Trouble understanding the Gamma function

[vcdfrexzaswq]

The Gamma function (http://en.wikipedia.org/wiki/Gamma_function) is the integral \Gamma(z)=\int_{0}^{\infty}{dt\, t^{z-1}e^{-t}} . 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^{z-1}e^{-t} is approximately equal to t^{z-1}.

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

[mjsd]

properties of the integrand may be seen by expanding it into a Laurent series

[gel]

The Gamma function (http://en.wikipedia.org/wiki/Gamma_function) is the integral \Gamma(z)=\int_{0}^{\infty}{dt\, t^{z-1}e^{-t}} .

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 \Gamma(z+1)=z\Gamma(z) holds everywhere.

[vcdfrexzaswq]

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 \Gamma(z+1)=z\Gamma(z) holds everywhere.

Ok thanks