Gamma function: Gamma(n) = integral(0 to inf) t^(n-1) * e^(-t) dt

• Gamma(n+1) = n * Gamma(n) (recurrence)
• Gamma(n+1) = n! for positive integers
• Gamma(1/2) = sqrt(pi)

Beta function: B(m,n) = integral(0 to 1) t^(m-1)(1-t)^(n-1) dt

• B(m,n) = Gamma(m)Gamma(n)/Gamma(m+n)
• B(m,n) = B(n,m) (symmetric)

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

JEE relevance: Rarely tested directly, but the Wallis formula results can be derived from Beta function. Understanding Gamma(1/2) = sqrt(pi) helps with Gaussian integral problems.