![Reciprocal-gamma-domain-coloring](https://upload.wikimedia.org/wikipedia/commons/3/3b/Reciprocal-gamma-domain-coloring.png)
In
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 ...
, the reciprocal gamma function is the
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 ...
:
where denotes the
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 ...
. Since the gamma function is
meromorphic
In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. The ...
and nonzero everywhere in the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
, its reciprocal 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 ...
. As an entire function, it is of order 1 (meaning that grows no faster than ), but of infinite type (meaning that grows faster than any multiple of , since its growth is approximately proportional to in the left-hand plane).
The reciprocal is sometimes used as a starting point for
numerical computation
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...
of the gamma function, and a few software libraries provide it separately from the regular gamma function.
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
called the reciprocal gamma function the "factorielle" and used it in his development of the
Weierstrass factorization theorem In mathematics, and particularly in the field of complex analysis, the Weierstrass factorization theorem asserts that every entire function can be represented as a (possibly infinite) product involving its zeroes. The theorem may be viewed as an e ...
.
Infinite product expansion
Following from the
infinite product In mathematics, for a sequence of complex numbers ''a''1, ''a''2, ''a''3, ... the infinite product
:
\prod_^ a_n = a_1 a_2 a_3 \cdots
is defined to be the limit of a sequence, limit of the Multiplication#Capital pi notation, partial products ''a' ...
definitions for the
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 ...
, due to
Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
and
Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern mathematical analysis, analysis". Despite leaving university without a degree, ...
respectively, we get the following infinite product expansion for the reciprocal gamma function:
:
where is the
Euler–Mascheroni constant
Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma ().
It is defined as the limiting difference between the harmonic series and the natural l ...
. These expansions are valid for all complex numbers .
Taylor series
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
expansion around 0 gives:
:
where is the
Euler–Mascheroni constant
Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma ().
It is defined as the limiting difference between the harmonic series and the natural l ...
. For , the coefficient for the term can be computed recursively as
:
where is the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
. An integral representation for these coefficients was recently found by Fekih-Ahmed (2014):
:
For small values, these give the following values:
Fekih-Ahmed (2014)
[ also gives an approximation for :
:
where and is the minus-first branch of the ]Lambert W function
In mathematics, the Lambert function, also called the omega function or product logarithm, is a multivalued function, namely the Branch point, branches of the converse relation of the function , where is any complex number and is the expone ...
.
The Taylor expansion around 1 has the same (but shifted) coefficients, i.e.:
:
(the reciprocal of Gauss' pi-function).
Asymptotic expansion
As goes to infinity at a constant we have:
:
Contour integral representation
An integral representation due to Hermann Hankel
Hermann Hankel (14 February 1839 – 29 August 1873) was a German mathematician. Having worked on mathematical analysis during his career, he is best known for introducing the Hankel transform and the Hankel matrix.
Biography
Hankel was born on ...
is
:
where is the Hankel contour
In mathematics, a Hankel contour is a path in the complex plane which extends from
(+∞,δ), around the origin counter clockwise and back to
(+∞,−δ), where δ is an arbitrarily small positive number. The contour thus remains arbitraril ...
, that is, the path encircling 0 in the positive direction, beginning at and returning to positive infinity with respect for the branch cut
In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that if the function is n-valued (has n values) at that point, a ...
along the positive real axis. According to Schmelzer & Trefethen, numerical evaluation of Hankel's integral is the basis of some of the best methods for computing the gamma function.
Integral representations at the positive integers
For positive integers , there is an integral for the reciprocal 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 given by
:
Similarly, for any real and we have the next integral for the reciprocal gamma function along the real axis in the form of :
:
where the particular case when provides a corresponding relation for the reciprocal double factorial
In mathematics, the double factorial or semifactorial of a number , denoted by , is the product of all the integers from 1 up to that have the same parity (odd or even) as . That is,
:n!! = \prod_^ (n-2k) = n (n-2) (n-4) \cdots.
For even , the ...
function,
Integral along the real axis
Integration of the reciprocal gamma function along the positive real axis gives the value
:
which is known as the Fransén–Robinson constant.
See also
* Bessel–Clifford function
* Inverse-gamma distribution
In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...
References
{{reflist, 25em
* Mette Lund
An integral for the reciprocal Gamma function
* Milton Abramowitz & Irene A. Stegun, ''''
* Eric W. Weisstein
Eric Wolfgang Weisstein (born March 18, 1969) is an American mathematician and encyclopedist who created and maintains the encyclopedias ''MathWorld'' and ''ScienceWorld''. In addition, he is the author of the '' CRC Concise Encyclopedia of M ...
,
Gamma Function
', MathWorld
Gamma and related functions
Analytic functions