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Gamma function

  1. Jul 31, 2009 #1
    [tex]\Gamma(z)=\int^{\infty}_0x^{z-1}e^{-x}dz[/tex]
    [tex]z\in\mathhad{C}[/tex]

    In which problems in statistical physics we need gamma functions of complex argument?

    I don't know how to calculate [tex]\Gamma(i)[/tex] for exaple?
     
    Last edited: Jul 31, 2009
  2. jcsd
  3. Jul 31, 2009 #2

    mathman

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    Your formula is wrong (typo) Should be

    [tex]\Gamma(z)=\int^{\infty}_0x^{z-1}e^{-x}dx[/tex]
     
  4. Jul 31, 2009 #3
    Yes mistake. I make corrcection!
     
  5. Aug 1, 2009 #4

    mathman

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    Not quite: you still have dz when it should be dx.
     
  6. Aug 2, 2009 #5
    [itex]\Gamma(i)[/itex]

    Well, the integral definition converges only if [itex]\Re z > 0[/itex], so in particular it does not converge at [itex]z=i[/itex]. So you need to use analytic continuation. But fortunately that is very easy for the [itex]\Gamma[/itex] function, unlike most other functions. Use the functional equation [itex]\Gamma(z+1) = z\Gamma(z)[/itex]. So to compute [itex]\Gamma(i)[/itex] we can compute [itex]\Gamma(1+i)[/itex] then apply the formula.

    You cannot expect a closed-form answer. [itex]\Gamma(1+i) \approx .4980156681-.1549498283i[/itex] so divide by [itex]i[/itex] to get [itex]\Gamma(i) \approx -.1549498283-.4980156681i[/itex].
     
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