A Gamma function

LagrangeEuler · May 15, 2016

I have two questions related Gamma functions

1. Finding ##\Gamma## analytically. Is that possible only for integers and halfintegers? Or is it possible mayble for some other numbers? For example is it possible to find analytically ##\Gamma(\frac{3}{4})##?

2. Integral ##\Gamma(x)=\int^{\infty}_0 \xi^{x-1}e^{-\xi}d \xi ## converge only for ##x>0## in real analysis. How can we then write ##\Gamma(\frac{1}{2})=\Gamma(-\frac{1}{2}+1)## when relation ##\Gamma(x+1)=x\Gamma(x)## is derived from partial integration?

fresh_42 · May 15, 2016

How about to use Gauß' formula for ##x \in ℂ \backslash \{0, -1, -2, \dots \}## instead:

$$Γ(x) =\lim_{n→\infty} \frac{n!n^x}{x(x+1) \cdots (x+n)}$$

Edit: It's sufficient to require ##Re(x) > 0## for the integral formula.

mathman · May 15, 2016

The gamma function has singularities at 0 and negative integers. Using analytic continuation the function can be defined elsewhere.

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A Gamma function

LagrangeEuler

fresh_42

Staff: Mentor

mathman

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