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plz sir provied me full information about gamma function.beta function, walli's function .how i solve question from these type of function

plz sir provied me full information about gamma function.beta function, walli's function .how i solve question from these type of function

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1 Answers

Badiuddin askIITians.ismu Expert
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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Badiuddin

 

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