and which are
Expressing the integrand in terms of these polynomials, performing a multidimensional Taylor expansion and integrating term-by-term...
The advantage of expanding about the mean value of the arguments is now apparent; it reduces identically to zero, and so eliminates all terms involving - which otherwise would be the most numerous.
An ascending series for may be found in a similar way. There is a slight difficulty because is not fully symmetric; its dependence on its fourth argument, , is different from its dependence on , and . This is overcome by treating as a fully symmetric function of five arguments, two of which happen to have the same value . The mean value of the arguments is therefore taken to be
and the differences , and defined by
The elementary symmetric polynomials in , , , and (again) are in full
However, it is possible to simplify the formulae for , and using the fact that . Expressing the integrand in terms of these polynomials, performing a multidimensional Taylor expansion and integrating term-by-term as before...
As with , by expanding about the mean value of the arguments, more than half the terms (those involving ) are eliminated.
In general, the arguments x, y, z of Carlson's integrals may not be real and negative, as this would place a branch point on the path of integration, making the integral ambiguous. However, if the second argument of , or the fourth argument, p, of is negative, then this results in a simple pole on the path of integration. In these cases the Cauchy principal value (finite part) of the integrals may be of interest; these are
which must be greater than zero for to be evaluated. This may be arranged by permuting x, y and z so that the value of y is between that of x and z.
The duplication theorem can be used for a fast and robust evaluation of the Carlson symmetric form of elliptic integrals and therefore also for the evaluation of Legendre-form of elliptic integrals. Let us calculate : first, define , and . Then iterate the series
until the desired precision is reached: if , and are non-negative, all of the series will converge quickly to a given value, say, . Therefore,
Evaluating is much the same due to the relation
- B. C. Carlson, John L. Gustafson 'Asymptotic approximations for symmetric elliptic integrals' 1993 arXiv
- B. C. Carlson 'Numerical Computation of Real Or Complex Elliptic Integrals' 1994 arXiv
- B. C. Carlson 'Elliptic Integrals:Symmetric Integrals' in Chap. 19 of Digital Library of Mathematical Functions. Release date 2010-05-07. National Institute of Standards and Technology.
- 'Profile: Bille C. Carlson' in Digital Library of Mathematical Functions. National Institute of Standards and Technology.
- Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 6.12. Elliptic Integrals and Jacobian Elliptic Functions", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8
- Fortran code from SLATEC for evaluating RF, RJ, RC, RD,