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The $\Gamma$ function is defined by

$$\Gamma(z)=\int_{0}^{+\infty}t^{z-1}e^{-t}dt$$

where $z$ is a complex number.

We know that if $z$ is real then the values of $\Gamma$ are also real. I am interested in the case where $z$ is complex and the values of $\Gamma$ are real. In particular where $0<Re(z)<1$.

Is this case possible?

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1 Answer 1

up vote 4 down vote accepted

Here's a picture of the contours for $\Im(\Gamma(z))=0$. As you can see, pure real values of the gamma function, and indeed most analytic functions in the complex plane, are quite common off the real line.

Gamma Function

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Is there some simple function that describes these contours? –  nbubis Dec 30 '12 at 9:31
    
@ Mario Carneiro : So, my case is possible. Can you indicate to me a reference about this. For your claim: "most analytic functions in the complex plane, are quite common off the real line". –  ZE1 Dec 30 '12 at 9:39
    
i don't understand can you please explain what is the contour is? –  Koushik Dec 30 '12 at 10:01

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