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The gamma function has immense applications in various fields. However sometimes you don't have access to a calculator (like in certain exams), so the question is :

Which all values of the gamma function can be calculated(by hand, no calculator) that might prove to be useful in certain problems? Also show how you'd calculate it (by hand of course, no calculators)

Now I know the value of $\Gamma(\frac{1}{2})$ (which can be calculated by various methods (See here for methods to do that)) so values like $\Gamma(\frac{3}{2})$, $\Gamma(\frac{5}{2})$..... can be found by $\Gamma(n) = (n-1).\Gamma(n-1)$, I am interested in other values that can be calculated by hand.

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    $\begingroup$ The Wikipedia article on particular values is likely a worthwhile starting point -- en.wikipedia.org/wiki/Particular_values_of_the_gamma_function $\endgroup$
    – PrincessEev
    Apr 13, 2020 at 21:52
  • $\begingroup$ Why would an exam that excludes calculators require the computation of $\Gamma (x)$ for $x \notin \mathbb{N}$ (or $2x \notin \mathbb{N}$)? Has that ever happened? $\endgroup$
    – David G. Stork
    Apr 13, 2020 at 21:55
  • $\begingroup$ @DavidG.Stork sometimes the gamma function isn't the only way to solve a particular problem, however it could be quicker or easier path in some case. $\endgroup$
    – Saket Gurjar
    Apr 13, 2020 at 22:04
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    $\begingroup$ Can you give one example from an actual test? $\endgroup$
    – David G. Stork
    Apr 13, 2020 at 22:12
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    $\begingroup$ @DavidG.Stork Honestly speaking, I do not have an example from an actual test (at least all tests I have given ), but for an example you could consider the integral:$\int_0^{\infty} x^2 e^{-x^2} \,dx$ $\endgroup$
    – Saket Gurjar
    Apr 13, 2020 at 22:40

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Well I’d recommend just to look at the Wikipedia article for particular values of the gamma function, there are many exact values you can recalculate by hand https://en.m.wikipedia.org/wiki/Particular_values_of_the_gamma_function

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