beta function
/ BAY-tuh; the symbol is the capital Greek letter B /
Many probability and geometry problems boil down to one shape of integral: integrate x to some power times (1 minus x) to some other power, over the interval from 0 to 1. The beta function packages exactly that family. B(p, q) = integral from 0 to 1 of x^{p-1} (1-x)^{q-1} dx — a single symbol for the area under that two-parameter hump on the unit interval.
Its defining feature is a clean link to the gamma function: B(p, q) = Gamma(p) Gamma(q) / Gamma(p+q). This identity is the workhorse — it lets you evaluate the integral instantly once you know gamma values, and conversely turns gamma identities into beta identities. There is also a trigonometric form, B(p, q) = 2 times integral from 0 to pi/2 of (sin theta)^{2p-1} (cos theta)^{2q-1} d theta, which is how powers-of-trig integrals over a quarter period get evaluated in closed form. The beta function is symmetric, B(p, q) = B(q, p), and converges whenever both p and q have positive real part.
The beta function is the normalising constant of the Beta probability distribution (ubiquitous in Bayesian statistics as the conjugate prior for a probability), and it appears in the volume of regions, in the original Veneziano amplitude that launched string theory, and throughout the theory of the gamma and hypergeometric functions. Practically, whenever a definite integral has the form 'power times one-minus-it to a power' or 'sine to a power times cosine to a power over a quarter turn', the beta function gives the answer without further integration.
B(2, 3) = Gamma(2) Gamma(3) / Gamma(5) = (1! times 2!) / 4! = 2/24 = 1/12, which matches integral from 0 to 1 of x (1-x)^2 dx = 1/12 computed directly.
The gamma-ratio formula turns a definite integral into a one-line arithmetic check.
The beta function is symmetric and built from gamma; it is not a probability by itself — the Beta distribution divides x^{p-1}(1-x)^{q-1} by B(p,q) to make a density.