Gamma Function: Gamma(n) = integral(0 to infinity) t^(n-1) * e^(-t) dt = (n-1)! for positive integers.

Beta Function: B(m,n) = integral(0 to 1) x^(m-1)(1-x)^(n-1) dx = Gamma(m)*Gamma$\frac{n}{Gamma}$(m+n)

Connection to Wallis: integral$\frac{0 to pi}{2}$ sin^(2m-1) x * cos^(2n-1) x dx = B$\frac{m,n}{2}$ = Gamma(m)Gamma$\frac{n}{(2*Gamma(m+n)}$)

JEE Relevance: While the Gamma function itself is beyond JEE scope, knowing Gamma$\frac{1}{2}$ = sqrt(pi) explains why integral$\frac{0 to pi}{2}$ sqrt(sin x cos x) dx involves sqrt(pi).