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Hankel contour integral

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Mathematics Subject Classification

30D30 no label found33B15 no label found

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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.

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