Beta function

The beta function is the two-argument integral B(x,y) = ∫₀¹ t^(x−1)(1−t)^(y−1) dt for Re(x), Re(y) > 0. It equals Γ(x)Γ(y)/Γ(x+y) and extends beyond this domain by analytic continuation.

Gamma function: A meromorphic extension of the factorial, satisfying Γ(z+1) = zΓ(z). The beta function is a quotient of three gamma values.
Complex number: A number of the form a+bi, with real a and b and i² = −1. The beta function's arguments and values are commonly complex.
Beta distribution: A probability distribution on [0,1] with density proportional to x^(α−1)(1−x)^(β−1). The beta function normalizes its density by supplying the integral of its kernel.
Multivariate beta function: The normalization function Γ(α₁)…Γ(αₖ)/Γ(α₁+…+αₖ) for positive parameters. It extends the two-argument gamma quotient to proportions with multiple components.
Leonhard Euler: An eighteenth-century mathematician whose work shaped analysis, number theory, and mechanics. Euler introduced the beta integral and related it to the gamma function.
Euler integral of the first kind: The integral representation of the beta function over the unit interval. This integral defines B(x,y) directly when both arguments have positive real parts.
Improper integral: An integral defined by a limit when an endpoint is infinite or the integrand is unbounded. The defining integrand can diverge at either endpoint while remaining integrable.
Dirichlet distribution: A probability distribution over vectors of nonnegative proportions that sum to one. Its normalization constant is a multivariate beta function built from gamma values.
Hypergeometric function: A function defined by a power series whose successive coefficient ratios are rational in the index. Beta integrals provide integral representations and coefficients for hypergeometric functions.
Euler integral: An integral representation associated with Euler's development of the gamma and beta functions. The beta function is one of the special functions defined through Euler's integrals.

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