# Gamma Function Integral

## Main Question or Discussion Point

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?

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.

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.
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.

mathman