Derivative of Gamma Function: Finding the Mistake?

[Lojzek]

I have tried to calculate the derivative of Gamma function and got a strange result, which
is obviously wrong. Can someone find the mistake?

Definition:
Gamma[z]=Integral[t^(z-1)exp(-t)dt]

Derivative:
(d/dz)Gamma[z]=Integral[(d/dz)t^(z-1)exp(-t)dt]=Integral[(z-1)t^(z-2)exp(-t)dt]=
(z-1)*Gamma[z-1]=Gamma[z]

Looks like gamma solves the equation f'=f, but this can't be true, since only
exponential function solves this equation.

 

[eok20]

Your problem is with (d/dz)t^(z-1). What you computed is really (d/dt)t^(z-1). You need to use logarithmic differentiation: (d/dz)t^(z-1) = t^(z-1) ln t

 

[Lojzek]

You are right, I derived with respect to the wrong variable. I wanted to calculate the derivative of gamma at least at integer parameter values, but it seems I won't be able to do this, since I can't calculate the resulting integral.