# Gamma Function Integral

1. May 22, 2013

### Mandelbroth

This is mostly calculus, but the question is computer based, I think.

The antiderivative of the gamma function is, fairly trivially, $\displaystyle \int_{0}^{+\infty}\frac{t^{z-1}}{e^{t}\ln{t}}-C$, where C is an arbitrary constant.

Why does Wolfram Alpha have trouble calculating the convergence of that integral at any given point?

2. May 22, 2013

### mathman

Definite integrals do not have a constant of integration. What values of z are you having trouble with? The integrand blows up at t = 0 for Re(z) <1 and the integral diverges for z = 0, -1, -2, etc.

3. May 22, 2013

### Mandelbroth

That's not what I'm saying. :tongue:

I'm saying that $\displaystyle \int \Gamma(z) dz = \int\int_{0}^{+\infty}\frac{t^{z-1}}{e^{t}}\, dt \ dz = \int_{0}^{+\infty}\frac{t^{z-1}}{e^{t}\ln{t}}-C$. I'm evaluating the antiderivative, or "indefinite" integral, of the Gamma function. It's fairly evident that I used Fubini's theorem, because it's the only sensible way to obtain that result.

I'm asking why the antiderivative of the Gamma function can't be evaluated by Wolfram Alpha.

4. May 23, 2013

### mathman

I now understand what you did. However I have never worked with Wolfram Alpha, so I can't help you there.