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A Gamma function

  1. May 15, 2016 #1
    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?
     
  2. jcsd
  3. May 15, 2016 #2

    fresh_42

    Staff: Mentor

    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.
     
  4. May 15, 2016 #3

    mathman

    User Avatar
    Science Advisor
    Gold Member

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