Toward Symbolic Integration of Elliptic Integrals
B. C. CARLSON1
2
Ames Laboratory and Department of Mathematics,
Iowa State University, Ames, Iowa 50011-3020
Abstract
A method is proposed by which elliptic integrals can be integrated symbolically
without the kind of information about limits of integration and branch points
of the integrand that is required in integral tables using Legendre's
integrals. However, it is assumed that when all polynomials in the integrand
have been factored symbolically into linear factors, the exponents of all
distinct linear factors are known. The recurrence relations are one-parameter
relations, all formulas are given explicitly, and the integral is
eventually expressed in terms of canonical R-functions, with no increase in
their number if neither limit of integration is a branch point of the
integrand. It is the use of R-functions rather than Legendre's integrals
that makes it possible to carry out the whole process symbolically.
If (possibly complex) numerical values of the symbols are known, there are
published algorithms for numerical computation of the R-functions.