Carlson symmetric form - AbsoluteAstronomy.com

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Carlson symmetric forms of elliptic integral
Elliptic integral
In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse. They were first studied by Giulio Fagnano and Leonhard Euler...

s 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 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.

Incomplete elliptic integrals

Incomplete elliptic integral

Elliptic integral

In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse. They were first studied by Giulio Fagnano and Leonhard Euler...

s can be calculated easily using Carlson symmetric forms:

(Note: the above are only valid for

and

)

Complete elliptic integrals

Complete elliptic integral

Elliptic integral

In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse. They were first studied by Giulio Fagnano and Leonhard Euler...

s can be calculated by substituting

Special cases

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,

Homogeneity

By substituting in the integral definitions

for any constant

, it is found that

Duplication theorem

where

and

Series Expansion

In obtaining a Taylor series

Taylor series

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....

expansion for

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

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 polynomial

Elementary symmetric polynomial

In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial P can be expressed as a polynomial in elementary symmetric polynomials: P can be given by an...

s 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 take to be

and the differences

and

defined by

The elementary symmetric polynomial

Elementary symmetric polynomial

s 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.

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

Branch point

In the mathematical field of complex analysis, a branch point of a multi-valued function is a point such that the function is discontinuous when going around an arbitrarily small circuit around this 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

Cauchy principal value

In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined.-Formulation:...

(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