Abstract
We consider the standard symmetric elliptic integral R_F(x,y,z) for complex x, y, z. We derive convergent expansions of R_F(x,y,z) in terms of elementary functions that hold uniformly for one of the three variables x, y or z in closed subsets (possibly unbounded) of mathbb {C}{setminus }(-infty ,0]. The expansions are accompanied by error bounds. The accuracy of the expansions and their uniform features are illustrated by means of some numerical examples.
Highlights
The family of elliptic integrals are integrals of the form R(x, y)d x, where R(x, y) is a rational function of the two variables x and y, and y2 is a polynomial of the third or fourth degree in x
The first complete elliptic integral plays an important role in the theory of iterated number sequences based on the arithmetic geometric mean [33, Sect. 12.1.2]
The period of a simple pendulum in a constant gravitational field can be computed in terms of the first complete elliptic integral [33, Sect. 12.1.1]; the zeros of these integrals can be used to determine an upper bound for the number of limit cycles of certain hamiltonian systems [35]; elliptic integrals play an important role in certain problems of electromagnetism [37]
Summary
The family of elliptic integrals are integrals of the form R(x, y)d x, where R(x, y) is a rational function of the two variables x and y, and y2 is a polynomial of the third or fourth degree in x. These functions cannot, in general, be expressed in terms of elementary functions when the polynomial y2 has not a repeated factor and R(x, y) contains some odd power of y. Legendre showed that all the elliptic integrals can be written in terms of three standard integrals, the so called Legendre’s normal elliptic integrals [23] (see [29] for further details)
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