Beta and Gamma functions and some useful properties

We here remind the definition of the Euler Gamma and Beta functions and derive their relation. In addition, we derive useful properties of summation and an asymptotic limit for the Beta function. The Gamma function is defined as: By integration by parts, we obtain the relation |$\Gamma(a) \,{=}\, (a\,{-}\,1) \Gamma(a\,{-}\,1)$|⁠, from which follows, when |$a$| is an integer, |$\Gamma(a) \,{=}\, (a\,{-}\,1)!$|⁠. The Beta function is defined as: The two functions are related by the identity: To demonstrate Eq. (F.3), we recall that the Laplace transform of a convolution of functions is equal to the product of the Laplace transform of the individual functions. The Laplace transform is defined as |$F(s) \,{=}\, \mathcal{L}(f(t)) \,{=}\, \int_{0}^{+\infty} e^{-st} f(t)dt$|⁠. In our case let us consider the functions |$f_1(t) \,{=}\, t^{a-1}$| and |$f_1(t) \,{=}\, t^{b-1}$|⁠, |$t\,{\gt}\,0$|⁠. We thus have: Let us make explicit the Laplace transform of |$f_1(t)$|⁠: where the solution is obtained by making the change of variable |$z \,{=}\, s t$|⁠. In an equivalent way, we can obtain |$\mathcal{L} (f_2(t)) \,{=}\, \Gamma(b)/s^b$|⁠. We now observe that since |$\Gamma(a)$| does not depend on |$s$|⁠, it holds for the inverse transform: Substituting in Eq. (F.4), we obtain: