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arXiv · 1509.05167

Computing the Kummer function U(a,b,z) for small values of the arguments

A. Gil·J. Segura·N. M. Temme
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Abstract

We describe methods for computing the Kummer function $U(a,b,z)$ for small values of $z$, with special attention to small values of $b$. For these values of $b$ the connection formula that represents $U(a,b,z)$ as a linear combination of two ${}_1F_1$-functions needs a limiting procedure. We use the power series of the ${}_1F_1$-functions and consider the terms for which this limiting procedure is needed. We give recursion relations for higher terms in the expansion, and we consider the derivative $U^\prime(a,b,z)$ as well. We also discuss the performance for small $\vert z\vert$ of an asymptotic approximation of the Kummer function in terms of modified Bessel functions.

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A. Gil, J. Segura, N. M. Temme. 2015-09-17. Computing the Kummer function U(a,b,z) for small values of the arguments. https://arxiv.org/abs/1509.05167

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