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$\frac{1}{2}$ = 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$\frac{n}{Gamma}$(m+n)
• B(m,n) = B(n,m) (symmetric)

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

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