# Gamma and Beta Functions

• ### ${\Gamma}$ Function

We are all familiar with the factorial notation. $x!$ is the product of first $x$ natural numbers. We also define $0!=1$. For all other values of $x$, $x!$ is undefined.

Consider the function:

${f(1)=1, f(x+1) = x \times f(x)}$

This is a recurrence relation, but is defined only for integers.

The $\Gamma$ function is an extension of the factorial function, with its argument shifted down by $1$. i.e.

${\Gamma (n)= (n-1)!}$

• ### Definition

For $n>0$, ${\Gamma (n) = \int \limits_0^{\infty} e^{-t}t^{n-1}dt}$

It can also be defined as ${\Gamma (n) = 2 \times \int \limits_0^{\infty} e^{-x^2}x^{2n-1}dx}$

• ### Useful Points about $\Gamma$ Function

I) Reduction Formula: $\Gamma(n+1)= n \Gamma (n)$

II) $\Gamma (1)=1$

III) $\Gamma (\frac 1 2) = \sqrt {\pi}$

IV) For $0 < a < 1$, $\Gamma (a) \times \Gamma (1-a) = \frac {\pi}{sin(a \pi)}$

V) $\Gamma (n), n\le 0, n \in \mathbb Z = \infty$

• ### Solving Problems using ${\Gamma}$ Function

I) For problems containing a function of $x$ in the exponent, i.e. $e^{-f(x)}$ (NOTE THE NEGATIVE SIGN), substitute $f(x)$ as $t$, and use integration by substitution.

II) For problems containing an exponential function of $x$, e.g. $a^{f(x)}$, substitute $a^{f(x)}$ in such a way that we get $e^{-t}$ in the numerator.

III) For problems involving $log (x)$, substitute $x$ as $e^{-t}$.

IV) For problems involving $sine$ and $cosine$ functions, use the Euler’s identity,

${e^{i \theta} = cos (\theta)+ i \sin(\theta), \ where \ i = \sqrt {-1}}$

V) Problems using the reduction formula $\Gamma (n+a) = n \Gamma (n)$

• ### ${\beta}$ Function

$\beta$ function is another important function in mathematics. French mathematician Jacques Binet gave it the name. It is defined as

${\beta (m,n) = \int \limits_0^1 x^{m-1} (1-x)^{n-1}dx, \ m>0,n>0}$

Alternatively,

${\beta (m,n)= 2 \times \int \limits_0^{\frac \pi 2} sin^{2m-1}\theta \times cos^{2n-1} \theta d \theta}$

• ### Properties

I) The function is symmetric, i.e. $\beta (m,n) = \beta (n,m)$

II) By substituting $x$ as $\frac {1}{1+t}$, we get

${\beta (m,n) = \int \limits_0^{\infty} \frac {t^{m-1}}{(1+x)^{m+n}}dt}$

III) ${\beta (m,n) = \frac {\Gamma (m) \Gamma (n)}{\Gamma (m+n)}}$

IV) Legendre’s duplication formula:

${\Gamma (m) \times \Gamma (m + \frac 1 2) = \frac {\sqrt \pi}{2^{2m-1}} \Gamma (2m)}$

• ### Solving Problems using ${\beta}$ Function

The problems are to be solved by using the definition of $\beta$ function, its relation to $\Gamma$ function etc.

For integrals of the kind $\int \limits_a^b (x-a)^l (b-x)^m dx$, substitute $(x-a)$ as $(b-a)t$