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147 Points
11 years ago

Dear nitish

# Gamma function

In mathematics, the Gamma function (represented by the capital Greek letter Γ) is an extension of the factorial function to realcomplex numbers. For a complex number z with positive real part the Gamma function is defined by and

$\Gamma(z) = \int_0^\infty t^{z-1} e^{-t}\,dt\;$

This definition can be extended by analytic continuation to the rest of the complex plane, except the non-positive integers.

If n is a positive integer, then

Γ(n) = (n − 1)!

showing the connection to the factorial function. Thus, the Gamma function extends the factorial function to the real and complex values of n.

# Beta function

In mathematics, the beta function, also called the Euler integral of the first kind, is a special function defined by

$\mathrm{\Beta}(x,y) = \int_0^1t^{x-1}(1-t)^{y-1}\,dt \!$

## Properties

The beta function is symmetric, meaning that

$\Beta(x,y) = \Beta(y,x). \!$

It has many other forms, including:

$\Beta(x,y)=\dfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)} \!$
$\Beta(x,y) = 2\int_0^{\pi/2}(\sin\theta)^{2x-1}(\cos\theta)^{2y-1}\,d\theta, \qquad \textrm{Re}(x)>0,\ \textrm{Re}(y)>0 \!$
$\Beta(x,y) = \int_0^\infty\dfrac{t^{x-1}}{(1+t)^{x+y}}\,dt, \qquad \textrm{Re}(x)>0,\ \textrm{Re}(y)>0 \!$
$\Beta(x,y) = \sum_{n=0}^\infty \dfrac{{n-y \choose n}} {x+n}, \!$
$\Beta(x,y) = \frac{x+y}{x y} \prod_{n=1}^\infty \left( 1+ \dfrac{x y}{n (x+y+n)}\right)^{-1}, \!$
$\Beta(x,y) \cdot \Beta(x+y,1-y) = \dfrac{\pi}{x \sin(\pi y)}, \!$

 Wallis Formula

The Wallis formula follows from the infinite product representation of the sine

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