Derivative of the Gamma function equals Psi times Gamma

[loto]

Hi gang,

I'm having trouble with doing a derivative of the Gamma function. I know both the definition of Gamma and Polygamma, but can't see how to get from the derivative of Gamma to Psi times Gamma. Any help or hints would be great.

Thanks!

 

[quasar987]

But isn't it the very definition of $\psi^{(0)}$ that $\Gamma'(z)=\psi^{(0)}\Gamma(z)$?

I think you just have to differentiate $\Gamma(z)$. Since everything is continuous, you can move the derivative inside the integral, etc.