mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, Hadamard's gamma function, named after

Jacques Hadamard Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry and partial differential equations. Biography The son of a teac ...

, is an extension of the

factorial In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \t ...

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...

, different from the classical

gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...

. This function, with its

argument An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...

shifted down by 1, interpolates the factorial and extends it to

real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...

and

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

s in a different way than Euler's gamma function. It is defined as: :

H(x) = \frac\,\dfrac \left \,

where denotes the classical gamma function. If is a positive integer, then: :

H(n) = \Gamma(n) = (n-1)!

Properties

Unlike the classical gamma function, Hadamard's gamma function is an

entire function In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any fin ...

, i.e. it has no

pole Pole may refer to: Astronomy *Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets *Pole star, a visible star that is approximately aligned with the ...

s in its domain. It satisfies the

functional equation In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...

H(x+1) = xH(x) + \frac,

with the understanding that

\tfrac

is taken to be for positive integer values of .

Representations

Hadamard's gamma can also be expressed as :

H(x)=\frac

and as :

H(x) = \Gamma(x) \left 1 + \frac \left \ \right

where denotes the

digamma function In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: :\psi(x)=\frac\ln\big(\Gamma(x)\big)=\frac\sim\ln-\frac. It is the first of the polygamma functions. It is strictly increasing and strictly ...

References

* * *{{cite web , website=The Wolfram Functions Site , url=http://functions.wolfram.com/GammaBetaErf/Gamma/introductions/Gamma/ShowAll.html , title=Introduction to the Gamma Function , publisher=Wolfram Research, Inc , access-date=27 February 2016 Gamma and related functions Analytic functions Special functions