11
Elliptic Integrals
Abstract
Main topic of this chapter are elliptic integrals, complete and incomplete of
first, second, and third kind for complex arguments, respectively, and complex
parameters in various representations: Legendre elliptic integrals, Carlson’s
elliptic integrals, and Bulirsch’s elliptic integrals. In addition, Jacobi’s zeta
function and Heumann’s Λ function in the complex domain are discussed.
The computation of the functions listed above is mainly based on Carlson’s
duplication formula. The corresponding m-code is available for download.
Main topic of this chapter are elliptic integrals, complete and incomplete of first,
second, and third kind for complex arguments, respectively, parameters in various
representations. Elliptic integrals find many applications, not only in physics; for
example, the elliptic integral of first kind occurs in the solution of the pendulum.
For its Legendre and its trigonometric form, see Eqs. (10.1b) and (10.1c); see also
Chap. 10.
11.1 Function Overview
ellipke The MATLAB function[K,E] = ellipke(M,tol) returns the complete elliptic integrals of first K(k) and second kind E(k), with “M” a real array
in the range 0 <= M <= 1 and “tol” (optional) the tolerance.
ellipInt The SPECFUNPHYS class ellipInt evaluates
• the complete and incomplete elliptic integral of first kind K(k) and F (φ, k),
the elliptic integral of second kind E(k) and E(φ, k), the elliptic integral of
third kind Π(n, k) and Π(φ, n, k), and the elliptic integral D(φ, k);
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
W. Schweizer, Special Functions in Physics with MATLAB,
https://doi.org/10.1007/978-3-030-64232-7_11
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