Gamma function explained

Let$$\backslash Gamma\_n(z)=\backslash frac$$and$$G\_n(z)=\backslash frac.$$Then$$\backslash Gamma\_n(z)=\backslash fracG\_n(z)$$and$$\backslash lim\_G\_(z)=\backslash lim\_G\_n(z)=\backslash lim\_\backslash Gamma\_n(z)=\backslash Gamma\; (z),$$therefore$$\backslash Gamma\; (z)=\backslash lim\_\backslash frac,\backslash quad\; z\backslash in\backslash mathbb\backslash setminus\backslash mathbb\_0^-.$$Then$$\backslash frac=\backslash frac=\backslash frac\backslash prod\_^n\; \backslash frac,\backslash quad\; z\backslash in\backslash mathbb\backslash setminus\; \backslash mathbb\_0^-$$and taking

Properties

General

Besides the fundamental property discussed above:$$\backslash Gamma(z+1)\; =\; z\backslash \; \backslash Gamma(z)$$other important functional equations for the gamma function are Euler's reflection formula$$\backslash Gamma(1-z)\; \backslash Gamma(z)\; =\; \backslash frac,\; \backslash qquad\; z\; \backslash not\backslash in\; \backslash Z$$which implies$$\backslash Gamma(z\; -\; n)\; =\; (-1)^\; \backslash ;\; \backslash frac,\; \backslash qquad\; n\; \backslash in\; \backslash Z$$and the Legendre duplication formula$$\backslash Gamma(z)\; \backslash Gamma\backslash left(z\; +\; \backslash tfrac12\backslash right)\; =\; 2^\; \backslash ;\; \backslash sqrt\; \backslash ;\; \backslash Gamma(2z).$$

Proof 1

We can use Euler's infinite product$$\backslash Gamma(z)\; =\; \backslash frac1z\; \backslash prod\_^\; \backslash frac$$to compute$$\backslash frac=\; \backslash frac=\; z\; \backslash prod\_^\; \backslash frac=\; z\; \backslash prod\_^\; \backslash left(1\; -\; \backslash frac\backslash right)=\; \backslash frac\backslash ,,$$where the last equality is a known result. A similar derivation begins with Weierstrass's definition.

Proof 2

First we prove that$$I=\backslash int\_^\backslash infty\; \backslash frac\backslash ,\; dx=\backslash int\_0^\backslash infty\; \backslash frac\backslash ,\; dv=\backslash frac,\backslash quad\; a\backslash in\; (0,1).$$Consider the positively oriented rectangular contour

The beta function can be represented as$$\backslash Beta\; (z\_1,z\_2)=\backslash frac=\backslash int\_0^1\; t^(1-t)^\; \backslash ,\; dt.$$

Setting

After the substitution

The function

Now assume$$\backslash Beta\; \backslash left(\backslash frac,z\backslash right)=\backslash int\_0^1\; t^(1-t)^\; \backslash ,\; dt,\; \backslash quad\; t=s^2.$$

Then$$\backslash Beta\; \backslash left(\backslash frac,z\backslash right)=2\backslash int\_0^1\; (1-s^2)^\; \backslash ,\; ds\; =\; 2\backslash int\_0^1\; (1-u^2)^\; \backslash ,\; du.$$

This implies$$2^\backslash Gamma^2(z)=\backslash Gamma\; (2z)\backslash Beta\; \backslash left(\backslash frac,z\backslash right).$$

Since$$\backslash Beta\; \backslash left(\backslash frac,z\backslash right)=\backslash frac,\; \backslash quad\; \backslash Gamma\; \backslash left(\backslash frac\backslash right)=\backslash sqrt,$$the Legendre duplication formula follows:$$\backslash Gamma\; (z)\backslash Gamma\; \backslash left(z+\backslash frac\backslash right)=2^\backslash sqrt\; \backslash ;\; \backslash Gamma\; (2z).$$

The duplication formula is a special case of the multiplication theorem (see  Eq. 5.5.6):$$\backslash prod\_^\backslash Gamma\backslash left(z\; +\; \backslash frac\backslash right)\; =\; (2\; \backslash pi)^\; \backslash ;\; m^\; \backslash ;\; \backslash Gamma(mz).$$

A simple but useful property, which can be seen from the limit definition, is:$$\backslash overline\; =\; \backslash Gamma(\backslash overline)\; \backslash ;\; \backslash Rightarrow\; \backslash ;\; \backslash Gamma(z)\backslash Gamma(\backslash overline)\; \backslash in\; \backslash mathbb\; .$$

In particular, with, this product is$$|\backslash Gamma(a+bi)|^2\; =\; |\backslash Gamma(a)|^2\; \backslash prod\_^\backslash infty\; \backslash frac$$

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

Second, consider the reflection formula applied to

Formulas for other values of

Perhaps the best-known value of the gamma function at a non-integer argument is$$\backslash Gamma\backslash left(\backslash tfrac12\backslash right)=\backslash sqrt,$$which can be found by setting $z\; =\; \backslash frac$ in the reflection or duplication formulas, by using the relation to the beta function given below with $z\_1\; =\; z\_2\; =\; \backslash frac$, or simply by making the substitution

It might be tempting to generalize the result that $\backslash Gamma\; \backslash left(\backslash frac\; \backslash right)\; =\; \backslash sqrt\backslash pi$ by looking for a formula for other individual values

The derivatives of the gamma function are described in terms of the polygamma function, :$$\backslash Gamma\text{'}(z)=\backslash Gamma(z)\backslash psi^(z).$$For a positive integer  the derivative of the gamma function can be calculated as follows:$$\backslash Gamma\text{'}(m+1)\; =\; m!\; \backslash left(-\; \backslash gamma\; +\; \backslash sum\_^m\backslash frac\; \backslash right)=\; m!\; \backslash left(-\; \backslash gamma\; +\; H(m)\; \backslash right)\backslash ,,$$where H(m) is the mth harmonic number and is the Euler–Mascheroni constant.

For

Using the identity$$\backslash Gamma^(1)=(-1)^n\; B\_n(\backslash gamma,\; 1!\; \backslash zeta(2),\; \backslash ldots,\; (n-1)!\; \backslash zeta(n))$$where

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

• For any two positive real numbers

• For any positive real number

Logarithmic convexity and Jensen's inequality together imply, for any positive real numbers

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

See main article: Stirling's approximation.

The behavior of

Another useful limit for asymptotic approximations for

When writing the error term as an infinite product, Stirling's formula can be used to define the gamma function: ^[9] $$\backslash Gamma\backslash left(x\backslash right)=\backslash sqrt\backslash left(\backslash frac\; \backslash right)^\backslash prod\_^e^\backslash left(1+\backslash frac\backslash right)^$$

Residues

The behavior for non-positive

For a function

For the simple pole

Minima and maxima

On the real line, the gamma function has a local minimum at ^[10] where it attains the value .^[11] 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

,

,

,

,

, etc.

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,^[12] $$\backslash Gamma\; (z)=\backslash int\_^\backslash infty\; e^\backslash ,\; dt$$and^[13] $$\backslash Gamma(z)\; =\; \backslash int\_0^1\; \backslash left(\backslash log\; \backslash frac\backslash right)^\backslash ,dt,$$$$\backslash Gamma(z)\; =\; 2\backslash int\_^t^e^\backslash ,dt$$where the three integrals respectively follow from the substitutions

Binet's first integral formula for the gamma function states that, when the real part of is positive, then:^[16] $$\backslash operatorname(z)\; =\; \backslash left(z\; -\; \backslash frac\backslash right)\backslash log\; z\; -\; z\; +\; \backslash frac\backslash log\; (2\backslash pi)\; +\; \backslash int\_0^\backslash infty\; \backslash left(\backslash frac\; -\; \backslash frac\; +\; \backslash frac\backslash right)\backslash frac\backslash ,dt.$$The integral on the right-hand side may be interpreted as a Laplace transform. That is,$$\backslash log\backslash left(\backslash Gamma(z)\backslash left(\backslash frac\backslash right)^z\backslash sqrt\backslash right)\; =\; \backslash mathcal\backslash left(\backslash frac\; -\; \backslash frac\; +\; \backslash frac\backslash right)(z).$$

Binet's second integral formula states that, again when the real part of is positive, then:^[17] $$\backslash operatorname(z)\; =\; \backslash left(z\; -\; \backslash frac\backslash right)\backslash log\; z\; -\; z\; +\; \backslash frac\backslash log(2\backslash pi)\; +\; 2\backslash int\_0^\backslash infty\; \backslash frac\backslash ,dt.$$

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

Continued fraction representation

The gamma function can also be represented by a sum of two continued fractions:^{[19] [20]} where

Fourier series expansion

The logarithm of the gamma function has the following Fourier series expansion for

Raabe's formula

In 1840 Joseph Ludwig Raabe proved that$$\backslash int\_a^\backslash operatorname(z)\backslash ,\; dz\; =\; \backslash tfrac12\backslash log2\backslash pi\; +\; a\backslash log\; a\; -\; a,\backslash quad\; a>0.$$In particular, if

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

Pi function

An alternative notation that was originally introduced by Gauss is the

Using the pi function the reflection formula takes on the form$$\backslash Pi(z)\; \backslash Pi(-z)\; =\; \backslash frac\; =\; \backslash frac$$where is the normalized sinc function, while the multiplication theorem takes on the form$$\backslash Pi\backslash left(\backslash frac\backslash right)\; \backslash ,\; \backslash Pi\backslash left(\backslash frac\backslash right)\; \backslash cdots\; \backslash Pi\backslash left(\backslash frac\backslash right)\; =\; (2\; \backslash pi)^\; m^\; \backslash Pi(z)\backslash \; .$$

We also sometimes find$$\backslash pi(z)\; =\; \backslash frac\backslash ,$$which is an entire function, defined for every complex number, just like the reciprocal gamma function. That

The volume of an -ellipsoid with radii can be expressed as$$V\_n(r\_1,\backslash dotsc,r\_n)=\backslash frac\; \backslash prod\_^n\; r\_k.$$

Relation to other functions

• In the first integral above, which defines the gamma function, the limits of integration are fixed. The upper and lower incomplete gamma functions are the functions obtained by allowing the lower or upper (respectively) limit of integration to vary.
• The gamma function is related to the beta function by the formula $$\backslash Beta(z\_1,z\_2)\; =\; \backslash int\_0^1\; t^(1-t)^\backslash ,dt\; =\; \backslash frac.$$
• The logarithmic derivative of the gamma function is called the digamma function; higher derivatives are the polygamma functions.
• The analog of the gamma function over a finite field or a finite ring is the Gaussian sums, a type of exponential sum.
• The reciprocal gamma function is an entire function and has been studied as a specific topic.
• The gamma function also shows up in an important relation with the Riemann zeta function,

• The gamma function is related to the stretched exponential function. For instance, the moments of that function are $$\backslash langle\backslash tau^n\backslash rangle\; \backslash equiv\; \backslash int\_0^\backslash infty\; t^\backslash ,\; e^\; \backslash ,\; \backslash mathrmt\; =\; \backslash frac\backslash Gamma\; \backslash left(\backslash right).$$

Particular values

See main article: Particular values of the gamma function. Including up to the first 20 digits after the decimal point, some particular values of the gamma function are:(These numbers can be found in the OEIS. The values presented here are truncated rather than rounded.)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 (and in the entire complex plane):$$\backslash frac\; =\; \backslash frac\; =\; \backslash frac\; =\; \backslash frac\; =\; 0.$$

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 logs instead of multiplying and dividing very large values. It is often defined as^[25] $$\backslash operatorname(z)\; =\; -\; \backslash gamma\; z\; -\; \backslash log\; z\; +\; \backslash sum\_^\backslash infty\; \backslash left[\backslash frac\; z\; k\; -\; \backslash log\; \backslash left(1\; +\; \backslash frac\; z\; k\; \backslash right)\; \backslash right].$$

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$$\backslash operatorname(z)\; =\; \backslash operatorname(z+1)\; -\; \backslash log\; z$$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 (1922) proposed for an approximation for large :$$\backslash operatorname(z)\; \backslash approx\; (z\; -\; \backslash tfrac)\; \backslash log\; z\; -\; z\; +\; \backslash tfrac\backslash log(2\backslash pi).$$

This can be used to accurately approximate for with a smaller via (P.E.Böhmer, 1939)$$\backslash operatorname(z-m)\; =\; \backslash operatorname(z)\; -\; \backslash sum\_^m\; \backslash log(z-k).$$

A more accurate approximation can be obtained by using more terms from the asymptotic expansions of and, which are based on Stirling's approximation.$$\backslash Gamma(z)\backslash sim\; z^\; e^\; \backslash sqrt\; \backslash left(1\; +\; \backslash frac\; +\; \backslash frac\; -\; \backslash frac\; -\; \backslash frac\; \backslash right)$$

as at constant . (See sequences and in the OEIS.)

In a more "natural" presentation:$$\backslash operatorname(z)\; =\; z\; \backslash log\; z\; -\; z\; -\; \backslash tfrac12\; \backslash log\; z\; +\; \backslash tfrac12\; \backslash log\; 2\backslash pi\; +\; \backslash frac\; -\; \backslash frac\; +\backslash frac\; +o\backslash left(\backslash frac1\backslash right)$$

as at constant . (See sequences and in the OEIS.)

The coefficients of the terms with of in the last expansion are simply$$\backslash frac$$where the are the Bernoulli numbers.

The gamma function also has Stirling Series (derived by Charles Hermite in 1900) equal to^[26] $$\backslash operatorname(1+x)=\backslash frac\; \backslash log(2)+\backslash frac\; (\backslash log(3)-2\backslash log(2))+\backslash cdots,\backslash quad\backslash Re\; (x)>\; 0.$$

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. for integer, $\backslash lim\_\; \backslash frac\; =\; 1$ 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:$$\backslash zeta(s)\; \backslash Gamma(s)\; =\; \backslash int\_0^\backslash infty\; \backslash frac\; \backslash ,\; \backslash frac$$we can find an integral representation for the log-gamma function:$$\backslash operatorname(z+1)=\; -\backslash gamma\; z\; +\; \backslash int\_0^\backslash infty\; \backslash frac\; \backslash ,\; dt$$or, setting to obtain an integral for, we can replace the term with its integral and incorporate that into the above formula, to get:$$\backslash operatorname(z+1)=\; \backslash int\_0^\backslash infty\; \backslash frac\; \backslash ,\; dt\backslash ,.$$

There also exist special formulas for the logarithm of the gamma function for rational . For instance, if

Integration over log-gamma

The integral$$\backslash int\_0^z\; \backslash operatorname\; (x)\; \backslash ,\; dx$$can be expressed in terms of the Barnes -function^{[28] [29]} (see Barnes -function for a proof):$$\backslash int\_0^z\; \backslash operatorname(x)\; \backslash ,\; dx\; =\; \backslash frac\; \backslash log\; (2\; \backslash pi)\; +\; \backslash frac\; +\; z\; \backslash operatorname(z)\; -\; \backslash log\; G(z+1)$$where .

It can also be written in terms of the Hurwitz zeta function:^{[30] [31]} $$\backslash int\_0^z\; \backslash operatorname(x)\; \backslash ,\; dx\; =\; \backslash frac\; \backslash log(2\; \backslash pi)\; +\; \backslash frac\; -\; \backslash zeta\text{'}(-1)\; +\; \backslash zeta\text{'}(-1,z)\; .$$

When

D. H. Bailey and his co-authors^[33] gave an evaluation for$$L\_n:=\backslash int\_0^1\; \backslash log^n\; \backslash Gamma(x)\; \backslash ,\; dx$$when

In addition, it is also known that^[34] $$\backslash lim\_\; \backslash frac=1.$$

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 goes to infinity.

The gamma function can be computed to fixed precision for

When and

A fast algorithm for calculation of the Euler gamma function for any algebraic argument (including rational) was constructed by E.A. Karatsuba.^{[35] [36] [37]}

For arguments that are integer multiples of, the gamma function can also be evaluated quickly using arithmetic–geometric mean iterations (see particular values of the gamma function).^[38]

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

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."^[40]

Integration problems

The gamma function finds application in such diverse areas as quantum physics, astrophysics and fluid dynamics.^[41] 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.^[42]

The primary reason for the gamma function's usefulness in such contexts is the prevalence of expressions of the type

$$\backslash int\_0^\backslash infty\; t^b\; e^\; \backslash ,dt\; =\; \backslash frac\; \backslash int\_0^\backslash infty\; u^b\; e^\; d\backslash left(\backslash frac\backslash right)\; =\; \backslash frac.$$

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 0 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. (The ordinary gamma function, obtained by integrating across the entire positive real line, is sometimes called the complete gamma function for contrast.)

An important category of exponentially decaying functions is that of Gaussian functions$$ae^$$and integrals thereof, such as the error function. There are many interrelations between these functions and the gamma function; notably, the factor

The integrals we have 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 write$$(1\; +\; z)^n\; =\; \backslash sum\_^\backslash infty\; \backslash frac\; z^k,$$which closely resembles the binomial coefficient when is a non-negative integer,$$(1\; +\; z)^n\; =\; \backslash sum\_^n\; \backslash frac\; z^k\; =\; \backslash sum\_^n\; \backslash binom\; z^k.$$

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, 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$$\backslash prod\_^b\; \backslash frac\; =\; \backslash left(\backslash prod\_^m\; \backslash frac\; \backslash right)\; \backslash left(\backslash prod\_^n\; \backslash frac\; \backslash right).$$

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:$$\backslash Gamma\backslash left(\backslash frac\backslash right)\backslash zeta(s)\backslash pi^\; =\; \backslash Gamma\backslash left(\backslash frac\backslash right)\backslash zeta(1-s)\backslash pi^.$$

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. Borwein et al. call this formula "one of the most beautiful findings in mathematics".^[43] Another contender for that title might be$$\backslash zeta(s)\; \backslash ;\; \backslash Gamma(s)\; =\; \backslash int\_0^\backslash infty\; \backslash frac\; \backslash ,\; \backslash frac.$$

Both formulas were derived by Bernhard Riemann in his seminal 1859 paper "Ueber die Anzahl der Primzahlen unter einer gegebenen Größe" ("On the Number of Primes Less Than a Given Magnitude"), 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."^[44]

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 representation^[45] $$x!=\backslash lim\_\backslash left(n+1+\backslash frac\backslash right)^\backslash prod\_^\backslash frac$$which 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,$$n!\; =\; \backslash prod\_^\backslash infty\; \backslash frac\backslash ,,$$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$$n!=\backslash int\_0^1\; (-\backslash log\; s)^n\backslash ,\; ds\backslash ,,$$which is valid when the real part of the complex number is strictly greater than (i.e.,

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

Carl Friedrich Gauss rewrote Euler's product as$$\backslash Gamma(z)\; =\; \backslash lim\_\backslash frac$$and 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.^[47] 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,$$\backslash Gamma(z)\; =\; \backslash frac\; \backslash prod\_^\backslash infty\; \backslash left(1\; +\; \backslash frac\backslash right)^\; e^\backslash frac,$$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" (some authors simply write "-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 does not appear to be known, and has been criticized as cumbersome by some (the 20th-century mathematician Cornelius Lanczos, for example, called it "void of any rationality" and would instead use).^[48] 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 $\backslash frac$ on the Lie group . Thus this normalization makes it clearer that the gamma function is a continuous analogue of a Gauss sum.^[49]

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.^[50] 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 (a function is logarithmically convex if its logarithm is convex). 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^[51] review three centuries of work on the gamma function.

Reference tables and software

Although the gamma function can be calculated virtually as easily as any mathematically simpler function with a modern computer—even with a programmable pocket calculator—this was of course not always the case. Until the mid-20th century, mathematicians relied on hand-made tables; in the case of the gamma function, notably a table computed by Gauss in 1813 and one computed by Legendre in 1825.^[52]

Tables of complex values of the gamma function, as well as hand-drawn graphs, were given in Tables of Functions With Formulas and Curves by Jahnke and, first published in Germany in 1909. According to Michael Berry, "the publication in J&E of a three-dimensional graph showing the poles of the gamma function in the complex plane acquired an almost iconic status."^[53]

There was in fact little practical need for anything but real values of the gamma function until the 1930s, when applications for the complex gamma function were discovered in theoretical physics. As electronic computers became available for the production of tables in the 1950s, several extensive tables for the complex gamma function were published to meet the demand, including a table accurate to 12 decimal places from the U.S. National Bureau of Standards.Double-precision floating-point implementations of the gamma function and its logarithm are now available in most scientific computing software and special functions libraries, for example TK Solver, Matlab, GNU Octave, and the GNU Scientific Library. The gamma function was also added to the C standard library (math.h). Arbitrary-precision implementations are available in most computer algebra systems, such as Mathematica and Maple. PARI/GP, MPFR and MPFUN contain free arbitrary-precision implementations. In some software calculators, e.g. Windows Calculator and GNOME Calculator, the factorial function returns Γ(x + 1) when the input x is a non-integer value.^{[54] [55]}

See also

• Ascending factorial
• Cahen–Mellin integral
• Elliptic gamma function
• Gauss's constant
• Pseudogamma function
• Hadamard's gamma function
• Inverse gamma function
• Lanczos approximation
• Multiple gamma function
• Multivariate gamma function
• -adic gamma function
• Pochhammer -symbol
• -gamma function
• Ramanujan's master theorem
• Spouge's approximation
• Stirling's approximation

Further reading

• Book: Milton . Abramowitz . Irene A. . Stegun . Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables . New York . Dover . 1972 . http://www.math.sfu.ca/~cbm/aands/page_253.htm . Chapter 6. Abramowitz and Stegun .
• Book: G. E. . Andrews . R. . Askey . Richard Askey . R. . Roy . Special Functions . Cambridge University Press . New York . 1999 . 978-0-521-78988-2 . Chapter 1 (Gamma and Beta functions).
• Book: Artin, Emil . Emil Artin . The Gamma Function . Rosen . Michael . Exposition by Emil Artin: a selection . History of Mathematics . 30 . Providence, RI . American Mathematical Society . 2006.
  • Birkhoff . George D. . George David Birkhoff . Note on the gamma function . Bull. Amer. Math. Soc. . 1913 . 20 . 1 . 1–10 . 1559418 . 10.1090/s0002-9904-1913-02429-7 . free .
• Book: Böhmer, P. E. . Differenzengleichungen und bestimmte Integrale . Differential Equations and Definite Integrals . Köhler Verlag . Leipzig . 1939.
• Philip J. . Davis . Leonhard Euler's Integral: A Historical Profile of the Gamma Function . . 66 . 10 . 849–869 . 1959 . 10.2307/2309786. 2309786 .
• Post . Emil . The Generalized Gamma Functions . Annals of Mathematics . Second Series . 1919 . 20 . 3 . 202–217 . 10.2307/1967871 . 1967871 . 5 March 2021.
• Book: Press . W. H. . Teukolsky . S. A. . Vetterling . W. T. . Flannery . B. P. . 2007 . Numerical Recipes: The Art of Scientific Computing . 3rd . Cambridge University Press . New York . 978-0-521-88068-8 . Section 6.1. Gamma Function . http://apps.nrbook.com/empanel/index.html?pg=256 .
• Book: Rocktäschel, O. R. . Methoden zur Berechnung der Gammafunktion für komplexes Argument . Methods for Calculating the Gamma Function for Complex Arguments . . Dresden . 1922.
• Book: Temme, Nico M. . Special Functions: An Introduction to the Classical Functions of Mathematical Physics . John Wiley & Sons . New York . 978-0-471-11313-3 . 1996.
• Book: E. T. Whittaker . E. T. . Whittaker . G. N. . Watson. G. N. Watson . . Cambridge University Press . 1927 . 978-0-521-58807-2.

External links

• NIST Digital Library of Mathematical Functions:Gamma function
• Pascal Sebah and Xavier Gourdon. Introduction to the Gamma Function. In PostScript and HTML formats.
• C++ reference for std::tgamma
• Examples of problems involving the gamma function can be found at Exampleproblems.com.
  • Wolfram gamma function evaluator (arbitrary precision)
  • Volume of n-Spheres and the Gamma Function at MathPages

Notes and References

1. Web site: Is the Gamma function misdefined? Or: Hadamard versus Euler — Who found the better Gamma function? .
2. Book: Special Functions: A Graduate Text . Richard . Beals . Roderick . Wong . Cambridge University Press . 2010 . 978-1-139-49043-6 . 28 . Extract of page 28
3. Book: Differential Equations: An Introduction with Mathematica . illustrated . Clay C. . Ross . Springer Science & Business Media . 2013 . 978-1-4757-3949-7 . 293 . Expression G.2 on page 293
4. Kingman. J. F. C.. A Convexity Property of Positive Matrices. The Quarterly Journal of Mathematics. 1961. 12. 1. 283–284. 10.1093/qmath/12.1.283. 1961QJMat..12..283K.
5. Davis . Philip . Leonhard Euler's Integral: A Historical Profile of the Gamma Function . maa.org.
6. Bonvini . Marco . October 9, 2010 . The Gamma function . Roma1.infn.it.
7. Waldschmidt . M. . 2006 . Transcendence of Periods: The State of the Art . https://web.archive.org/web/20060506050646/http://www.math.jussieu.fr/~miw/articles/pdf/TranscendencePeriods.pdf . 2006-05-06 . live . Pure Appl. Math. Quart. . 2 . 2 . 435–463 . 10.4310/pamq.2006.v2.n2.a3. free.
8. Web site: How to obtain the Laurent expansion of gamma function around $z=0$? . 2022-08-17 . Mathematics Stack Exchange . en.
9. Book: Artin . Emil . The Gamma Function . 2015 . 24. Dover .
10. Decimal expansion of real number x such that y = Gamma(x) is a minimum.
11. Decimal expansion of real number y such that y = Gamma(x) is a minimum.
12. Book: Gradshteyn . I. S.. Ryzhik . I. M. . Table of Integrals, Series, and Products . Seventh . Academic Press . 2007 . 978-0-12-373637-6. 893.
13. Whittaker and Watson, 12.2 example 1.
14. Detlef . Gronau . Why is the gamma function so as it is? . Imsc.uni-graz.at.
15. Pascal Sebah . Xavier Gourdon . Introduction to the Gamma Function . Numbers Computation . 30 January 2023 . 30 January 2023 . https://web.archive.org/web/20230130155521/https://www.csie.ntu.edu.tw/~b89089/link/gammaFunction.pdf . dead .
16. Whittaker and Watson, 12.31.
17. Whittaker and Watson, 12.32.
18. Whittaker and Watson, 12.22.
19. Web site: Exponential integral E: Continued fraction representations (Formula 06.34.10.0005) .
20. Web site: Exponential integral E: Continued fraction representations (Formula 06.34.10.0003) .
21. Book: Harry . Bateman . Arthur . Erdélyi . Higher Transcendental Functions . McGraw-Hill . 1955.
22. Book: H. M. . Srivastava . J. . Choi . Series Associated with the Zeta and Related Functions . Kluwer Academic . The Netherlands . 2001.
23. 10.1007/s11139-013-9528-5 . Iaroslav V. . Blagouchine . Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results . Ramanujan J. . 35 . 1 . 21–110 . 2014 . 120943474 .
24. 10.1007/s11139-015-9763-z . Iaroslav V. . Blagouchine . Erratum and Addendum to "Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results" . Ramanujan J. . 42 . 3 . 777–781 . 2016 . 125198685 .
25. Web site: Log Gamma Function . Wolfram MathWorld . 3 January 2019.
26. Web site: Leonhard Euler's Integral: An Historical Profile of the Gamma Function . https://web.archive.org/web/20140912213629/http://www.maa.org/sites/default/files/pdf/upload_library/22/Chauvenet/Davis.pdf . 2014-09-12 . live . 11 April 2022.
27. Blagouchine . Iaroslav V. . 2015 . A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations . Journal of Number Theory . 148 . 537–592 . 1401.3724 . 10.1016/j.jnt.2014.08.009.
28. W. P. . Alexejewsky . Über eine Classe von Funktionen, die der Gammafunktion analog sind . On a class of functions analogous to the gamma function . Leipzig Weidmannsche Buchhandlung . 46 . 1894 . 268–275.
29. E. W. . Barnes . The theory of the G-function . Quart. J. Math. . 31 . 1899 . 264–314.
30. Victor S. . Adamchik . Polygamma functions of negative order . J. Comput. Appl. Math. . 100 . 2 . 1998 . 191–199 . 10.1016/S0377-0427(98)00192-7. free .
31. R. W. . Gosper .

    	m/6
    style\int
    	n/4

    logF(z)dz

    in special functions, q-series and related topics . J. Am. Math. Soc. . 14 . 1997.
32. Olivier . Espinosa. Victor H. . Moll. On Some Integrals Involving the Hurwitz Zeta Function: Part 1. The Ramanujan Journal . 6 . 2002 . 2. 159–188 . 10.1023/A:1015706300169. 128246166.
33. David H. . Bailey. David . Borwein. Jonathan M.. Borwein. On Eulerian log-gamma integrals and Tornheim-Witten zeta functions. The Ramanujan Journal . 36 . 2015 . 1–2. 43–68 . 10.1007/s11139-012-9427-1. 7335291.
34. T. . Amdeberhan. Mark W.. Coffey. Olivier. Espinosa. Christoph. Koutschan. Dante V.. Manna. Victor H.. Moll. Integrals of powers of loggamma. Proc. Amer. Math. Soc.. 139. 2 . 2011 . 535–545 . 10.1090/S0002-9939-2010-10589-0. free.
35. E.A. Karatsuba, Fast evaluation of transcendental functions. Probl. Inf. Transm. Vol.27, No.4, pp. 339–360 (1991).
36. E.A. Karatsuba, On a new method for fast evaluation of transcendental functions. Russ. Math. Surv. Vol.46, No.2, pp. 246–247 (1991).
37. E.A. Karatsuba "Fast Algorithms and the FEE Method".
38. Borwein . J. M. . Zucker . I. J. . Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals of the first kind . IMA Journal of Numerical Analysis . 1992 . 12 . 4 . 519–526 . 10.1093/IMANUM/12.4.519.
39. Helmut. Werner. Robert. Collinge. Chebyshev approximations to the Gamma Function. Math. Comput.. 1961. 195–197. 15. 74. 10.1090/S0025-5718-61-99220-1 . 2004230.
40. Michon, G. P. "Trigonometry and Basic Functions ". Numericana. Retrieved 5 May 2007.
41. Chaudry, M. A. & Zubair, S. M. (2001). On A Class of Incomplete Gamma Functions with Applications. p. 37
42. Rice, J. A. (1995). Mathematical Statistics and Data Analysis (Second Edition). p. 52–53
43. Book: Borwein, J. . Bailey, D. H. . Girgensohn, R. . amp . 2003 . Experimentation in Mathematics . A. K. Peters . 133 . 978-1-56881-136-9 .
44. Davis . P. J. . 1959 . Leonhard Euler's Integral: A Historical Profile of the Gamma Function . . 66 . 10 . 849–869 . 3 December 2016 . 10.2307/2309786 . 2309786 . 7 November 2012 . https://web.archive.org/web/20121107190256/http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=3104 . dead .
45. Web site: Interpolating the natural factorial n! or The birth of the real factorial function (1729 - 1826) .
46. Euler's paper was published in Commentarii academiae scientiarum Petropolitanae 5, 1738, 36–57. See E19 -- De progressionibus transcendentibus seu quarum termini generales algebraice dari nequeunt, from The Euler Archive, which includes a scanned copy of the original article.
47. Book: Remmert, R. . Kay . L. D. . Classical Topics in Complex Function Theory . Springer . 2006 . 978-0-387-98221-2 .
48. Lanczos . C. . 1964 . A precision approximation of the gamma function . J. SIAM Numer. Anal. Ser. B . 1. 1 . 86 . 10.1137/0701008 . 1964SJNA....1...86L .
49. Book: Notes from the International Autumn School on Computational Number Theory . Ilker Inam . Engin Büyükaşşk . Springer . 2019 . 978-3-030-12558-5 . 205 . Extract of page 205
50. Book: Knuth, D. E. . The Art of Computer Programming, Volume 1 (Fundamental Algorithms) . Addison-Wesley . 1997 .
51. Borwein. Jonathan M. . Jonathan Borwein. Corless. Robert M. . Gamma and Factorial in the Monthly. American Mathematical Monthly. en. Mathematical Association of America. 2017. 125. 5 . 400–24. 1703.05349. 2017arXiv170305349B. 10.1080/00029890.2018.1420983. 119324101 .
52. Web site: What's the history of Gamma_function? . 2022-11-05 . yearis.com.
53. News: Berry . M. . Why are special functions special? . Physics Today . April 2001.
54. Web site: microsoft/calculator. 2020-12-25. GitHub. en.
55. Web site: gnome-calculator. 2023-03-03. GNOME.org. en.