Question

what is gamma function?

Answer 1

 the gamma function (represented by Γ, the capital Greek alphabet letter gamma) is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers. If n is a positive integer,

Gamma (n)=(n-1)!}

The gamma function is defined for all complex numbers except the non-positive integers. For complex numbers with a positive real part, it is defined via a convergent improper integral:

Gamma (z) = {0}^{\infinty }x^{z-1}e^{-x}.dx}

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Answer 2

$\large\text{[Gamma function]}$

The previous step was on the extent of the factorial function to the whole number.

If we choose a number from the set of natural numbers, it will produce a graph. A graph that consists of disjoints between each whole number.

Gauss defined a function that satisfies the three conditions.

$\begin{gather\large\text{ \boxed{\begin{aligned}&\text{(1) }x!=x\cdot(x-1)!\\\\&\text{(2) No disjoint points on the graph}\\\\&\text{(3) A graph in reals}\end{aligned}}\\\\ \longrightarrow\boxed{\Gamma(x)=(x-1)!} }$

If we evaulate $\dfrac{1}{2}!$we get $\dfrac{\sqrt{\pi}}{2}$