I'm interested in the Special Functions especially the Gamma Function. I decided to write a Bachelor Thesis about it but I do not know what kind of theorem(s) in the Gamma Function that is (are) very important or popular. Can you suggest some ideas?
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That it is a proper extension? – Zelos Malum Dec 24 '15 at 16:28
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1I would start with whatever Emil Artin thought was interesting about the Gamma function. See Rosen's Expositions by Emil Artin for a copy of his writeup. – Jyrki Lahtonen Dec 24 '15 at 21:12
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1Julian Havil's Gamma: Exploring Euler's Constant contains much that will be of interest to you. – John Dawkins Dec 24 '15 at 22:25
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There are some milestones:
- The Bohr-Mollerup theorem;
- The integral representation for the $\Gamma$ function;
- Euler's and Weierstrass products.
You may further investigate the Wallis product and Gautchi's inequality, the Laplace transform of $x^u$ (and the bound for the least quadratic non-residue given by Ankeny through that technique, very interesting), problems related with the digamma function, relations between some values of the $\Gamma$ function and some elliptic integrals, irrationality or trascendence of some values of the $\Gamma$ function and much more. It is a very nice topic.
Jack D'Aurizio
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