0

There are almost no tutorials how to compute the gamma function by hand. If I try to do it myself, I fall into all sorts of pitholes, for example finding a limit that is indeterminate no matter how many times I differentiate the both sides of the fraction, or integrating by parts in an infinite loop, etc.

There must be some tricks to calculate the gamma function by hand.

So how I would for example compute the gamma function of $\frac12$?

KKZiomek
  • 4,012
  • That is the same as asking how to compute $\sqrt{\pi}$ by hand. I guess you have to know a few digits of $\pi$ in advance to perform that. – Jack D'Aurizio Feb 16 '17 at 19:13

1 Answers1

2

$$\Gamma(1/2)=\int_0^\infty x^{-1/2}\exp(-x)dx.$$

Let $u=\sqrt{x}$. We get:

$$\int_0^\infty \frac{1}{u}e^{-u^2}2udu=2\int_0^\infty e^{-u^2}du.$$

The last integral is quite famous and easy to compute. Can you finish it from here?

Alex R.
  • 32,771
  • Unfortunately i am learning calculus myself, so no, I don't recognize this integral. Can you walk through the whole computation if you can? – KKZiomek Feb 16 '17 at 17:58
  • @KKZiomek This integral is so famous it has a wiki page: https://en.wikipedia.org/wiki/Gaussian_integral . I would not recommend getting into the Gamma Function if you have still not met this integral: try working a bit of multivariable calculus first. – Lonidard Feb 16 '17 at 18:00
  • See e.g. this – Robert Israel Feb 16 '17 at 18:01