Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric -functions

Author:
B. C. Carlson

Journal:
Math. Comp. **75** (2006), 1309-1318

MSC (2000):
Primary 33E05, 33C75; Secondary 33C70, 33C65.

DOI:
https://doi.org/10.1090/S0025-5718-06-01838-2

Published electronically:
March 13, 2006

MathSciNet review:
2219030

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Abstract | References | Similar Articles | Additional Information

Abstract: Any product of real powers of Jacobian elliptic functions can be written in the form . If all three 's are even integers, the indefinite integral of this product with respect to is a constant times a multivariate hypergeometric function with half-odd-integral 's and , showing it to be an incomplete elliptic integral of the second kind unless all three 's are 0. Permutations of c, d, and n in the integrand produce the same permutations of the variables }, allowing as many as six integrals to take a unified form. Thirty -functions of the type specified, incorporating 136 integrals, are reduced to a new choice of standard elliptic integrals obtained by permuting , , and in , which is symmetric in its first two variables and has an efficient algorithm for numerical computation.

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Additional Information

**B. C. Carlson**

Affiliation:
Ames Laboratory and Department of Mathematics, Iowa State University, Ames, Iowa 50011-3020

Email:
bcarlson@scl.ameslab.gov

DOI:
https://doi.org/10.1090/S0025-5718-06-01838-2

Keywords:
Jacobian elliptic function,
hypergeometric $R$-function,
elliptic integral.

Received by editor(s):
May 5, 2005

Published electronically:
March 13, 2006

Additional Notes:
This manuscript has been authored by Iowa State University of Science and Technology under contract No. W-7405-ENG-82 with the US Department of Energy.

Article copyright:
© Copyright 2006
American Mathematical Society