کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
4642860 | 1341359 | 2007 | 21 صفحه PDF | دانلود رایگان |
We find two convergent series expansions for Legendre's first incomplete elliptic integral F(λ,k)F(λ,k) in terms of recursively computed elementary functions. Both expansions are valid at every point of the unit square 0<λ,k<10<λ,k<1. Truncated expansions yield asymptotic approximations for F(λ,k)F(λ,k) as λλ and/or k tend to unity, including the case when logarithmic singularity λ=k=1λ=k=1 is approached from any direction. Explicit error bounds are given at every order of approximation. For the reader's convenience we present explicit expressions for low-order approximations and numerical examples to illustrate their accuracy. Our derivation is based on rearrangements of some known double series expansions, hypergeometric summation algorithms and inequalities for hypergeometric functions.
Journal: Journal of Computational and Applied Mathematics - Volume 205, Issue 1, 1 August 2007, Pages 186–206