Gamma function


In mathematics, the gamma function 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 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 the 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, 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 which interpolates the shifted factorial :
But this still does not give a unique solution, since it allows for multiplication by any periodic function with and, such as .
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 convex function|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.
The value can be calculated as
Integrating by parts, one sees thatRecognizing that as and as,
Thus we have shown that for any positive integer by induction.
The identity 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, 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 :
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:
Using the identitywhere 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:
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 formulawhere the symbol means asymptotic convergence: the ratio of the two sides converges to 1 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 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, isand the denominatorSo 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.
The gamma function must alternate sign between its poles at the non-positive integers because the product in the forward recurrence contains an odd number of negative factors if the number of poles between and is odd, and an even number if the number of poles is even. The values at the local extrema of the gamma function along the real axis between the non-positive integers are:

Integral representations

There are many formulas, besides the Euler integral of the second kind, that express the gamma function as an integral. For instance, when the real part of is positive,
and
where the three integrals respectively follow from the substitutions, and in Euler's second integral. The last integral in particular makes clear the connection between the gamma function at half integer arguments and the Gaussian integral: if we get
Binet's first integral formula for the gamma function states that, when the real part of is positive, then:The integral on the right-hand side may be interpreted as a Laplace transform. That is,
Binet's second integral formula states that, again when the real part of is positive, then:
Let be a Hankel contour, meaning a path that begins and ends at the point on the Riemann sphere, whose unit tangent vector converges to at the start of the path and to at the end, which has winding number 1 around, and which does not cross. Fix a branch of by taking a branch cut along and by taking to be real when is on the negative real axis. If is not an integer, then Hankel's formula for the gamma function is:where is interpreted as. The reflection formula leads to the closely related expressionwhich is valid whenever.

Continued fraction representation

The gamma function can also be represented by a sum of two continued fractions:
where.

Fourier series expansion

The logarithm of the gamma function has the following Fourier series expansion for
which was for a long time attributed to Ernst Kummer, who derived it in 1847. However, Iaroslav Blagouchine discovered that Carl Johan Malmsten first derived this series in 1842.

Raabe's formula

In 1840 Joseph Ludwig Raabe proved that In particular, if then
The latter can be derived taking the logarithm in the above multiplication formula, which gives an expression for the Riemann sum of the integrand. Taking the limit for gives the formula.

Pi function

An alternative notation introduced by Gauss is the -function, a shifted version of the gamma function:
so that for every non-negative integer.
Using the pi function, the reflection formula is:
using the normalized sinc function; while the multiplication theorem becomes:
The shifted reciprocal gamma function is sometimes denoted, an entire function.
The volume of an -ellipsoid with radii can be expressed as

Relation to other functions

Particular values

Including up to the first 20 digits after the decimal point, some particular values of the gamma function are:
The complex-valued gamma function is undefined for non-positive integers, but in these cases the value can be defined in the Riemann sphere as. The reciprocal gamma function is well defined and analytic at these values :

Log-gamma function

Because the gamma and factorial functions grow so rapidly for moderately large arguments, many computing environments include a function that returns the natural logarithm of the gamma function, often given the name lgamma or lngamma in programming environments or gammaln in spreadsheets. This grows much more slowly, and for combinatorial calculations allows adding and subtracting logarithmic values instead of multiplying and dividing very large values. It is often defined as
The digamma function, which is the derivative of this function, is also commonly seen.
In the context of technical and physical applications, e.g. with wave propagation, the functional equation
is often used since it allows one to determine function values in one strip of width 1 in from the neighbouring strip. In particular, starting with a good approximation for a with large real part one may go step by step down to the desired. Following an indication of Carl Friedrich Gauss, Rocktaeschel proposed for an approximation for large :
This can be used to accurately approximate for with a smaller via
A more accurate approximation can be obtained by using more terms from the asymptotic expansions of and, which are based on Stirling's approximation.
In a more "natural" presentation,
The coefficients of the terms with of in the last expansion are simplywhere the are the Bernoulli numbers.
The gamma function also has Stirling Series equal to

Properties

The Bohr–Mollerup theorem states that among all functions extending the factorial functions to the positive real numbers, only the gamma function is log-convex, that is, its natural logarithm is convex on the positive real axis. Another characterisation is given by the Wielandt theorem.
The gamma function is the unique function that simultaneously satisfies
  1. ,
  2. for all complex numbers except the non-positive integers, and,
  3. for integer, for all complex numbers.
In a certain sense, the log-gamma function is the more natural form; it makes some intrinsic attributes of the function clearer. A striking example is the Taylor series of around 1:
with denoting the Riemann zeta function at.
So, using the following property:
an integral representation for the log-gamma function is:
or, setting to obtain an integral for, we can replace the term with its integral and incorporate that into the above formula, to get:
There also exist special formulas for the logarithm of the gamma function for rational.
For instance, if and are integers with and, then
This formula is sometimes used for numerical computation, since the integrand decreases very quickly.

Integration over log-gamma

The integral
can be expressed in terms of the Barnes -function :
where.
It can also be written in terms of the Hurwitz zeta function:
When it follows that
and this is a consequence of Raabe's formula as well. O. Espinosa and V. Moll derived a similar formula for the integral of the square of :
where is.
D. H. Bailey and his co-authors gave an evaluation for
when in terms of the Tornheim–Witten zeta function and its derivatives.
In addition, it is also known that

Approximations

Complex values of the gamma function can be approximated using Stirling's approximation or the Lanczos approximation,This is precise in the sense that the ratio of the approximation to the true value approaches 1 in the limit as.
The gamma function can be computed to fixed precision for by applying integration by parts to Euler's integral. For any positive number the gamma function can be written
When and, the absolute value of the last integral is smaller than. By choosing a large enough, this last expression can be made smaller than for any desired value. Thus, the gamma function can be evaluated to bits of precision with the above series.
A fast algorithm for calculation of the Euler gamma function for any algebraic argument was constructed by E.A. Karatsuba.
For arguments that are integer multiples of, the gamma function can also be evaluated quickly using arithmetic–geometric mean iterations.

Practical implementations

Unlike many other functions, such as a Normal Distribution, no obvious fast, accurate implementation that is easy to implement for the Gamma Function is easily found. Therefore, it is worth investigating potential solutions. For the case that speed is more important than accuracy, published tables for are easily found in an Internet search, such as the Online Wiley Library. Such tables may be used with linear interpolation. Greater accuracy is obtainable with the use of cubic interpolation at the cost of more computational overhead. Since tables are usually published for argument values between 1 and 2, the property may be used to quickly and easily translate all real values and into the range, such that only tabulated values of between 1 and 2 need be used.
If interpolation tables are not desirable, then the Lanczos approximation mentioned above works well for 1 to 2 digits of accuracy for small, commonly used values of z. If the Lanczos approximation is not sufficiently accurate, the Stirling's formula for the Gamma Function may be used.

Applications

One author describes the gamma function as "Arguably, the most common special function, or the least 'special' of them. The other transcendental functions are called 'special' because you could conceivably avoid some of them by staying away from many specialized mathematical topics. On the other hand, the gamma function is most difficult to avoid."

Integration problems

The gamma function finds application in such diverse areas as quantum physics, astrophysics and fluid dynamics. The gamma distribution, which is formulated in terms of the gamma function, is used in statistics to model a wide range of processes; for example, the time between occurrences of earthquakes.
The primary reason for the gamma function's usefulness in such contexts is the prevalence of expressions of the type which describe processes that decay exponentially in time or space. Integrals of such expressions can occasionally be solved in terms of the gamma function when no elementary solution exists. For example, if is a power function and is a linear function, a simple change of variables gives the evaluation
The fact that the integration is performed along the entire positive real line might signify that the gamma function describes the cumulation of a time-dependent process that continues indefinitely, or the value might be the total of a distribution in an infinite space.
It is of course frequently useful to take limits of integration other than and to describe the cumulation of a finite process, in which case the ordinary gamma function is no longer a solution; the solution is then called an incomplete gamma function.
An important category of exponentially decaying functions is that of Gaussian functions
and integrals thereof, such as the error function. There are many interrelations between these functions and the gamma function; notably, the factor obtained by evaluating is the "same" as that found in the normalizing factor of the error function and the normal distribution.
The integrals discussed so far involve transcendental functions, but the gamma function also arises from integrals of purely algebraic functions. In particular, the arc lengths of ellipses and of the lemniscate, which are curves defined by algebraic equations, are given by elliptic integrals that in special cases can be evaluated in terms of the gamma function. The gamma function can also be used to calculate "volume" and "area" of -dimensional hyperspheres.

Calculating products

The gamma function's ability to generalize factorial products immediately leads to applications in many areas of mathematics; in combinatorics, and by extension in areas such as probability theory and the calculation of power series. Many expressions involving products of successive integers can be written as some combination of factorials, the most important example perhaps being that of the binomial coefficient. For example, for any complex numbers and, with, we can writewhich closely resembles the binomial coefficient when is a non-negative integer,
The example of binomial coefficients motivates why the properties of the gamma function when extended to negative numbers are natural. A binomial coefficient gives the number of ways to choose elements from a set of elements; if, there are of course no ways. If, then is the factorial of a negative integer and hence infinite if we use the gamma function definition of factorials—dividing by infinity gives the expected value of 0.
We can replace the factorial by a gamma function to extend any such formula to the complex numbers. Generally, this works for any product wherein each factor is a rational function of the index variable, by factoring the rational function into linear expressions. If and are monic polynomials of degree and with respective roots and , we have
If we have a way to calculate the gamma function numerically, it is very simple to calculate numerical values of such products. The number of gamma functions in the right-hand side depends only on the degree of the polynomials, so it does not matter whether equals 5 or 105. By taking the appropriate limits, the equation can also be made to hold even when the left-hand product contains zeros or poles.
By taking limits, certain rational products with infinitely many factors can be evaluated in terms of the gamma function as well. Due to the Weierstrass factorization theorem, analytic functions can be written as infinite products, and these can sometimes be represented as finite products or quotients of the gamma function. We have already seen one striking example: the reflection formula essentially represents the sine function as the product of two gamma functions. Starting from this formula, the exponential function as well as all the trigonometric and hyperbolic functions can be expressed in terms of the gamma function.
More functions yet, including the hypergeometric function and special cases thereof, can be represented by means of complex contour integrals of products and quotients of the gamma function, called Mellin–Barnes integrals.

Analytic number theory

An application of the gamma function is the study of the Riemann zeta function. A fundamental property of the Riemann zeta function is its functional equation:
Among other things, this provides an explicit form for the analytic continuation of the zeta function to a meromorphic function in the complex plane and leads to an immediate proof that the zeta function has infinitely many so-called "trivial" zeros on the real line. Another powerful formula is
Both formulas were derived by Bernhard Riemann in his seminal 1859 paper "Ueber die Anzahl der Primzahlen unter einer gegebenen Größe", one of the milestones in the development of analytic number theory—the branch of mathematics that studies prime numbers using the tools of mathematical analysis.

History

The gamma function has caught the interest of some of the most prominent mathematicians of all time. Its history, notably documented by Philip J. Davis in an article that won him the 1963 Chauvenet Prize, reflects many of the major developments within mathematics since the 18th century. In the words of Davis, "each generation has found something of interest to say about the gamma function. Perhaps the next generation will also."

18th century: Euler and Stirling

The problem of extending the factorial to non-integer arguments was apparently first considered by Daniel Bernoulli and Christian Goldbach in the 1720s. In particular, in a letter from Bernoulli to Goldbach dated 6 October 1729 Bernoulli introduced the product representationwhich is well defined for real values of other than the negative integers.
Leonhard Euler later gave two different definitions: the first was not his integral but an infinite product that is well defined for all complex numbers other than the negative integers,of which he informed Goldbach in a letter dated 13 October 1729. He wrote to Goldbach again on 8 January 1730, to announce his discovery of the integral representation
which is valid when the real part of the complex number is strictly greater than . By the change of variables, this becomes the familiar Euler integral. Euler published his results in the paper "De progressionibus transcendentibus seu quarum termini generales algebraice dari nequeunt", submitted to the St. Petersburg Academy on 28 November 1729. Euler further discovered some of the gamma function's important functional properties, including the reflection formula.
James Stirling, a contemporary of Euler, also attempted to find a continuous expression for the factorial and came up with what is now known as Stirling's formula. Although Stirling's formula gives a good estimate of, also for non-integers, it does not provide the exact value. Extensions of his formula that correct the error were given by Stirling himself and by Jacques Philippe Marie Binet.

19th century: Gauss, Weierstrass, and Legendre

rewrote Euler's product asand used this formula to discover new properties of the gamma function. Although Euler was a pioneer in the theory of complex variables, he does not appear to have considered the factorial of a complex number, as instead Gauss first did. Gauss also proved the multiplication theorem of the gamma function and investigated the connection between the gamma function and elliptic integrals.
Karl Weierstrass further established the role of the gamma function in complex analysis, starting from yet another product representation,where is the Euler–Mascheroni constant. Weierstrass originally wrote his product as one for, in which case it is taken over the function's zeros rather than its poles. Inspired by this result, he proved what is known as the Weierstrass factorization theorem—that any entire function can be written as a product over its zeros in the complex plane; a generalization of the fundamental theorem of algebra.
The name gamma function and the symbol were introduced by Adrien-Marie Legendre around 1811; Legendre also rewrote Euler's integral definition in its modern form. 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". The alternative "pi function" notation due to 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 instead of simply using. Consider that the notation for exponents, , has been generalized from integers to complex numbers without any change. Legendre's motivation for the normalization is not known, and has been criticized as cumbersome by some. Legendre's normalization does simplify some formulae, but complicates others. From a modern point of view, the Legendre normalization of the gamma function is the integral of the additive character against the multiplicative character with respect to the Haar measure on the Lie group. Thus this normalization makes it clearer that the gamma function is a continuous analogue of a Gauss sum.

19th–20th centuries: characterizing the gamma function

It is somewhat problematic that a large number of definitions have been given for the gamma function. Although they describe the same function, it is not entirely straightforward to prove the equivalence. Stirling never proved that his extended formula corresponds exactly to Euler's gamma function; a proof was first given by Charles Hermite in 1900. Instead of finding a specialized proof for each formula, it would be desirable to have a general method of identifying the gamma function.
One way to prove equivalence would be to find a differential equation that characterizes the gamma function. Most special functions in applied mathematics arise as solutions to differential equations, whose solutions are unique. However, the gamma function does not appear to satisfy any simple differential equation. Otto Hölder proved in 1887 that the gamma function at least does not satisfy any algebraic differential equation by showing that a solution to such an equation could not satisfy the gamma function's recurrence formula, making it a transcendentally transcendental function. This result is known as Hölder's theorem.
A definite and generally applicable characterization of the gamma function was not given until 1922. Harald Bohr and Johannes Mollerup then proved what is known as the Bohr–Mollerup theorem: that the gamma function is the unique solution to the factorial recurrence relation that is positive and logarithmically convex for positive and whose value at 1 is 1. Another characterisation is given by the Wielandt theorem.
The Bohr–Mollerup theorem is useful because it is relatively easy to prove logarithmic convexity for any of the different formulas used to define the gamma function. Taking things further, instead of defining the gamma function by any particular formula, we can choose the conditions of the Bohr–Mollerup theorem as the definition, and then pick any formula we like that satisfies the conditions as a starting point for studying the gamma function. This approach was used by the Bourbaki group.
Borwein & Corless review three centuries of work on the gamma function.