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}


{\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


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