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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 $\frac12$?

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    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. – 2017-02-16

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

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?

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    Unfortunately i am learning calculus myself, so no, I don't recognize this integral. Can you walk through the whole computation if you can? – 2017-02-16
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    @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. – 2017-02-16
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    See e.g. [this](http://math.stackexchange.com/questions/9286/proving-int-0-infty-mathrme-x2-dx-dfrac-sqrt-pi2) – 2017-02-16