In mathematics, the gamma function, denoted by (capital Greek letter gamma), is the most common extension of the factorial function to complex numbers. First studied by Daniel Bernoulli, the gamma function is defined for all complex numbers except non-positive integers, and satisfies for every positive integer . The gamma function can be defined via a convergent improper integral for complex numbers with positive real part:
The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which is holomorphic except at zero and the negative integers, where it has simple poles.
Since the gamma function has no zeros, its reciprocal is an entire function. In fact, the gamma function corresponds to the Mellin transform of exponential decay:
Other extensions of the factorial function do exist, but the gamma function is the most popular and useful. It appears as a factor in various probability-distribution functions and other formulas in the fields of probability, statistics, analytic number theory, and combinatorics.
Motivation
The gamma function can be seen as a solution to the interpolation problem of finding a smooth curve that connects the points of the factorial sequence: for all positive integer values of . The simple formula for the factorial, is only valid when is a positive integer, and no elementary function has this property, but a good solution is the gamma function .
The gamma function is not only smooth but analytic (except at the non-positive integers), and it can be defined in several explicit ways. However, it is not the only analytic function that extends the factorial, as one may add any analytic function that is zero on the positive integers, such as for an integer . Such a function is known as a pseudogamma function, the most famous being the Hadamard function.
A more restrictive requirement is the functional equation that interpolates the shifted factorial : Expression G.2 on page 293
But this still does not give a unique solution, since it allows for multiplication by any periodic function with and , such as for any integer.
One way to resolve the ambiguity is the Bohr–Mollerup theorem, which shows that is the unique interpolating function for the factorial, defined over the positive reals, which is logarithmically convex, meaning that is convex.
Definition
Main definition The notation is due to Legendre. If the real part of the complex number is strictly positive (), then the integral converges absolutely, and is known as the Euler integral of the second kind. (Euler's integral of the first kind is the beta function.)
The relationship to the factorial can be shown by induction. The value can be calculated as
Integrating by parts, one sees that
Recognizing that when either (if ) or ,
Thus we have shown that for any positive integer by induction.
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 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 There are many equivalent definitions.
Euler's definition as an infinite product For a fixed integer , as the integer increases, we have that
If is not an integer then this equation is meaningless since, in this section, the factorial of a non-integer has not been defined yet. However, in order to define the Gamma function for non-integers, let us assume that this equation continues to hold when is replaced by an arbitrary complex number :
Multiplying both sides by gives
This infinite product, which is due to Euler, converges for all complex numbers except the non-positive integers, which fail because of a division by zero. In fact, the above assumption produces a unique definition of as .
Intuitively, this formula indicates that is approximately the result of computing for some large integer , multiplying by to approximate , and then using the relationship backwards times to get an approximation for , and furthermore that this approximation becomes exact as increases to infinity.
The infinite product for the reciprocal
is an entire function, converging for every complex number .
Weierstrass's definition The definition for the gamma function due to Weierstrass is also valid for all complex numbers except non-positive integers:
where is the Euler–Mascheroni constant. This is the Hadamard product of in a rewritten form.
Equivalence of the integral definition and Weierstrass definition
By the integral definition, the relation and Hadamard factorization theorem,
for some constants , since is an entire function of order . Since as , (or an integer multiple of ) and since ,
where for some integer . Since for , we have and
Equivalence of the Weierstrass definition and Euler definition
Properties
General
Besides the fundamental property discussed above,
Other important functional equations for the gamma function are Euler's reflection formula,
which implies
and the Legendre duplication formula
Proof 1
With Euler's infinite product compute
where the last equality is a known result. A similar derivation begins with Weierstrass's definition.
Proof 2
First prove that
Consider the positively oriented rectangular contour with vertices at , , and where . Then by the residue theorem,
Let
and let be the analogous integral over the top side of the rectangle. Then as and . If denotes the right vertical side of the rectangle, then
for some constant and since , the integral tends to as . Analogously, the integral over the left vertical side of the rectangle tends to as . Therefore
from which
Then
and
Proving the reflection formula for all proves it for all by analytic continuation.
The beta function can be represented as
Setting yields
After the substitution :
The function is even, hence
Now
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:
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 formula, by using the relation to the beta function given below with , or simply by making the substitution in the integral definition of the gamma function, resulting in a Gaussian integral. In general, for non-negative integer values of we have:
where the double factorial . See Particular values of the gamma function 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 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 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, :
For a positive integer the derivative of the gamma function can be calculated as follows:
where is the th harmonic number and is the Euler–Mascheroni constant.
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 .)
Using the identity
where is the Riemann zeta function, and is the th Bell polynomial, 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 ,
For any two positive real numbers and , and >
For any positive real number ,
The last of these statements is, essentially by definition, the same as the statement that , where is the polygamma function of order 1. To prove the logarithmic convexity of the gamma function, it therefore suffices to observe that has a series representation which, for positive real , consists of only positive terms.
Logarithmic convexity and Jensen's inequality together imply, for any positive real numbers and ,
There are also bounds on ratios of gamma functions. The best-known is Gautschi's inequality, which says that for any positive real number and any ,
Stirling's formula
The behavior of for an increasing positive real variable is given by Stirling's formula
where the symbol means asymptotic convergence: the ratio of the two sides converges to in the limit . This growth is faster than exponential, , for any fixed value of .
Another useful limit for asymptotic approximations for is:
When writing the error term as an infinite product, Stirling's formula can be used to define the gamma function:
Extension to negative, non-integer values Although the main definition of the gamma function—the Euler integral of the second kind—is only valid (on the real axis) for positive arguments, its domain can be extended with analytic continuation to negative arguments by shifting the negative argument to positive values by using either the Euler's reflection formula,
or the fundamental property,
when . For example,
Residues The behavior for non-positive is more intricate. Euler's integral does not converge for , but the function it defines in the positive complex half-plane has a unique analytic continuation to the negative half-plane. One way to find that analytic continuation is to use Euler's integral for positive arguments and extend the domain to negative numbers by repeated application of the recurrence formula,
choosing such that is positive. The product in the denominator is zero when equals any of the integers . Thus, the gamma function must be undefined at those points to avoid division by zero; it is a meromorphic function with simple poles at the non-positive integers.
For a function of a complex variable , at a simple pole , the residue of is given by:
For the simple pole , the recurrence formula can be rewritten as:
The numerator at , is
and the denominator
So the residues of the gamma function at those points are:
The gamma function is non-zero everywhere along the real line, although it comes arbitrarily close to zero as . There is in fact no complex number for which , and hence the reciprocal gamma function is an entire function, with zeros at .
Minima and maxima On the real line, the gamma function has a local minimum at where it attains the value . The gamma function rises to either side of this minimum. The solution to is and the common value is . The positive solution to is , the golden ratio, and the common value is .