The

gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except t ...

is an important

special function Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined ...

in mathematics. Its particular values can be expressed in closed form for

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...

and

half-integer In mathematics, a half-integer is a number of the form :n + \tfrac, where n is an whole number. For example, :, , , 8.5 are all ''half-integers''. The name "half-integer" is perhaps misleading, as the set may be misunderstood to include numbers ...

arguments, but no simple expressions are known for the values at

rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abili ...

points in general. Other fractional arguments can be approximated through efficient infinite products, infinite series, and recurrence relations.

Integers and half-integers

For positive integer arguments, the gamma function coincides with the

factorial In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) ...

. That is, :

\Gamma(n) = (n-1)!,

and hence :

\begin
\Gamma(1) &= 1, \\
\Gamma(2) &= 1, \\
\Gamma(3) &= 2, \\
\Gamma(4) &= 6, \\
\Gamma(5) &= 24,
\end

and so on. For non-positive integers, the gamma function is not defined. For positive half-integers, the function values are given exactly by :

\Gamma \left (\tfrac \right) = \sqrt \pi \frac\,,

or equivalently, for non-negative integer values of : :

\begin
\Gamma\left(\tfrac12+n\right) &= \frac\, \sqrt = \frac \sqrt \\
\Gamma\left(\tfrac12-n\right) &= \frac\, \sqrt = \frac \sqrt 
\end

where denotes the

double factorial In mathematics, the double factorial or semifactorial of a number , denoted by , is the product of all the integers from 1 up to that have the same parity (odd or even) as . That is, :n!! = \prod_^ (n-2k) = n (n-2) (n-4) \cdots. For even , th ...

. In particular, : and by means of the

reflection formula In mathematics, a reflection formula or reflection relation for a function ''f'' is a relationship between ''f''(''a'' − ''x'') and ''f''(''x''). It is a special case of a functional equation, and it is very common in the literature t ...

, :

General rational argument

In analogy with the half-integer formula, :

\begin
\Gamma \left(n+\tfrac13 \right) &= \Gamma \left(\tfrac13 \right) \frac \\
\Gamma \left(n+\tfrac14 \right) &= \Gamma \left(\tfrac14 \right ) \frac \\
\Gamma \left(n+\tfrac \right ) &= \Gamma \left(\tfrac \right ) \frac \\
\Gamma \left(n+\tfrac \right) &= \Gamma \left(\tfrac\right) \frac \prod _^n (k q+p-q)
\end

where denotes the th multifactorial of . Numerically, :

\Gamma\left(\tfrac13\right) \approx 2.678\,938\,534\,707\,747\,6337

\Gamma\left(\tfrac14\right) \approx 3.625\,609\,908\,221\,908\,3119

\Gamma\left(\tfrac15\right) \approx 4.590\,843\,711\,998\,803\,0532

\Gamma\left(\tfrac16\right) \approx 5.566\,316\,001\,780\,235\,2043

\Gamma\left(\tfrac17\right) \approx 6.548\,062\,940\,247\,824\,4377

\Gamma\left(\tfrac18\right) \approx 7.533\,941\,598\,797\,611\,9047

. As

n

tends to infinity, :

\Gamma\left(\tfrac1n\right) \sim n+\gamma-1

where

\gamma

is the

Euler–Mascheroni constant Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma (). It is defined as the limiting difference between the harmonic series and the natural l ...

and

\sim

denotes asymptotic equivalence. It is unknown whether these constants are transcendental in general, but and were shown to be transcendental by G. V. Chudnovsky. has also long been known to be transcendental, and Yuri Nesterenko proved in 1996 that , , and are algebraically independent. The number is related to the

lemniscate constant In mathematics, the lemniscate constant p. 199 is a transcendental mathematical constant that is the ratio of the perimeter of Bernoulli's lemniscate to its diameter, analogous to the definition of for the circle. Equivalently, the perimeter ...

by :

\Gamma\left(\tfrac14\right) = \sqrt,

and it has been conjectured by Gramain that :

\Gamma \left (\tfrac14 \right ) = \sqrt /math>

where  is the Masser–Gramain constant, although numerical work by Melquiond et al. indicates that this conjecture is false.

Borwein and Zucker have found that  can be expressed algebraically in terms of , , , , and  where  is a

complete elliptic integral of the first kind In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising i ...

. This permits efficiently approximating the gamma function of rational arguments to high precision using quadratically convergent arithmetic–geometric mean iterations. For example: :

\sqrt K\left(3-2 \sqrt\right) \\ \frac &= \frac \end

No similar relations are known for or other denominators. In particular, where AGM() is the arithmetic–geometric mean, we have :

\Gamma\left(\tfrac13\right) = \frac

\Gamma\left(\tfrac14\right) = \sqrt \frac

\Gamma\left(\tfrac16\right) = \frac.

Other formulas include the

infinite product In mathematics, for a sequence of complex numbers ''a''1, ''a''2, ''a''3, ... the infinite product : \prod_^ a_n = a_1 a_2 a_3 \cdots is defined to be the limit of the partial products ''a''1''a''2...''a'n'' as ''n'' increases without bound. ...

s :

\Gamma\left(\tfrac14\right) = (2 \pi)^\frac34 \prod_^\infty \tanh \left( \frac \right)

and :

\Gamma\left(\tfrac14\right) = A^3 e^ \sqrt 2^\frac16 \prod_^\infty \left(1-\frac\right)^

where is the

Glaisher–Kinkelin constant In mathematics, the Glaisher–Kinkelin constant or Glaisher's constant, typically denoted , is a mathematical constant, related to the -function and the Barnes -function. The constant appears in a number of sums and integrals, especially those ...

and is

Catalan's constant In mathematics, Catalan's constant , is defined by : G = \beta(2) = \sum_^ \frac = \frac - \frac + \frac - \frac + \frac - \cdots, where is the Dirichlet beta function. Its numerical value is approximately : It is not known whether is ir ...

. The following two representations for were given by I. Mező :

\sqrt\frac=i\sum_^\infty e^\theta_1\left(\frac(2k-1),e^\right),

and :

\sqrt\frac=\sum_^\infty\frac,

where and are two of the Jacobi theta functions.

Products

Some product identities include: :

\prod_^2 \Gamma\left(\tfrac\right) = \frac \approx 3.627\,598\,728\,468\,435\,7012

\prod_^3 \Gamma\left(\tfrac\right) = \sqrt \approx 7.874\,804\,972\,861\,209\,8721

\prod_^4 \Gamma\left(\tfrac\right) = \frac \approx 17.655\,285\,081\,493\,524\,2483

\prod_^5 \Gamma\left(\tfrac\right) = 4\sqrt \approx 40.399\,319\,122\,003\,790\,0785

\prod_^6 \Gamma\left(\tfrac\right) = \frac \approx 93.754\,168\,203\,582\,503\,7970

\prod_^7 \Gamma\left(\tfrac\right) = 4\sqrt \approx 219.828\,778\,016\,957\,263\,6207

In general: :

\prod_^n \Gamma\left(\tfrac\right) = \sqrt

From those products can be deduced other values, for example, from the former equations for

\prod_^3 \Gamma\left(\tfrac\right)

\Gamma\left(\tfrac\right)

and

\Gamma\left(\tfrac\right)

, can be deduced:

\Gamma\left(\tfrac\right) =\left(\tfrac \right) ^ ^

Other rational relations include :

\frac = \frac

\frac = \frac

\frac = \frac

and many more relations for where the denominator d divides 24 or 60. Gamma quotients with algebraic values must be "poised" in the sense that the sum of arguments is the same (modulo 1) for the denominator and the numerator. A more sophisticated example: :

\frac = \frac

Imaginary and complex arguments

The gamma function at the

imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition and ...

gives , : :

\Gamma(i) = (-1+i)! \approx -0.1549 - 0.4980i.

It may also be given in terms of the Barnes -function: :

\Gamma(i) = \frac = e^.

Curiously enough,

\Gamma(i)

appears in the below integral evaluation:The webpage of István Mező
/ref> :

\int_0^\\,dx=1-\frac+\frac\log\left(\frac\right).

Here

\

denotes the fractional part. Because of the Euler Reflection Formula, and the fact that

\Gamma(\bar)=\bar(z)

, we have an expression for the

modulus squared In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2, and is denoted by a superscript 2; for instance, the square ...

of the gamma function evaluated on the imaginary axis: :

\left, \Gamma(i\kappa)\^2=\frac

The above integral therefore relates to the phase of

\Gamma(i)

. The gamma function with other complex arguments returns :

\Gamma(1 + i) = i\Gamma(i) \approx 0.498 - 0.155i

\Gamma(1 - i) = -i\Gamma(-i) \approx 0.498 + 0.155i

\Gamma(\tfrac12 + \tfrac12 i) \approx 0.818\,163\,9995 - 0.763\,313\,8287\, i

\Gamma(\tfrac12 - \tfrac12 i) \approx 0.818\,163\,9995 + 0.763\,313\,8287\, i

\Gamma(5 + 3i) \approx 0.016\,041\,8827 - 9.433\,293\,2898\, i

\Gamma(5 - 3i) \approx 0.016\,041\,8827 + 9.433\,293\,2898\, i.

Other constants

The gamma function has a

local minimum In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ran ...

on the positive real axis :

x_ = 1.461\,632\,144\,968\,362\,341\,262\ldots\,

with the value :

\Gamma\left(x_\right) = 0.885\,603\,194\,410\,888\ldots\,

. Integrating the

reciprocal gamma function In mathematics, the reciprocal gamma function is the function :f(z) = \frac, where denotes the gamma function. Since the gamma function is meromorphic and nonzero everywhere in the complex plane, its reciprocal is an entire function. As an ...

along the positive real axis also gives the Fransén–Robinson constant. On the negative real axis, the first local maxima and minima (zeros of the

digamma function In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: :\psi(x)=\frac\ln\big(\Gamma(x)\big)=\frac\sim\ln-\frac. It is the first of the polygamma functions. It is strictly increasing and strictl ...

) are:

References