In
mathematics, the Carlson symmetric forms of
elliptic integral
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in ...
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 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 integral
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in ...
s can be calculated easily using Carlson symmetric forms:
:
:
:
(Note: the above are only valid for
and
)
Complete elliptic integrals
Complete
elliptic integral
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in ...
s can be calculated by substituting φ = π:
:
:
:
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,
:
Properties
Homogeneity
By substituting in the integral definitions
for any constant
, it is found that
:
:
Duplication theorem
:
where
.
:
where
and
Series Expansion
In obtaining a
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
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 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 can be expressed as a polynomial in elementary sy ...
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 taken to be
:
and the differences
,
and
defined by
:
The
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 can be expressed as a polynomial in elementary sy ...
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
In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that if the function is n-valued (has n values) at that 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
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
Depending on the type of singularity in the integrand ...
(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
:
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
:
References and External links
B. C. Carlson, John L. Gustafson 'Asymptotic approximations for symmetric elliptic integrals' 1993 arXivB. C. Carlson 'Numerical Computation of Real Or Complex Elliptic Integrals' 1994 arXivB. 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.*{{Citation , last1=Press , first1=WH , last2=Teukolsky , first2=SA , last3=Vetterling , first3=WT , last4=Flannery , first4=BP , year=2007 , title=Numerical Recipes: The Art of Scientific Computing , edition=3rd , publisher=Cambridge University Press , publication-place=New York , isbn=978-0-521-88068-8 , chapter=Section 6.12. Elliptic Integrals and Jacobian Elliptic Functions , chapter-url=http://apps.nrbook.com/empanel/index.html#pg=309
*
Fortran code from
SLATEC
SLATEC Common Mathematical Library is a FORTRAN 77 library of over 1400 general purpose mathematical and statistical routines. The code was developed at US Government research laboratories and is therefore public domain software.
"SLATEC" is an a ...
for evaluatin
RFRJRCRD
Elliptic functions