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

Carlson, B.

doi:10.1090/S0025-5718-06-01838-2

Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric $R$-functions
HTML articles powered by AMS MathViewer

by B. C. Carlson PDF

Math. Comp. 75 (2006), 1309-1318 Request permission

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 $R_D(x,y,z) =R_{-3/2}(\frac {1}{2},\frac {1}{2}, \frac {3}{2}; x,y,z)$, which is symmetric in its first two variables and has an efficient algorithm for numerical computation.

References

Paul F. Byrd and Morris D. Friedman, Handbook of elliptic integrals for engineers and scientists, Die Grundlehren der mathematischen Wissenschaften, Band 67, Springer-Verlag, New York-Heidelberg, 1971. Second edition, revised. MR 0277773
Billie Chandler Carlson, Special functions of applied mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1977. MR 0590943
B. C. Carlson, Numerical computation of real or complex elliptic integrals, Numer. Algorithms 10 (1995), no. 1-2, 13–26. Special functions (Torino, 1993). MR 1345407, DOI 10.1007/BF02198293
B. C. Carlson, Symmetry in c, d, n of Jacobian elliptic functions, J. Math. Anal. Appl. 299 (2004), no. 1, 242–253. MR 2091285, DOI 10.1016/j.jmaa.2004.06.049
W. J. Nellis and B. C. Carlson, Reduction and evaluation of elliptic integrals, Math. Comp. 20 (1966), 223–231. MR 215497, DOI 10.1090/S0025-5718-1966-0215497-8
Eric Harold Neville, Jacobian Elliptic Functions, Oxford, at the Clarendon Press, 1951. 2d ed. MR 0041934