A Question About the Gamma Function

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A Question About the Gamma Function

Postby philt » Wed Oct 27, 2010 12:32 pm UTC

I've got a problem for a class that requires me to modify the gamma function from its usual form:
[math]\Gamma(s) = \int_0^\infty x^{s-1} e^{-x}\,{\rm d}x\,[/math]
to the form
[math]\Gamma(s) = f(s)\int_0^\infty e^{-A(y)/ \xi(s)}\,{\rm d}y\,[/math]
through 'a suitable change of the integration variable'. It's not too hard to get the right form of the exponential (eg [imath]y = e^{-x}[/imath]), but I never manage to keep the limits the same. I can't find anything online or in my textbooks, and would really appreciate any help at all.

EDIT: done it! And only a few hours before the deadline :) If anyone's interested the substitution is (annoyingly simple) [imath]x = (s-1)y[/imath]. If anyone wants to see where this comes from (it's not just random guessing!), let me know and I'll type it up.

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