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:[1]
, all elliptic integrals can ultimately be evaluated in terms of just
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.
Incomplete elliptic integrals can be calculated easily using Carlson symmetric forms: (Note: the above are only valid for
) Complete elliptic integrals can be calculated by substituting φ = 1⁄2π: When any two, or all three of the arguments of
renders the integrand rational.
The integral can then be expressed in terms of elementary transcendental functions.
are the same, By substituting in the integral definitions
In obtaining a Taylor series expansion for
it proves convenient to expand about the mean value of the several arguments.
are defined with this sign (such that they are subtracted), in order to be in agreement with Carlson's papers.
and its integral can be expressed as functions of the elementary symmetric polynomials in
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
is not fully symmetric; its dependence on its fourth argument,
as a fully symmetric function of five arguments, two of which happen to have the same value
defined by The elementary symmetric polynomials in
are in full However, it is possible to simplify the formulae for
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
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.
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
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.
Then iterate the series until the desired precision is reached: if
are non-negative, all of the series will converge quickly to a given value, say,