A very vague question:(adsbygoogle = window.adsbygoogle || []).push({});

What is the derivative of the gamma function?

Here's what I've got, using differentiation under the integral. Can anybody tell me if I'm on the right track? What does my answer mean?

[tex]\Gamma(z) = \int_0^{\infty} t^{z - 1} \: e^{-t} \; dt[/tex]

The integrand can be expressed as a function [itex]f(z, t) = t^{z - 1} \: e^{-t}[/itex]:

[tex]\frac{d}{dz} \int_{y_0}^{y_1} f(z, t) \; dt = \int_{y_0}^{y_1} \frac{\partial}{\partial z} f(z, t) \; dt [/tex]

Because [itex]e^{-t}[/itex] is a constant of sorts (correct me if I'm wrong), with respect to the partial derivative toz, all this turns into

[tex]\frac{d}{dz} \Gamma(z) = \int_0^{\infty} t^{z-1} \: e^{-t} \: \ln(t) \; dt[/tex]

But this is useless! :yuck: Does anybody have any thoughts?

(I'm trying to find all the extrema of the gamma function, ... they look like the follow an exponential curve and I want to see if there is an expression for it)

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Derivative of the Gamma Function

**Physics Forums | Science Articles, Homework Help, Discussion**