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(n)/Gamma(m+n)

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

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