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Asymptotic approximations for the first incomplete elliptic integral near logarithmic singularity. (English) Zbl 1118.33010
The authors obtain two convergent series expansions for the incomplete elliptic integral of the first kind $$F(\lambda,k)=\int_0^\lambda\frac{dt}{\sqrt{(1-t^2)(1-k^2t^2)}}$$ valid at any point in the unit square $0<\lambda, k<1$. These expansions are expressed in terms of recursively computed elementary functions. The expansions are truncated after $N$ terms and, by expressing the tails as integrals combined with use of bounds for certain hypergeometric functions, explicit bounds for the remainders $R_N$ are obtained. The truncated expansions yield asymptotic approximations for $F(\lambda,k)$ as $\lambda$ and/or $k$ approach unity. The approximations also remain valid as the logarithmic singularity $\lambda=k=1$ is approached in any direction. The first two approximations complete with error bounds are presented explicitly and numerical calculations are given to illustrate their accuracy.

##### MSC:
 33E05 Elliptic functions and integrals 33C75 Elliptic integrals as hypergeometric functions 33F05 Numerical approximation and evaluation of special functions
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##### References:
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