Gamma function
The gamma function extends the factorial to complex arguments, with poles at nonpositive integers and Γ(n) = (n − 1)! for positive integers n.
Euler integral of the first kind: An improper integral that defines the gamma function for complex arguments with positive real part. This integral provides the standard analytic definition of Γ(z) before continuation extends it further.
Factorial: The product of the positive integers up to a given integer, with 0! defined as 1. The factorial is the integer-valued pattern that Γ extends by shifting its argument.
Gamma distribution: A continuous probability distribution on positive real values parameterized by shape and scale. Its density uses Γ to normalize a family of waiting-time and positive-valued models.
Leonhard Euler: An eighteenth-century mathematician who made foundational contributions to analysis, number theory, and mechanics. Euler introduced the integral that now defines the gamma function.
Reciprocal gamma function: The entire function 1/Γ(z), which has zeros at the nonpositive integers. Taking the reciprocal turns Γ’s poles into zeros and removes its singularities.
Gamma recurrence relation: The identity Γ(z + 1) = zΓ(z), which generalizes the factorial step from n! to (n + 1)!. Repeated use of this identity gives factorial values and reveals the function’s poles.
Complex analysis: The study of functions of complex variables, including their differentiability, integrals, and singularities. The gamma function’s continuation and poles are naturally described using complex analysis.
Beta function: A two-argument integral function related to gamma values by B(x,y) = Γ(x)Γ(y)/Γ(x + y). This identity converts beta integrals into gamma ratios used in probability and analysis.
Adrien-Marie Legendre: A French mathematician whose work shaped number theory, analysis, and mathematical physics. Legendre introduced the Γ notation and developed the function’s theory.
q-gamma function: A deformation of the gamma function that depends on a parameter q and relates to q-factorials. It replaces the ordinary recurrence with a q-dependent analogue.