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# Hankel contour integral

Hankel’s contour integral is a unit (and nilpotent) for gamma function over $\mathbb{C}$. That is,

$\left(\frac{i}{2\pi}\int_{\mathcal{C}}(-t)^{{-z}}e^{{-t}}dt\right)\Gamma(z)=1,% \qquad|z|<\infty.$ |

Hankel’s integral is holomorphic with simple zeros in $\mathbb{Z}_{{\leq 0}}$. Its path of integration starts on the positive real axis *ad infinitum*, rounds the origin counterclockwise and returns to $+\infty$. As an example of application of Hankel’s integral, we have

$\frac{i}{2\pi}\int_{\mathcal{C}}(-t)^{{-\frac{1}{2}}}e^{{-t}}dt=\frac{1}{\sqrt% {\pi}}\,,$ |

where the path of integration is the one above mentioned.

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## Mathematics Subject Classification

30D30*no label found*33B15

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## Comments

## Help with the proof of Hnakel's contour integral

Hello, I was wondering if you have a proof of the Hankel's contour integral. In the web I found one using the theory of Laplace transforms, but the book that is recommended for the subject that I am taking, have a lists of steps to proof the formula.

1st is to proof that the integral in the formula is entire

2nd is estimate the integral to see that improper integral converges uniformly to later say that the Hankel integral is an entire function.

3rd Use Cauchy's Theorem to proof that the values is independent from z and the radio of the circle (in the Hankel's contour)

then use that arg(-t)=-\pi in the upper side of the real axis and \pi in the other side. See that the radio is going to zero and use the Identity theorem to conclude that have sense that the formulas are the same in both sides in z~=0,1,2,...

I really need help please... I need som explanation of this steps or another proof.