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How it can be shown that $$ _1F_1(a;b;z) = \frac{\Gamma(b)}{\Gamma(a)}\, e^{z} \, z^{a-b}\, [1+ O(\mid z\mid^{-1})]; \quad (\Re(z)>0)$$ or $$ _1F_1(a;b;z) = \frac{\Gamma(b)}{\Gamma(b-a)}\, (-z)^{-a}\, [1+ O(\mid z\mid^{-1})]; \quad (\Re(z)<0)$$ where $ _1F_1(a;b;z)$ is the confluent hypergeometric function given by $$ _1F_1(a;b;z)=\sum_{n=0}^{\infty} \frac{(a)_n}{(b)_n } \frac{z^n}{n!} $$ Thanks you in advance

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    $\begingroup$ This is shown in most serious Special function book (e.g. Lebedev, Ch. 9.12; Temme, Ch. 7.1). The treatment in A. Erdélyi et al., Higher Transcendental Functions Vol. I, Ch. 6.13 Asymptotic behavior is available from en.wikipedia.org/wiki/Bateman_Manuscript_Project. Most derivations use an integral representation (either for $M$ or the Tricomi function $U$) and substitute a power series for $(1-x)^y$. $\endgroup$ – gammatester Dec 17 '15 at 10:07

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