Gamma function

Gamma function. The Gamma function is a fundamental special function that extends the factorial to complex numbers, excluding the non-positive integers. It was first introduced by Leonhard Euler in the 18th century and later studied in depth by Adrien-Marie Legendre, who gave it its name and symbol. The function is ubiquitous in mathematical analysis, probability theory, and theoretical physics, providing a continuous interpolation of the factorial sequence.

Definition

The most common definition for a complex argument with a positive real part is via the convergent integral known as Euler's integral of the second kind. This definition was pioneered by Leonhard Euler and can be connected to the Gaussian integral through a simple substitution. For arguments with a real part greater than zero, it is also equivalent to the Weierstrass factorization theorem representation, an infinite product definition independently developed by Karl Weierstrass. Another pivotal representation, valid for all complex numbers except the non-positive integers, is given by the Hankel contour integral, a result deeply connected to the theory of complex analysis.

Properties

A fundamental property is its functional equation, which directly relates it to the factorial for positive integers. This equation is a cornerstone in proving its analytic continuation to the entire complex plane. The function satisfies the reflection formula, a elegant result proven by Leonhard Euler that involves the sine function. The multiplication theorem, generalized by Carl Friedrich Gauss, and the Stirling's approximation for its asymptotic behavior, discovered by Abraham de Moivre and refined by James Stirling, are other critical properties. Its logarithmic derivative is the digamma function, which plays a key role in the calculation of series and appears in the Riemann zeta function.

Special values

For positive integers, its value is precisely the factorial of one less than the integer, a result foundational to combinatorics. At the argument of one-half, its value is the square root of pi, a result derived from the Gaussian integral and pivotal in normal distribution theory. The value at one-quarter is related to elliptic integrals and the lemniscate constant, studied by Carl Friedrich Gauss. The Euler–Mascheroni constant appears prominently in the series expansion of its logarithmic derivative, the digamma function, at unity. The function has simple poles at all non-positive integers, a feature central to the theory of meromorphic functions.

Applications

In probability theory and statistics, it is essential in defining the gamma distribution and the chi-squared distribution, which are fundamental to hypothesis testing. It appears in the normalization constants of the Student's t-distribution and the F-distribution. Within number theory, it is intimately connected to the functional equation of the Riemann zeta function, a central object in the study of prime numbers. In theoretical physics, it arises in calculations of Feynman diagrams in quantum field theory and in the Bohr–Mollerup theorem related to thermodynamic potentials. The beta function, used in Bayesian statistics, is expressible as a ratio of two such functions.

Related functions

The digamma function is its logarithmic derivative and is studied in the context of harmonic series. The polygamma functions are higher-order derivatives, useful in summation of series. The incomplete gamma functions, both upper and lower, are generalizations crucial in survival analysis and the chi-squared distribution. The beta function, closely related, is fundamental in integral calculus and was studied by Leonhard Euler. The Barnes G-function, a generalization involving multiple parameters, appears in the study of determinants of Laplacians. The q-gamma function, a q-analogue, extends its definition within the field of quantum groups and combinatorics.

Category:Special functions Category:Mathematical analysis Category:Leonhard Euler