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 R_C and R_D are special cases of R_F and R_J, all elliptic integrals can ultimately be evaluated in terms of just R_F, R_J, and R_G. 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 R_F(x,y,z) 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:

Special cases

When any two, or all three of the arguments of R_F 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 R_J are the same,

Properties

Homogeneity

By substituting in the integral definitions for any constant \kappa, it is found that

Duplication theorem

where. where and

Series Expansion

In obtaining a Taylor series expansion for R_{F} or R_{J} it proves convenient to expand about the mean value of the several arguments. So for R_{F}, letting the mean value of the arguments be, and using homogeneity, define \Delta x, \Delta y and \Delta z by that is etc. The differences \Delta x, \Delta y and \Delta z 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 x, y and z, it is also symmetric in the quantities \Delta x, \Delta y and \Delta z. It follows that both the integrand of R_{F} and its integral can be expressed as functions of the elementary symmetric polynomials in \Delta x, \Delta y and \Delta z 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 E_{1} identically to zero, and so eliminates all terms involving E_{1} - which otherwise would be the most numerous. An ascending series for R_{J} may be found in a similar way. There is a slight difficulty because R_{J} is not fully symmetric; its dependence on its fourth argument, p, is different from its dependence on x, y and z. This is overcome by treating R_{J} as a fully symmetric function of five arguments, two of which happen to have the same value p. The mean value of the arguments is therefore taken to be and the differences \Delta x, \Delta y \Delta z and \Delta p defined by The elementary symmetric polynomials in \Delta x, \Delta y, \Delta z, \Delta p and (again) \Delta p are in full However, it is possible to simplify the formulae for E_{2}, E_{3} and E_{4} using the fact that E_{1} = 0. Expressing the integrand in terms of these polynomials, performing a multidimensional Taylor expansion and integrating term-by-term as before... As with R_{J}, by expanding about the mean value of the arguments, more than half the terms (those involving E_{1}) are eliminated.

Negative arguments

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 R_C, or the fourth argument, p, of R_J 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 and where 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.

Numerical evaluation

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 R_F(x,y,z): first, define x_0=x, y_0=y and z_0=z. Then iterate the series until the desired precision is reached: if x, y and z are non-negative, all of the series will converge quickly to a given value, say, \mu. Therefore, Evaluating R_C(x,y) is much the same due to the relation

References and External links