In
mathematics, the gamma function (represented by , the capital letter
gamma from the
Greek alphabet
The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BCE. It is derived from the earlier Phoenician alphabet, and was the earliest known alphabetic script to have distinct letters for vowels as w ...
) is one commonly used extension of the
factorial function to
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 fo ...
s. The gamma function is defined for all complex numbers except the non-positive integers. For every
positive integer ,
Derived by
Daniel Bernoulli, for complex numbers with a positive real part, the gamma function is defined via a convergent
improper integral
In mathematical analysis, an improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number or positive or negative infinity; or in some instances as both endpoin ...
:
The gamma function then is defined as the
analytic continuation of this integral function to a
meromorphic function
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 ...
that is
holomorphic
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivati ...
in the whole complex plane except zero and the negative integers, where the function has simple
poles
Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in C ...
.
The gamma function has no zeroes, so the
reciprocal 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 ...
. In fact, the gamma function corresponds to the
Mellin transform In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is
often used ...
of the negative
exponential function:
Other extensions of the factorial function do exist, but the gamma function is the most popular and useful. It is a component in various probability-distribution functions, and as such it is applicable in the fields of
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speakin ...
and
statistics, as well as
combinatorics.
Motivation
The gamma function can be seen as a solution to the following
interpolation problem:
: "Find a
smooth curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
that connects the points given by at the positive integer values for ."
A plot of the first few factorials makes clear that such a curve can be drawn, but it would be preferable to have a formula that precisely describes the curve, in which the number of operations does not depend on the size of . The simple formula for the factorial, , cannot be used directly for non-integer values of since it is only valid when is a
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal ...
(or positive integer). There are, relatively speaking, no such simple solutions for factorials; no finite combination of sums, products, powers,
exponential functions, or
logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
s will suffice to express ; but it is possible to find a general formula for factorials using tools such as
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...
s and
limits
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
from
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
. A good solution to this is the gamma function.
There are infinitely many continuous extensions of the factorial to non-integers: infinitely many curves can be drawn through any set of isolated points. The gamma function is the most useful solution in practice, being
analytic (except at the non-positive integers), and it can be defined in several equivalent ways. However, it is not the only analytic function which extends the factorial, as adding to it any analytic function which is zero on the positive integers, such as for an integer , will give another function with that property.
Such a function is known as a
pseudogamma function, the most famous being the
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 ...
function.
A more restrictive property than satisfying the above interpolation is to satisfy the
recurrence relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
defining a translated version of the factorial function,
:
:
for any positive real number . But this would allow for multiplication by any function satisfying both for all real numbers and , such as the function . One of several ways to resolve the ambiguity comes from the
Bohr–Mollerup theorem
In mathematical analysis, the Bohr–Mollerup theorem is a theorem proved by the Danish mathematicians Harald Bohr and Johannes Mollerup. The theorem characterizes the gamma function, defined for by
:\Gamma(x)=\int_0^\infty t^ e^\,dt
as the '' ...
. It states that when the condition that be
logarithmically convex (or "super-convex,"
meaning that
is
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
) is added, it uniquely determines for positive, real inputs. From there, the gamma function can be extended to all real and complex values (except the negative integers and zero) by using the unique
analytic continuation of .
Definition
Main definition
The notation
is due to
Legendre.
If the real part of the complex number is strictly positive (
), then the
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...
:
converges absolutely
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is said ...
, and is known as the Euler integral of the second kind. (Euler's integral of the first kind is the
beta function
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral
: \Beta(z_1,z_2) = \int_0^1 t^( ...
.
) Using
integration by parts, one sees that:
:
Recognizing that
as
:
We can calculate
:
Given that
and
:
for all positive integers . This can be seen as an example of
proof by induction
Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help ...
.
The identity
can be used (or, yielding the same result,
analytic continuation can be used) to uniquely extend the integral formulation for
to a
meromorphic function
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 ...
defined for all complex numbers , except integers less than or equal to zero.
It is this extended version that is commonly referred to as the gamma function.
Alternative definitions
Euler's definition as an infinite product
When seeking to approximate
for a complex number
, it is effective to first compute
for some large integer
. Use that to approximate a value for
, and then use the recursion relation
backwards
times, to unwind it to an approximation for
. Furthermore, this approximation is exact in the limit as
goes to infinity.
Specifically, for a fixed integer
, it is the case that
:
If
is not an integer then it is not possible to say whether this equation is true because we have not yet (in this section) defined the factorial function for non-integers. However, we do get a unique extension of the factorial function to the non-integers by insisting that this equation continue to hold when the arbitrary integer
is replaced by an arbitrary complex number
.
:
Multiplying both sides by
gives
:
This
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 the partial products ''a''1''a''2...''a'n'' as ''n'' increases without bound. ...
converges for all complex numbers
except the negative integers, which fail because trying to use the recursion relation
backwards through the value
involves a division by zero.
Similarly for the gamma function, the definition as an infinite product due to
Euler is valid for all complex numbers
except the non-positive integers:
:
By this construction, the gamma function is the unique function that simultaneously satisfies
,
for all complex numbers
except the non-positive integers, and
for all complex numbers
.
Weierstrass's definition
The definition for the gamma function due to
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 ...
is also valid for all complex numbers except the non-positive integers:
:
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 ...
.
This is the
Hadamard product of
in a rewritten form. Indeed, since
is
entire of genus 1 with a simple zero at
, we have the product representation
:
where the product is over the zeros
of
. Since
has simple poles at the non-positive integers, it follows
has simple zeros at the nonpositive integers, and so the equation above becomes Weierstrass's formula with
in place of
. The derivation of the constants
and
is somewhat technical, but can be accomplished by using some identities involving the
Riemann zeta function (see
this identity, for instance). See also the
Weierstrass factorization theorem.
Properties
General
Other important functional equations for the gamma function are
Euler's reflection formula
In mathematics, a reflection formula or reflection relation for a function ''f'' is a relationship between ''f''(''a'' − ''x'') and ''f''(''x''). It is a special case of a functional equation, and it is very common in the literature ...
:
which implies
:
and the
Legendre duplication formula
:
Since
the gamma function can be represented as
:
Integrating by parts
times yields
:
which is equal to
:
This can be rewritten as
:
We can use this to evaluate the left-hand side of the reflection formula:
:
It can be
proved that
:
Then
:
Euler's reflection formula follows:
:
The
beta function
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral
: \Beta(z_1,z_2) = \int_0^1 t^( ...
can be represented as
:
Setting
yields
:
After the substitution
we get
:
The function
is even, hence
:
Now assume
:
Then
:
This implies
:
Since
:
the Legendre duplication formula follows:
:
The duplication formula is a special case of the
multiplication theorem (See,
Eq. 5.5.6)
:
A simple but useful property, which can be seen from the limit definition, is:
:
In particular, with , this product is
:
If the real part is an integer or a half-integer, this can be finitely expressed in
closed form:
:
First, consider the reflection formula applied to
.
:
Applying the recurrence relation to the second term, we have
:
which with simple rearrangement gives
:
Second, consider the reflection formula applied to
.
:
Formulas for other values of
for which the real part is integer or half-integer quickly follow by
induction using the recurrence relation in the positive and negative directions.
Perhaps the best-known value of the gamma function at a non-integer argument is
:
which can be found by setting
in the reflection or duplication formulas, by using the relation to the
beta function
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral
: \Beta(z_1,z_2) = \int_0^1 t^( ...
given below with
, or simply by making the substitution
in the integral definition of the gamma function, resulting in a
Gaussian integral
The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function f(x) = e^ over the entire real line. Named after the German mathematician Carl Friedrich Gauss, the integral is
\int_^\infty e^\,dx = \s ...
. In general, for non-negative integer values of
we have:
:
where the
double factorial . See
Particular values of the gamma function
The gamma function is an important special function in mathematics. Its particular values can be expressed in closed form for integer and half-integer arguments, but no simple expressions are known for the values at rational points in general. Othe ...
for calculated values.
It might be tempting to generalize the result that
by looking for a formula for other individual values
where
is rational, especially because according to
Gauss's digamma theorem, it is possible to do so for the closely related
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 strict ...
at every rational value. However, these numbers
are not known to be expressible by themselves in terms of elementary functions. It has been proved that
is a
transcendental number and
algebraically independent
In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non-trivial polynomial equation with coefficients in K.
In particular, a one element set \ is algebraically in ...
of
for any integer
and each of the fractions
. In general, when computing values of the gamma function, we must settle for numerical approximations.
The derivatives of the gamma function are described in terms of the
polygamma function
In mathematics, the polygamma function of order is a meromorphic function on the complex numbers \mathbb defined as the th derivative of the logarithm of the gamma function:
:\psi^(z) := \frac \psi(z) = \frac \ln\Gamma(z).
Thus
:\psi^(z) ...
, :
:
For a positive integer the derivative of the gamma function can be calculated as follows:
:
where H(m) is the mth
harmonic number and 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
th derivative of the gamma function is:
:
(This can be derived by differentiating the integral form of the gamma function with respect to
, and using the technique of
differentiation under the integral sign
In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form
\int_^ f(x,t)\,dt,
where -\infty < a(x), b(x) < \infty and the integral are .)
Using the identity
:
where
is the
Riemann zeta function, and
is a
partition of
given by
:
we have in particular the
Laurent series expansion of the gamma function
:
Inequalities
When restricted to the positive real numbers, the gamma function is a strictly
logarithmically convex function. This property may be stated in any of the following three equivalent ways:
* For any two positive real numbers
and
, and for any