I'm looking into the asymptotic expansion for confluent hypergeometric function $_1F_1(a;b;z) \equiv M(a;b;z)$ and I've two quick questions regarding its asymptotic behavior for real values $x,$ i.e. I'm interested in the asymptotic behavior of $_1F_1(a;b;x) \equiv M(a;b;x), x \in \mathbb{R}, x \to \infty.$

A comment on notation: Below $M(a;b;z), \textbf{M}(a;b;z)$ are indeed different and they're related by $M(a;b;z) = \Gamma(b) \textbf{M}(a;b;z), \Gamma(.)$ denoting the Gamma function, link below.

(1) From this link, it seems $M(a;b;z) \sim e^{z}z^{a-b} \frac{\Gamma(b)}{\Gamma(a)}, $ look at the part "§13.2(iv) Limiting Forms as $z \to \infty.$" which states $\textbf{M}(a;b;z) \sim e^{z}z^{a-b}/\Gamma(a)$ and also the equation 13.2.4 which states $M(a;b;z) = \Gamma(b) \textbf{M}(a;b;z)$. I've not seen this exact expression elsewhere, so I'd like to check if it's correct, where $f(z) \sim g(z)$ means $\frac{f(z)}{g(z)} \to 1, |z| \to \infty.$ Do you know any other reference that cites this asymptotic result?

(2) Next, if true, is the asymptotic formula $M(a;b;z) \sim e^{z}z^{a-b} \frac{\Gamma(b)}{\Gamma(a)}, |z| \to \infty$ uniform w.r.t. real $z=x \in \mathbb{R}?$ (or better for complex $z \in \mathbb{C}?$). Could you give the proof or at least cite a reference?



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