Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric -functions |
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Authors: | B C Carlson |
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Institution: | Ames Laboratory and Department of Mathematics, Iowa State University, Ames, Iowa 50011-3020 |
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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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Keywords: | Jacobian elliptic function hypergeometric $R$-function elliptic integral |
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