Trouble understanding the Gamma function

[vcdfrexzaswq]

The 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 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