Relationship between Beta and Gamma Functions:
The Beta and Gamma function are related by
Proof: We have
Substitute x = t2, dx = 2t dt, we get
Replacing n by m and x by y, we have
We shall transform the double integral into polar coordinates. Substitute x = r cosθ, y = r sinθ then we have dx dy = drdθ. As x and y varies from 0 to infinity the region of integration entire first quadrant. Hence θ varies from 0 to π/2 and r varies from 0 to infinite and also x2 y2 = r2