Calculate Gamma12 and Gamma32 using Gaussian integralSolutio

Calculate Gamma(1/2) and Gamma(3/2) using Gaussian integral

a) Gamma (1/2) :

For any integer n 0,

We have , n ! = tⁿ * e^-t dt , for x>0

x = t^x * e^-t dt / t (Here range is from 0 to infinity) So n = (n-1)! when n 1

Using integration by parts,

(x+1) = x* x

The property if gamma function is,

(x * y) / x+y = t^x-1 * (1 - t )^y-1 dt. (range is from 0 to 1)

Set here x = y = 1/2.

(1/2)² = t^x-1 * ( 1 - t )^y-1 dt (range is from 0 to 1)

Note that , 1/2 = sqrt(t) * e^-t dt / t (sqrt(t) means square root of t ) (from o to infinity)

= e^-t / sqrt(t) dt. (From 0 to infinity) ____a)

Put t = x² ==> x = sqrt(t)

take derivative with respect to x,

dt = 2*x dx

Put this in equation a),

= e^{-x^2} /x 2x dx (from 0 to infinity)

= 2 * e^{-x ^2} dx = 2K (from o to infinity)

So 4K² = dt / sqrt(t*(1 - t)) (from 0 to 1)

Substitute t = sin²

differentiate with respect to ,

dt = 2 sin*cos d

4K² = 2 sin*cos d / (sin*cos) (from 0 to / 2)

= 2 * 1 d (from 0 to / 2)

= 2*[] (from 0 to / 2)

= 2* ( / 2 - 0)

K = sqrt() / 2

1/2 = sqrt()

b) Gamma (3/2) :

We can write 3 / 2 = (1 + 1/2)

And we have result that x+1 = x * x

Using this result,

(1/2 + 1) = 1/2 * 1/2

And we know that 1/2 = sqrt ()

(1/2 + 1) = 1/2 * sqrt ()

(1/2 + 1) = sqrt () / 2

3 / 2 = sqrt() / 2

1 / 2 = sqrt()

3 / 2 = sqrt() / 2