Gamma function

The gamma function is a mathematical function that extends the domain of factorials to non-integers. The factorial of a positive integer n, written n!, is the product 1·2·3···n. The gamma function, denoted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma} , is defined to satisfy Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma(n) = (n-1)!} for all positive integers n and to smoothly interpolate the factorial between the integers. The gamma function is one of the most commonly occurring examples of a nonelementary function; that is, a function that cannot be expressed in finite terms using algebraic operations, exponentials, and logarithms. Its study dates back to Leonhard Euler, who gave a formula for its calculation in 1729.

Definition and basic properties

The product Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1\cdot2\cdot...\cdot n} cannot be expressed in closed form using only regular arithmetic operations and elementary functions. This is perhaps surprising, since the analogous sum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1+2+...+n} is simply Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n(n+1)/2} , a triangular number. To find a closed-form expression for factorials, it is necessary to deploy tools from calculus. In modern treatments, the gamma function is usually defined using Euler's integral,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma(z)=\int_{0}^{\infty}e^{-t}t^{z-1}dt.}

It is a slight technicality to prove that this integral defines a well-behaved function. The integral converges uniformly if z is a positive real number or a complex number with positive real part, and hence the gamma function is a continuous function in this domain. In fact, the integral defines an analytic function in the positive complex half-plane. The integral is only valid for positive z, but the gamma function can be defined for negative numbers as well using analytic continuation. We will give the continuation explicitly in a moment.

To see why we are justified in calling the gamma function as defined above an extension of the factorial, we insert z+1 instead of z and perform an integration by parts to obtain

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma(z+1)=\left[-e^{-t}t^{z}\right]_{0}^{\infty}+z\int_{0}^{\infty}e^{-t}t^{z-1}dt=z\Gamma(z).}

This relation is called the functional equation of the gamma function. It is analogous to the recurrence formula for the factorial, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n!=n(n-1)!} , the only difference being that the function argument has been shifted by 1. More generally, a repeated application of the functional equation gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma(z+n)=z(z+1)(z+2)\cdots(z+n-1)\Gamma(z).}

With the initial value Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textstyle \Gamma(1)=\int_{0}^{\infty}e^{-t}dt=1=0!} , this establishes that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma(n)=1\cdot2\cdot3\ldots(n-1)=(n-1)!}

for positive natural numbers n, or equivalently, that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma(n+1)=n!} . We may use these formulas to explicitly calculate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma(n)} , or conversely, to define z! for non-integers in terms of the gamma function. The functional equation allows us to extend our definition to the entire complex plane — including negative numbers — by rewriting the functional equation as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma(z)=\frac{\Gamma(z+n)}{z(z+1)\cdots(z+n-1)},}

which we might call the forward recurrence relation, and choosing n such that z+n lies in the positive half-plane. The right-hand side blows up for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=0,-1,-2,\ldots} , so the gamma function must be undefined at these points, but our extension defines an analytic function everywhere else. Hence, the gamma function is a meromorphic function with poles at the nonpositive integers. The following image shows the graph of the gamma function along the real line:

The gamma function is nonzero everywhere along the real line, although it grows arbitrarily close along the negative half-line. There is in fact no complex number z for which Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma(z) = 0} , and hence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/\Gamma} is an entire function. We see that the gamma function has a local minimum at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{\mathrm{min}}\approx1.46163} where it attains the value Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma(x_{\mathrm{min}})\approx0.885603} . The gamma function must alternate sign between the poles because the product in the forward recurrence contains an odd number of negative factors if the number of poles between z and z+n is odd, and an even number if the number of poles is even.

Plotting the gamma function in the complex plane yields beautiful graphs:

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  Absolute value

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  Real part

• 

  Imaginary part

A final basic but important property of the gamma function is the reflection formula

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin\pi z}}

which gives a concise relation between the gamma function of positive and negative numbers. The division by a sine, which is periodically zero, again indicates the existence of the gamma function's periodically occurring poles. Further, inserting z = 1/2 reveals the surprising fact that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma(1/2)=\sqrt{\pi}.}

Hence, by the recurrence formula, the gamma function or factorial of any half-integer is a rational multiple of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{\pi}} .

History, notation, and alternative definitions

The problem of extending the factorial to non-integer arguments was apparently first proposed by Daniel Bernoulli and Christian Goldbach in the 1720s. Leonhard Euler wrote to Goldbach on October 13, 1729 to announce that he had found a solution,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n! = \lim_{m\to\infty}\frac{m^{n}m!}{(n+1)(n+2)\cdots(n+m)}.}

The following year, on January 8, 1730, he again wrote to Goldbach to reveal that, for n > 0, the formula

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n!=\int_{0}^{1}(-\log t)^{n}\, dt}

is a solution. This is in fact simply Euler's integral with a change of variables.

The problem of finding a continuous expression for the factorial was also considered by James Stirling, who in Methodus Differentialis (1730) published the famous Stirling's formula

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n!\approx\sqrt{2\pi n}\left(\frac{n}{e}\right)^{n}}

Although Stirling's formula gives a good estimate of n!, also for non-integers, it does not provide the exact value. Stirling made several attempts to refine his approximation, and eventually found a solution, although he never managed to prove that the extended version of his formula indeed corresponds exactly to the factorial. A proof was first given by Charles Hermite in 1900.

In conclusion, there have been several independent attempts to characterize the gamma function. We used Euler's integral as a starting point in this article, but it is not entirely uncommon for authors to start with Euler's original (1729) representation. Are these approaches compatible? Euler himself does not appear to have proved that he had found a unique solution.

There are in fact infinitely many ways to extend the factorial, but the gamma function is the unique analytic solution, subject to the additional condition of being logarithmically convex for positive arguments. This fact was first proved by Harald Bohr and Johannes Mollerup in 1922 and is now known as the Bohr-Mollerup theorem.

The name gamma function and the symbol Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma} were introduced by Adrien-Marie Legendre around 1811. Although the symbol is an upper-case Greek Gamma, there is no accepted standard for whether the function name should be written "Gamma function" or "gamma function" (some authors simply write "Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma} -function"). The alternative "Pi function" notation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Pi(z)=z!} due to Carl Friedrich Gauss is sometimes encountered in older literature, but Legendre's notation is dominant in modern works.

It is justified to ask why we distinguish between the "ordinary factorial" and the gamma function by using distinct symbols, and particularly why the gamma function should be normalized to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma(n+1)=n!} instead of simply using "Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma(n)=n!} ". Legendre's motivation for the normalization does not appear to be known, and has been criticized as cumbersome by some mathematicians (Cornelius Lanczos, for example, called it "void of any rationality" and used Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z!} instead). The normalization does simplify some formulas, but complicates others.

Derivatives, products, and connection to Euler's constant

Many of the important special functions in mathematics, such as Bessel functions and orthogonal polynomials, are defined in terms of differential equations. The gamma function is characterized by functional equations such as the recurrence formula, but it is unusual among the classical special functions in that it does not appear to satisfy any simple differential equation. The term "simple" can be qualified: Otto Hölder proved in 1887 that the gamma function does not satisfy any algebraic differential equation whose coefficients are rational functions. This theorem, known as Hölder's theorem, has later been given alternative proofs and generalized by several other mathematicians.

Nonetheless, the derivatives of the gamma function are important special functions in their own right. The logarithmic derivatives of the gamma function are called polygamma functions (of order m), denoted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi^{m}} . They are given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\frac{d}{dz}\right)^{m} \log \Gamma(z) = \psi^{(m-1)}(z).}

In particular, the first derivative of the gamma function is given in terms of the 0-th order polygamma function (also called the digamma function) by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma'(z) = \Gamma(z) \, \psi^{(0)}(z)}

and the derivative at an integer is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma'(n+1) = n!\,\left(1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n} - \gamma\right) = \left|S_n^{(2)}\right| - n! \,\gamma}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma} is Euler's constant 0.57721566... and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_n^{(2)}} denotes a Stirling number of the first kind. As a special case, we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma'(1) = -\gamma} , which provides a geometric interpretation of Euler's constant as the slope of the gamma function's graph at 1.

Use in integration

Two frequently encountered generalizations of the gamma function are the incomplete gamma functions

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma(a,x) = \int_x^{\infty} t^{a-1}\,e^{-t}\,dt\,\!}

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma(a,x) = \int_0^x t^{a-1}\,e^{-t}\,dt.\,\!}

A variety of definite and indefinite integrals whose solutions are nonelementary can be expressed in terms of some combination of gamma functions and incomplete gamma functions. An example is that the Laplace transform of the power function x^c reduces to the Euler integral and hence the gamma function. Watson's triple integrals provide a more complicated example.

Integrands involving powers are particularly often susceptible to evaluation in terms of gamma functions, as it may be possible to consider the exponent an integer, eliminate it by repeated integration by parts to obtain a factorial times a solvable integral, and generalizing the result to arbitrary complex exponents by replacing the factorial with a gamma function. At other times, it is possible to directly transform an integral into an Euler integral by an appropriate change of variables.

Viewed as a non-elementary integral of an elementary function, the gamma function belongs to the same category of special functions as the error function, the exponential integral, the logarithmic integral, and the sine integral. There are many known interrelations between these functions and the gamma function.

The gamma function is also important in the theory of hypergeometric functions, the Meijer G-function and the Fox H-function. The G and H-functions are defined in terms of Mellin-Barnes integrals which are certain complex contour integrals of products of gamma functions.

Miscellaneous applications

An illustrative example of the consequences of being able to extend the factorial to non-integers is that of the fractional derivative. According to Cauchy's integral formula, the ordinary nth derivative of a function f is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{(n)}(a) = {n! \over 2\pi i} \oint_C {f(z) \over (z-a)^{n+1}}\, dz.}

Replacing n! with Γ(n+1), it becomes possible to calculate "derivatives" of fractional order. The gamma function is also used to express Bessel functions of fractional order and gives a general formula for area and volume of an n-dimensional hypersphere.

The gamma function can be used to express many types of products besides the ordinary factorial, such as multiple factorials, Pochhammer symbols and binomial coefficients. For a monic polynomial P with roots p₁...p_n, one has the general formula

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \prod_{j=a}^b P(j) = \prod_{k=1}^n \frac{\Gamma(b-p_k+1)}{\Gamma(a-p_k)},}

which in turn can be generalized to a formula for quotients of polynomials. Summation of series where each term is a rational function of the running index can be performed in an analogous manner using polygamma functions.

The gamma function is also used in statistics to define the gamma distribution, the inverse-gamma distribution, and the beta distribution.

Calculation

Although the gamma function cannot be expressed exactly in terms of elementary functions, the graphs and numerical values given in this article imply the existence of methods for its numerical calculation. One straightforward approach would be to perform numerical integration of Euler's integral, but there are more efficient methods. The most popular is an extended version of Stirling's formula known as Stirling's series

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma(z+1) \sim \sqrt{2\pi z}\left(\frac{z}{e}\right)^{z}\left(1+\frac{1}{12z}+\frac{1}{288z^{2}}-\cdots\right)}

where the coefficients in the series can be calculated analytically in terms of Bernoulli numbers. The "Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sim} " sign in this formula denotes an asymptotic equality: the series diverges for every z, but yields arbitrarily accurate approximations of the gamma function as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z\to\infty} if truncated appropriately. A practical way to calculate the gamma function is to calculate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma(z+n)} for some large integer n using Stirling's series and then repeatedly apply the recurrence formula to obtain Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma(z)} . Taking Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=10} and including the three first terms in Stirling's series, we obtain

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma(1) \approx 1.00000267}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma(2) \approx 1.00000201}

This particular approximation is good to five decimal places on the interval [1, 2]. With larger n and more terms, Stirling's formula can be used to calculate the gamma function with arbitrary precision. Other practical methods include the Lanczos approximation and Spouge's approximation.

In general, the following properties of the gamma function are useful to have in mind for numerical calculations:

• Due to the recurrence formula, the gamma function can be calculated anywhere in the complex plane if it can be calculated in some region with real part in $[x,x+1]$.
• If the gamma function can be calculated in the positive half-plane, the reflection formula permits its computation in the negative half-plane as well.
• It may be convenient to work with the logarithm of the gamma function to avoid overflow for large arguments. It is common to encounter a quotient of two gamma functions, which are best computed by subtracting logarithms.

Double-precision (16-digit) floating-point implementations of the gamma function are available in most scientific computing software and special functions libraries, for example Matlab, GNU Octave, and the GNU Scientific Library. The gamma function was also added to the C mathematics library (math.h) as part of the C99 standard, but is not implemented by all C compilers. Arbitrary-precision implementations are available in most computer algebra systems, such as Mathematica and Maple. Pari/GP and MPFR provide free arbitrary-precision implementations.