# Two questions on asymptotic expansion of confluent hypergeometric functions for real variable $x, |x| \to \infty$

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?