Trouble understanding the Gamma function

  • #1
The http://en.wikipedia.org/wiki/Gamma_function" [Broken] is the integral [tex]\Gamma(z)=\int_{0}^{\infty}{dt\, t^{z-1}e^{-t}}[/tex] . It has poles for integers of z less than 1 and is finite everywhere else. But to me it seems like it should be infinite for non integer values of z less than 0.

My reasoning: when t is close to 0, the function t[tex]^{z-1}[/tex]e[tex]^{-t}[/tex] is approximately equal to t[tex]^{z-1}[/tex].

The integral [tex]\int_{t_1}^{t_2}{dt\, t^{z-1}}[/tex] is equal to ([tex]t_2^{z}[/tex] - [tex]t_1^{z}[/tex])/z. When z<0 and [tex]t_1 = 0[/tex], [tex]t_1^{z}[/tex] is infinite. Thus it seems the integral [tex]\int_{0}^{\infty}{dt\, t^{z-1}e^{-t}}[/tex] has an infinite contribution right after t=0 for z<0.
 
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Answers and Replies

  • #2
mjsd
Homework Helper
726
3
properties of the integrand may be seen by expanding it into a Laurent series
 
  • #3
gel
533
5
The http://en.wikipedia.org/wiki/Gamma_function" [Broken] is the integral [tex]\Gamma(z)=\int_{0}^{\infty}{dt\, t^{z-1}e^{-t}}[/tex] .

Only when z has positive real part. It is extended to the rest of the complex plane by analytic continuation or, equivalently, by enforcing that [itex]\Gamma(z+1)=z\Gamma(z)[/itex] holds everywhere.
 
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  • #4
Only when z has positive real part. It is extended to the rest of the complex plane by analytic continuation or, equivalently, by enforcing that [itex]\Gamma(z+1)=z\Gamma(z)[/itex] holds everywhere.

Ok thanks
 

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