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Let $$U(\alpha,\delta,z)=\frac{\Gamma(1-\delta)}{\Gamma(\alpha-\delta+1} {}_1\!F_1(\alpha,\delta,z) + z^{1-\delta}\frac{\Gamma(\delta-1)}{\Gamma(\delta)}\!F_1(\alpha-\delta+1, 2-\delta,z)$$ be the Tricomi confluent hypergeometric function. the function $a\to U(a,1,z)$ is it holomorphic on $C$?

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