Today we’re going to discuss one of the non-elementary functions called gamma function and consider some of its properties. Gamma function is of great importance, it’s widely applied in math (in particular, when integrating certain types of expression gamma function helps greatly, we’ll see that later in examples), also gamma function is used in probability theory (possibly, you’ve heard about gamma distribution), etc. One of the most common representations of gamma function is the following:
\Gamma(p)=\int_0^{\infty}{e^{-x}x^{p-1}dx}, p>0
This is only true for positive values of parameter p. So to obtain value of gamma function in a certain point p we need to integrate over x the expression e^{-x} x^{(p-1)}, leaving p as a free parameter. Notice limits of integration – zero and infinity. This means that we’re dealing with improper integral. (more…)