How would I find the gamma function of i+1 or any other complex number?
I tried using integration by parts, but it just went on forever.
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There is not a simpler closed form for $\Gamma(i+1)$ other than just writing $\Gamma(i+1)$. It does have a decimal expansion $0.498015668118356042713691117462...$ though.
If you want a numerical answer, Riemann Sums will do, or even Integration by Parts like you tried with error estimates. Stirling's Approximation would also help you for large values of $z$.
Here is the most basic solution:
Pretty basic and doable with a basic calculator.