Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Gamma function

  1. Jul 31, 2009 #1

    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


    User Avatar
    Science Advisor

    Your formula is wrong (typo) Should be

  4. Jul 31, 2009 #3
    Yes mistake. I make corrcection!
  5. Aug 1, 2009 #4


    User Avatar
    Science Advisor

    Not quite: you still have dz when it should be dx.
  6. Aug 2, 2009 #5

    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].
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook