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

Author: B. C. Carlson
Journal: Math. Comp. 75 (2006), 1309-1318
MSC (2000): Primary 33E05, 33C75; Secondary 33C70, 33C65.
Published electronically: March 13, 2006
MathSciNet review: 2219030
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Abstract: Any product of real powers of Jacobian elliptic functions can be written in the form $ \textrm{cs}^{m_1}(u,k)\,\textrm{ds}^{m_2}(u,k)\,\textrm{ns}^{m_3}(u,k)$. If all three $ m$'s are even integers, the indefinite integral of this product with respect to $ u$ is a constant times a multivariate hypergeometric function $ R_{-a}(b_1,b_2,b_3;\,x,y,z)$ with half-odd-integral $ b$'s and $ -a+b_1+b_2+b_3=1$, showing it to be an incomplete elliptic integral of the second kind unless all three $ m$'s are 0. Permutations of c, d, and n in the integrand produce the same permutations of the variables $ \{x,y,z\} =\{\textrm{cs}^2,\textrm{ds}^2,\textrm{ns}^2$}, allowing as many as six integrals to take a unified form. Thirty $ R$-functions of the type specified, incorporating 136 integrals, are reduced to a new choice of standard elliptic integrals obtained by permuting $ x$, $ y$, and $ z$ in $ R_D(x,y,z) =R_{-3/2}({\textstyle\frac{1}{2}},\frac{1}{2}, \frac{3}{2};\,x,y,z)$, 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

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