The confluent hyper geometric function of the first kind (or the Kummer's function) is defined as
$${\mathbf{M}}\left(a,b,z\right)=\frac{1}{\Gamma\left(a\right)\Gamma\left(b-a% \right)}\int_{0}^{1}e^{zt}t^{a-1}(1-t)^{b-a-1}\,\mathrm{d}t$$
Also the confluent hyper geometric function of the second kind (or the Tricomi's function)
$$U\left(a,b,z\right)=\frac{1}{\Gamma\left(a\right)}\int_{0}^{\infty}e^{-zt}t^{a% -1}(1+t)^{b-a-1}\,\mathrm{d}t$$
I have some integrals that can be expressed in terms of these functions. However my integral requires the value of $b$ to be integer, specifically the values $b=0,1$. I know that both $M,U$ are analytical functions of $a,b,z$ except when $b=0,-1,-2,\cdots,$ is a negative integer. However, I was wondering, can I use the value of these functions in neighborhood of $b=0,1$ to approximate the values of my integrals, for example instead of $U(a,0,z)$ use $U(a,\epsilon,z)$ for small values of $\epsilon=10^{-4}$?
I would appreciate hints or references if possible!
Thanks in advance
