  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.
