## Relationship between Beta and Gamma Functions

**Relationship between Beta and Gamma Functions:**

The Beta and Gamma function are related by

**Proof:** We have

Substitute x = t^{2}, 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 x^{2} y^{2 }= r^{2}