Incomplete gamma function

  • Thread starter bruno67
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  • #1
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Main Question or Discussion Point

How do I calculate the integral

[tex]\int_{ix}^{i\infty} e^{-t} t^{-s-1}dt,[/tex]
where [itex]x>0[/itex], [itex]s>0[/itex]? Mathematica gives [itex]\Gamma(-s,ix)[/itex], where [itex]\Gamma(\cdot,\cdot)[/itex] is the incomplete gamma function, but I am not sure how to justify this formally.
 

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  • #2
mathman
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Substitute u = -it, so the integral is from x to inf.
 
  • #3
Mute
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The incomplete Gamma function is defined by the integral

[tex]\Gamma(s,x) = \int_{x}^\infty dt~t^{s-1}e^{-t}.[/tex]

Replacing x with ix formally gives [itex]\Gamma(s,ix)[/itex]. However, the meaning of the integral with lower bound ix is really just formal, I think. You identify the integral with the incomplete Gamma function, and then you determine the "integral's" value by using the analytic continuation of the incomplete Gamma function for complex arguments.
 
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