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What's the asymptotic behavior of the Tricomi confluent hypergeometric function $U(a,b,z)$ when $|z|\to0$ and $b$ is complex but with $Re(b)=1$. The Abramowitz and Stegun handbook does not seem to include this case; They have the cases when $Re(b)>1$ and when $b=1$, meaning when $b$ is purely real. Any references?

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There are a lot of formulas for $U(a,b,z)$ at the Wolfram function site. The simplest for $z\rightarrow 0$ is this $$U(a,b,z) = \frac{\Gamma(1-b)}{\Gamma(a-b+1)}\Big(1+(O(z)\Big)+ \frac{\Gamma(b-1)}{\Gamma(a)}z^{1-b}\Big(1+O(z)\Big), \;b\not \in \mathbb{Z}, z\rightarrow 0.$$ Other cases for $b\not \in \mathbb{Z}$ are in the section General case and for $b \in \mathbb{Z}$ see the entries in Logarithmic case.

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