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

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