My brain is spinning around in circles trying to reconcile three distinct contour-integral representation of the confluent hypergeometric function $_1F_1(a,b,z)$ for $b \in \mathbb{Z}_+$:

From K.T.Hecht QM (2000) Chapter 42, $$_1F_1(a,b,z)=\frac{\Gamma(b)}{2\pi i}\oint_{C_1} dt\,e^t\, t^{a-b}\, (t-z)^{-a}\,,\qquad\qquad (1)$$ where the contour $C_1$ "surrounds the branch cut from $t=0$ to $t=z$."

From Messiah, QM (1961) Appendix B, $$_1F_1(a,b,z)=(1-e^{-2\pi i a})^{-1}\frac{\Gamma(b)}{\Gamma(a)\Gamma(b-a)}\oint_{C_2} dt\,e^{zt}\,t^{a-1}\,(1-z)^{b-a-1}\,,\qquad\qquad (2)$$ where the contour $C_2$ "surrounds the points $t=0$ and $t=1$."

And finally, from NIST Handbook of mathematical functions §13.4(ii), (either eqn 13.4.9, 13.4.10, or 13.4.11) which I don't even know how to read: $$_1F_1(a,b,z)=\frac{\Gamma(1+a-b)}{2\pi i\,\Gamma(a)\Gamma(b)}\int_0^{(1+)}dt\,e^{zt}\,t^{a-1}\,(t-1)^{b-a-1} \qquad\qquad (3)$$

I badly need help in showing the equivalence of these three functions. My highest priority is showing equivalence of (1) and (3).


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