Computing the Kummer function U(a,b,z) for small values of the arguments
This work provides practical computational tools for researchers and engineers who need to evaluate the Kummer function in regimes where standard methods fail.
The paper presents methods for computing the Kummer function U(a,b,z) for small z, particularly when b is small, using power series expansions and recursion relations. It also evaluates an asymptotic approximation in terms of modified Bessel functions.
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.