Gamma function

The notation ${\displaystyle \Gamma (z)}$ is due to Legendre.[1] If the real part of the complex number z is positive (${\displaystyle \Re (z)>0}$), then the integral

${\displaystyle \Gamma (z)=\int _{0}^{\infty }x^{z-1}e^{-x}\,dx}$

converges absolutely, and is known as the Euler integral of the second kind. (Euler's integral of the first kind is the beta function.[1]) Using integration by parts, one sees that:

${\displaystyle {\begin{aligned}\Gamma (z+1)&=\int _{0}^{\infty }x^{z}e^{-x}\,dx\\&={\Big [}-x^{z}e^{-x}{\Big ]}_{0}^{\infty }+\int _{0}^{\infty }zx^{z-1}e^{-x}\,dx\\&=\lim _{x\to \infty }(-x^{z}e^{-x})-(-0^{z}e^{-0})+z\int _{0}^{\infty }x^{z-1}e^{-x}\,dx.\end{aligned}}}$

Recognizing that ${\displaystyle -x^{z}e^{-x}\to 0}$ as ${\displaystyle x\to \infty ,}$

${\displaystyle {\begin{aligned}\Gamma (z+1)&=z\int _{0}^{\infty }x^{z-1}e^{-x}\,dx\\&=z\Gamma (z).\end{aligned}}}$

We can calculate ${\displaystyle \Gamma (1){\text{:}}}$

${\displaystyle {\begin{aligned}\Gamma (1)&=\int _{0}^{\infty }x^{1-1}e^{-x}\,dx\\&={\Big [}-e^{-x}{\Big ]}_{0}^{\infty }\\&=\lim _{x\to \infty }(-e^{-x})-(-e^{-0})\\&=0-(-1)\\&=1.\end{aligned}}}$

Given that ${\displaystyle \Gamma (1)=1}$ and ${\displaystyle \Gamma (n+1)=n\Gamma (n),}$

${\displaystyle \Gamma (n)=1\cdot 2\cdot 3\cdots (n-1)=(n-1)!}$

for all positive integers n. This can be seen as an example of proof by induction.

The identity ${\textstyle \Gamma (z)={\frac {\Gamma (z+1)}{z}}}$ can be used (or, yielding the same result, analytic continuation can be used) to uniquely extend the integral formulation for ${\displaystyle \Gamma (z)}$ to a meromorphic function defined for all complex numbers z, except integers less than or equal to zero.[1] It is this extended version that is commonly referred to as the gamma function.[1]

When seeking to approximate ${\displaystyle z!}$ for a complex number ${\displaystyle z}$, it is effective to first compute ${\displaystyle n!}$ for some large integer ${\displaystyle n}$. Use that to approximate a value for ${\displaystyle (n+z)!}$, and then use the recursion relation ${\displaystyle m!=m(m-1)!}$ backwards ${\displaystyle n}$ times, to unwind it to an approximation for ${\displaystyle z!}$. Furthermore, this approximation is exact in the limit as ${\displaystyle n}$ goes to infinity.

Specifically, for a fixed integer ${\displaystyle m}$, it is the case that

${\displaystyle \lim _{n\to \infty }{\frac {n!\;(n+1)^{m}}{(n+m)!}}=1\,.}$

If ${\displaystyle m}$ is not an integer then it is not possible to say whether this equation is true because we have not yet (in this section) defined the factorial function for non-integers. However, we do get a unique extension of the factorial function to the non-integers by insisting that this equation continue to hold when the arbitrary integer ${\displaystyle m}$ is replaced by an arbitrary complex number ${\displaystyle z}$.

${\displaystyle \lim _{n\to \infty }{\frac {n!\;(n+1)^{z}}{(n+z)!}}=1\,.}$

Multiplying both sides by ${\displaystyle z!}$ gives

${\displaystyle {\begin{aligned}z!&=\lim _{n\to \infty }n!{\frac {z!}{(n+z)!}}(n+1)^{z}\\[8pt]&=\lim _{n\to \infty }(1\cdots n){\frac {1}{(1+z)\cdots (n+z)}}\left(\left({\frac {2}{1}}\right)\left({\frac {3}{2}}\right)\cdots \left({\frac {n+1}{n}}\right)\right)^{z}\\[8pt]&=\prod _{n=1}^{\infty }\left[{\frac {1}{1+{\frac {z}{n}}}}\left(1+{\frac {1}{n}}\right)^{z}\right].\end{aligned}}}$

This infinite product converges for all complex numbers ${\displaystyle z}$ except the negative integers, which fail because trying to use the recursion relation ${\displaystyle m!=m(m-1)!}$ backwards through the value ${\displaystyle m=0}$ involves a division by zero.

Similarly for the gamma function, the definition as an infinite product due to Euler is valid for all complex numbers ${\displaystyle z}$ except the non-positive integers:

${\displaystyle \Gamma (z)={\frac {1}{z}}\prod _{n=1}^{\infty }{\frac {\left(1+{\frac {1}{n}}\right)^{z}}{1+{\frac {z}{n}}}}\,.}$

By this construction, the gamma function is the unique function that simultaneously satisfies ${\displaystyle \Gamma (1)=1}$, ${\displaystyle \Gamma (z+1)=z\Gamma (z)}$ for all complex numbers ${\displaystyle z}$ except the non-positive integers, and ${\textstyle \lim _{n\to \infty }{\frac {\Gamma (n+z)}{\Gamma (n)\;n^{z}}}=1}$ for all complex numbers ${\displaystyle z}$.[1]

The definition for the gamma function due to Weierstrass is also valid for all complex numbers z except the non-positive integers:

${\displaystyle \Gamma (z)={\frac {e^{-\gamma z}}{z}}\prod _{n=1}^{\infty }\left(1+{\frac {z}{n}}\right)^{-1}e^{z/n},}$

where ${\displaystyle \gamma \approx 0.577216}$ is the Euler–Mascheroni constant.[1] This is the Hadamard Product of ${\displaystyle 1/\Gamma (z)}$ in a rewritten form. Indeed, since ${\displaystyle 1/\Gamma (z)}$ is entire of genus 1 with a simple zero at ${\displaystyle z=0}$, we have the product representation

${\displaystyle {\frac {1}{\Gamma (z)}}=ze^{Az+B}\prod _{\rho }\left(1-{\frac {z}{\rho }}\right)e^{z/\rho },}$

where the product is over the zeros ${\displaystyle \rho \neq 0}$ of ${\displaystyle 1/\Gamma (z)}$. Since ${\displaystyle \Gamma (z)}$ has simple poles at the non-positive integers, it follows ${\displaystyle 1/\Gamma (z)}$ has simple zeros at the nonpositive integers, and so the equation above becomes Weierstrass's formula with ${\displaystyle -Az-B}$ in place of ${\displaystyle -\gamma z}$. The derivation of the constants ${\displaystyle A=\gamma }$ and ${\displaystyle B=0}$ is somewhat technical, but can be accomplished by using some identities involving the Riemann zeta function (see this identity, for instance). See also the Weierstrass factorization theorem.

A representation of the incomplete gamma function in terms of generalized Laguerre polynomials is

${\displaystyle \Gamma (z,x)=x^{z}e^{-x}\sum _{n=0}^{\infty }{\frac {L_{n}^{(z)}(x)}{n+1}},}$

which converges for ${\displaystyle \Re (z)>-1}$ and ${\displaystyle x>0}$.[6]

Other important functional equations for the gamma function are Euler's reflection formula

${\displaystyle \Gamma (1-z)\Gamma (z)={\pi \over \sin(\pi z)},\qquad z\not \in \mathbb {Z} }$

which implies

${\displaystyle \Gamma (\varepsilon -n)=(-1)^{n-1}\;{\frac {\Gamma (-\varepsilon )\Gamma (1+\varepsilon )}{\Gamma (n+1-\varepsilon )}},}$

and the Legendre duplication formula

${\displaystyle \Gamma (z)\Gamma \left(z+{\tfrac {1}{2}}\right)=2^{1-2z}\;{\sqrt {\pi }}\;\Gamma (2z).}$

The duplication formula is a special case of the multiplication theorem (See[7], Eq. 5.5.6)

${\displaystyle \prod _{k=0}^{m-1}\Gamma \left(z+{\frac {k}{m}}\right)=(2\pi )^{\frac {m-1}{2}}\;m^{{\frac {1}{2}}-mz}\;\Gamma (mz).}$

A simple but useful property, which can be seen from the limit definition, is:

${\displaystyle {\overline {\Gamma (z)}}=\Gamma ({\overline {z}})\;\Rightarrow \;\Gamma (z)\Gamma ({\overline {z}})\in \mathbb {R} .}$

In particular, with z = a + bi, this product is

${\displaystyle |\Gamma (a+bi)|^{2}=|\Gamma (a)|^{2}\prod _{k=0}^{\infty }{\frac {1}{1+{\frac {b^{2}}{(a+k)^{2}}}}}}$

If the real part is an integer or a half-integer, this can be finitely expressed in closed form:

${\displaystyle {\begin{aligned}|\Gamma (bi)|^{2}&={\frac {\pi }{b\sinh(\pi b)}}\\[6pt]|\Gamma \left({\tfrac {1}{2}}+bi\right)|^{2}&={\frac {\pi }{\cosh(\pi b)}}\\|\Gamma \left(1+bi\right)|^{2}&={\frac {\pi b}{\sinh(\pi b)}}\\|\Gamma \left(1+n+bi\right)|^{2}&={\frac {\pi b}{\sinh(\pi b)}}\prod _{k=1}^{n}\left(k^{2}+b^{2}\right),\quad n\in \mathbb {N} \\|\Gamma \left(-n+bi\right)|^{2}&={\frac {\pi }{b\sinh(\pi b)}}\prod _{k=1}^{n}\left(k^{2}+b^{2}\right)^{-1},\quad n\in \mathbb {N} \\|\Gamma \left({\tfrac {1}{2}}\pm n+bi\right)|^{2}&={\frac {\pi }{\cosh(\pi b)}}\prod _{k=1}^{n}\left(\left(k-{\tfrac {1}{2}}\right)^{2}+b^{2}\right)^{\pm 1},\quad n\in \mathbb {N} \end{aligned}}}$

Perhaps the best-known value of the gamma function at a non-integer argument is

${\displaystyle \Gamma \left({\tfrac {1}{2}}\right)={\sqrt {\pi }},}$

which can be found by setting ${\textstyle z={\frac {1}{2}}}$ in the reflection or duplication formulas, by using the relation to the beta function given below with ${\textstyle x=y={\frac {1}{2}}}$, or simply by making the substitution ${\displaystyle u={\sqrt {x}}}$ in the integral definition of the gamma function, resulting in a Gaussian integral. In general, for non-negative integer values of ${\displaystyle n}$ we have:

${\displaystyle {\begin{aligned}\Gamma \left({\tfrac {1}{2}}+n\right)&={(2n)! \over 4^{n}n!}{\sqrt {\pi }}={\frac {(2n-1)!!}{2^{n}}}{\sqrt {\pi }}={\binom {n-{\frac {1}{2}}}{n}}n!{\sqrt {\pi }}\\[8pt]\Gamma \left({\tfrac {1}{2}}-n\right)&={(-4)^{n}n! \over (2n)!}{\sqrt {\pi }}={\frac {(-2)^{n}}{(2n-1)!!}}{\sqrt {\pi }}={\frac {\sqrt {\pi }}{{\binom {-1/2}{n}}n!}}\end{aligned}}}$

where ${\displaystyle n!!}$ denotes the double factorial of n and, when ${\displaystyle n=0}$, ${\displaystyle n!!=1}$. See Particular values of the gamma function for calculated values.

It might be tempting to generalize the result that ${\textstyle \Gamma \left({\frac {1}{2}}\right)={\sqrt {\pi }}}$ by looking for a formula for other individual values ${\displaystyle \Gamma (r)}$ where ${\displaystyle r}$ 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 ${\displaystyle \Gamma (r)}$ are not known to be expressible by themselves in terms of elementary functions. It has been proved that ${\displaystyle \Gamma (n+r)}$ is a transcendental number and algebraically independent of ${\displaystyle \pi }$ for any integer ${\displaystyle n}$ and each of the fractions ${\textstyle r={\frac {1}{6}},{\frac {1}{4}},{\frac {1}{3}},{\frac {2}{3}},{\frac {3}{4}},{\frac {5}{6}}}$.[8] In general, when computing values of the gamma function, we must settle for numerical approximations.

Another useful limit for asymptotic approximations is:

${\displaystyle \lim _{n\to \infty }{\frac {\Gamma (n+\alpha )}{\Gamma (n)n^{\alpha }}}=1,\qquad \alpha \in \mathbb {C} .}$

The derivatives of the gamma function are described in terms of the polygamma function. For example:

${\displaystyle \Gamma '(z)=\Gamma (z)\psi _{0}(z).}$

For a positive integer m the derivative of the gamma function can be calculated as follows (here ${\displaystyle \gamma }$ is the Euler–Mascheroni constant):

${\displaystyle \Gamma '(m+1)=m!\left(-\gamma +\sum _{k=1}^{m}{\frac {1}{k}}\right)\,.}$

For ${\displaystyle \Re (x)>0}$ the ${\displaystyle n}$th derivative of the gamma function is:

Derivative of the function Γ(z)

${\displaystyle {\frac {d^{n}}{dx^{n}}}\Gamma (x)=\int _{0}^{\infty }t^{x-1}e^{-t}(\ln t)^{n}\,dt.}$

(This can be derived by differentiating the integral form of the gamma function with respect to ${\displaystyle x}$, and using the technique of differentiation under the integral sign.)

Using the identity

${\displaystyle \Gamma ^{(n)}(1)=(-1)^{n}n!\sum \limits _{\pi \,\vdash \,n}\,\prod _{i=1}^{r}{\frac {\zeta ^{*}(a_{i})}{k_{i}!\cdot a_{i}}}\qquad \zeta ^{*}(x):={\begin{cases}\zeta (x)&x\neq 1\\\gamma &x=1\end{cases}}}$

where ${\displaystyle \zeta (z)}$ is the Riemann zeta function, and ${\displaystyle \pi }$ is a partition of ${\displaystyle n}$ given by

${\displaystyle \pi =\underbrace {a_{1}+\cdots +a_{1}} _{k_{1}{\text{ terms}}}+\cdots +\underbrace {a_{r}+\cdots +a_{r}} _{k_{r}{\text{ terms}}},}$

we have in particular

${\displaystyle \Gamma (z)={\frac {1}{z}}-\gamma +{\tfrac {1}{2}}\left(\gamma ^{2}+{\frac {\pi ^{2}}{6}}\right)z-{\tfrac {1}{6}}\left(\gamma ^{3}+{\frac {\gamma \pi ^{2}}{2}}+2\zeta (3)\right)z^{2}+O(z^{3}).}$

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 ${\displaystyle x_{1}}$ and ${\displaystyle x_{2}}$, and for any ${\displaystyle t\in [0,1]}$,

${\displaystyle \Gamma (tx_{1}+(1-t)x_{2})\leq \Gamma (x_{1})^{t}\Gamma (x_{2})^{1-t}.}$

• For any two positive real numbers x and y with y > x,

${\displaystyle \left({\frac {\Gamma (y)}{\Gamma (x)}}\right)^{\frac {1}{y-x}}>\exp \left({\frac {\Gamma '(x)}{\Gamma (x)}}\right).}$

• For any positive real number ${\displaystyle x}$,

${\displaystyle \Gamma ''(x)\Gamma (x)>\Gamma '(x).}$

The last of these statements is, essentially by definition, the same as the statement that ${\displaystyle \psi ^{(1)}(x)>0}$, where ${\displaystyle \psi ^{(1)}}$ is the polygamma function of order 1. To prove the logarithmic convexity of the gamma function, it therefore suffices to observe that ${\displaystyle \psi ^{(1)}}$ has a series representation which, for positive real x, consists of only positive terms.

Logarithmic convexity and Jensen's inequality together imply, for any positive real numbers ${\displaystyle x_{1},\ldots ,x_{n}}$ and ${\displaystyle a_{1},\ldots ,a_{n}}$,

${\displaystyle \Gamma \left({\frac {a_{1}x_{1}+\cdots +a_{n}x_{n}}{a_{1}+\cdots +a_{n}}}\right)\leq {\bigl (}\Gamma (x_{1})^{a_{1}}\cdots \Gamma (x_{n})^{a_{n}}{\bigr )}^{\frac {1}{a_{1}+\cdots +a_{n}}}.}$

There are also bounds on ratios of gamma functions. The best-known is Gautschi's inequality, which says that for any positive real number x and any s ∈ (0, 1),

${\displaystyle x^{1-s}<{\frac {\Gamma (x+1)}{\Gamma (x+s)}}<(x+1)^{1-s}.}$

Representation of the gamma function in the complex plane. Each point ${\displaystyle z}$ is colored according to the argument of ${\displaystyle \Gamma (z)}$. The contour plot of the modulus ${\displaystyle |\Gamma (z)|}$ is also displayed.

3-dimensional plot of the absolute value of the complex gamma function

The behavior of ${\displaystyle \Gamma (z)}$ for an increasing positive variable is simple. It grows quickly, faster than an exponential function in fact. Asymptotically as ${\displaystyle z\to \infty }$, the magnitude of the gamma function is given by Stirling's formula

${\displaystyle \Gamma (z+1)\sim {\sqrt {2\pi z}}\left({\frac {z}{e}}\right)^{z},}$

where the symbol ${\displaystyle \sim }$ implies asymptotic convergence. In other words, the ratio of the two sides converges to 1 as ${\textstyle z\to +\infty }$.[1]

The behavior for non-positive ${\displaystyle z}$ is more intricate. Euler's integral does not converge for ${\displaystyle z\leq 0}$, 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,[1]

${\displaystyle \Gamma (z)={\frac {\Gamma (z+n+1)}{z(z+1)\cdots (z+n)}},}$

choosing ${\displaystyle n}$ such that ${\displaystyle z+n}$ is positive. The product in the denominator is zero when ${\displaystyle z}$ equals any of the integers ${\displaystyle 0,-1,-2,\cdots }$. 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.[1]

For a function ${\displaystyle f}$ of a complex variable ${\displaystyle z}$, at a simple pole ${\displaystyle c}$, the residue of ${\displaystyle f}$ is given by:

${\displaystyle \operatorname {Res} (f,c)=\lim _{z\to c}(z-c)f(z).}$

For the simple pole ${\displaystyle z=-n,}$ we rewrite recurrence formula as:

${\displaystyle (z+n)\Gamma (z)={\frac {\Gamma (z+n+1)}{z(z+1)\cdots (z+n-1)}}.}$

The numerator at ${\displaystyle z=-n,}$ is

${\displaystyle \Gamma (z+n+1)=\Gamma (1)=1}$

and the denominator

${\displaystyle z(z+1)\cdots (z+n-1)=(-1)^{n}n!.}$

So the residues of the gamma function at those points are:

${\displaystyle \operatorname {Res} (\Gamma ,-n)={\frac {(-1)^{n}}{n!}}.}$[9]

The gamma function is non-zero everywhere along the real line, although it comes arbitrarily close to zero as z → −∞. There is in fact no complex number ${\displaystyle z}$ for which ${\displaystyle \Gamma (z)=0}$, and hence the reciprocal gamma function ${\textstyle {\frac {1}{\Gamma (z)}}}$ is an entire function, with zeros at ${\displaystyle z=0,-1,-2,\cdots }$.[1]

The gamma function has a local minimum at z_min ≈ +1.46163214496836234126 (truncated) where it attains the value Γ(z_min) ≈ +0.88560319441088870027 (truncated). 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 ${\displaystyle z}$ and ${\displaystyle z+n}$ is odd, and an even number if the number of poles is even.[9]

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 z is positive,[10]

${\displaystyle \Gamma (z)=\int _{0}^{1}\left(\log {\frac {1}{t}}\right)^{z-1}\,dt.}$

Binet's first integral formula for the gamma function states that, when the real part of z is positive, then:[11]

${\displaystyle \log \Gamma (z)=\left(z-{\frac {1}{2}}\right)\log z-z+{\frac {1}{2}}\log(2\pi )+\int _{0}^{\infty }\left({\frac {1}{2}}-{\frac {1}{t}}+{\frac {1}{e^{t}-1}}\right){\frac {e^{-tz}}{t}}\,dt.}$

The integral on the right-hand side may be interpreted as a Laplace transform. That is,

${\displaystyle \log \left(\Gamma (z)\left({\frac {e}{z}}\right)^{z}{\sqrt {2\pi z}}\right)={\mathcal {L}}\left({\frac {1}{2t}}-{\frac {1}{t^{2}}}+{\frac {1}{t(e^{t}-1)}}\right)(z).}$

Binet's second integral formula states that, again when the real part of z is positive, then:[12]

${\displaystyle \log \Gamma (z)=\left(z-{\frac {1}{2}}\right)\log z-z+{\frac {1}{2}}\log(2\pi )+2\int _{0}^{\infty }{\frac {\arctan(t/z)}{e^{2\pi t}-1}}\,dt.}$

Let C be a Hankel contour, meaning a path that begins and ends at the point ∞ on the Riemann sphere, whose unit tangent vector converges to −1 at the start of the path and to 1 at the end, which has winding number 1 around 0, and which does not cross [0, ∞). Fix a branch of ${\displaystyle \log(-t)}$ by taking a branch cut along [0, ∞) and by taking ${\displaystyle \log(-t)}$ to be real when t is on the negative real axis. Assume z is not an integer. Then Hankel's formula for the gamma function is:[13]

${\displaystyle \Gamma (z)=-{\frac {1}{2i\sin \pi z}}\int _{C}(-t)^{z-1}e^{-t}\,dt,}$

where ${\displaystyle (-t)^{z-1}}$ is interpreted as ${\displaystyle \exp((z-1)\log(-t))}$. The reflection formula leads to the closely related expression

${\displaystyle {\frac {1}{\Gamma (z)}}={\frac {i}{2\pi }}\int _{C}(-t)^{-z}e^{-t}\,dt,}$

again valid whenever z is not an integer.

The logarithm of the gamma function has the following Fourier series expansion for ${\displaystyle 0<z<1:}$

${\displaystyle \ln \Gamma (z)=\left({\frac {1}{2}}-z\right)(\gamma +\ln 2)+(1-z)\ln \pi -{\frac {1}{2}}\ln \sin(\pi z)+{\frac {1}{\pi }}\sum _{n=1}^{\infty }{\frac {\ln n}{n}}\sin(2\pi nz),}$

which was for a long time attributed to Ernst Kummer, who derived it in 1847.[14][15] However, Iaroslav Blagouchine discovered that Carl Johan Malmsten first derived this series in 1842.[16][17]

In 1840 Joseph Ludwig Raabe proved that

${\displaystyle \int _{a}^{a+1}\ln \Gamma (z)\,dz={\tfrac {1}{2}}\ln 2\pi +a\ln a-a,\quad a>0.}$

In particular, if ${\displaystyle a=0}$ then

${\displaystyle \int _{0}^{1}\ln \Gamma (z)\,dz={\tfrac {1}{2}}\ln 2\pi .}$

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 ${\displaystyle m\rightarrow \infty }$ gives the formula.

An alternative notation which was originally introduced by Gauss and which was sometimes used is the ${\displaystyle \Pi }$-function, which in terms of the gamma function is

${\displaystyle \Pi (z)=\Gamma (z+1)=z\Gamma (z)=\int _{0}^{\infty }e^{-t}t^{z}\,dt,}$

so that ${\displaystyle \Pi (n)=n!}$ for every non-negative integer ${\displaystyle n}$.

Using the pi function the reflection formula takes on the form

${\displaystyle \Pi (z)\Pi (-z)={\frac {\pi z}{\sin(\pi z)}}={\frac {1}{\operatorname {sinc} (z)}}}$

where sinc is the normalized sinc function, while the multiplication theorem takes on the form

${\displaystyle \Pi \left({\frac {z}{m}}\right)\,\Pi \left({\frac {z-1}{m}}\right)\cdots \Pi \left({\frac {z-m+1}{m}}\right)=(2\pi )^{\frac {m-1}{2}}m^{-z-{\frac {1}{2}}}\Pi (z).}$

We also sometimes find

${\displaystyle \pi (z)={\frac {1}{\Pi (z)}},}$

which is an entire function, defined for every complex number, just like the reciprocal gamma function. That ${\displaystyle \pi \left(z\right)}$ is entire entails it has no poles, so ${\displaystyle \Pi \left(z\right)}$, like ${\displaystyle \Gamma \left(z\right)}$, has no zeros.

The volume of an n-ellipsoid with radii r₁, …, r_n can be expressed as

${\displaystyle V_{n}(r_{1},\dotsc ,r_{n})={\frac {\pi ^{\frac {n}{2}}}{\Pi \left({\frac {n}{2}}\right)}}\prod _{k=1}^{n}r_{k}.}$

• 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

${\displaystyle \mathrm {B} (x,y)=\int _{0}^{1}t^{x-1}(1-t)^{y-1}\,dt={\frac {\Gamma (x)\,\Gamma (y)}{\Gamma (x+y)}}.}$

• 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, ${\displaystyle \zeta (z)}$.

${\displaystyle \pi ^{-{\frac {z}{2}}}\;\Gamma \left({\frac {z}{2}}\right)\zeta (z)=\pi ^{-{\frac {1-z}{2}}}\;\Gamma \left({\frac {1-z}{2}}\right)\;\zeta (1-z).}$

It also appears in the following formula:

${\displaystyle \zeta (z)\Gamma (z)=\int _{0}^{\infty }{\frac {u^{z}}{e^{u}-1}}\,{\frac {du}{u}},}$

which is valid only for ${\displaystyle \Re (z)>1}$.

The logarithm of the gamma function satisfies the following formula due to Lerch:

${\displaystyle \log \Gamma (x)=\zeta _{H}'(0,x)-\zeta '(0),}$

where ${\displaystyle \zeta _{H}}$ is the Hurwitz zeta function, ${\displaystyle \zeta }$ is the Riemann zeta function and the prime (′) denotes differentiation in the first variable.

• The gamma function is related to the stretched exponential function. For instance, the moments of that function are

${\displaystyle \langle \tau ^{n}\rangle \equiv \int _{0}^{\infty }dt\,t^{n-1}\,e^{-\left({\frac {t}{\tau }}\right)^{\beta }}={\frac {\tau ^{n}}{\beta }}\Gamma \left({n \over \beta }\right).}$

Including up to the first 20 digits after the decimal point, some particular values of the gamma function are:

${\displaystyle {\begin{array}{rcccl}\Gamma \left(-{\tfrac {3}{2}}\right)&=&{\tfrac {4}{3}}{\sqrt {\pi }}&\approx &+2.36327\,18012\,07354\,70306\\\Gamma \left(-{\tfrac {1}{2}}\right)&=&-2{\sqrt {\pi }}&\approx &-3.54490\,77018\,11032\,05459\\\Gamma \left({\tfrac {1}{2}}\right)&=&{\sqrt {\pi }}&\approx &+1.77245\,38509\,05516\,02729\\\Gamma (1)&=&0!&=&+1\\\Gamma \left({\tfrac {3}{2}}\right)&=&{\tfrac {1}{2}}{\sqrt {\pi }}&\approx &+0.88622\,69254\,52758\,01364\\\Gamma (2)&=&1!&=&+1\\\Gamma \left({\tfrac {5}{2}}\right)&=&{\tfrac {3}{4}}{\sqrt {\pi }}&\approx &+1.32934\,03881\,79137\,02047\\\Gamma (3)&=&2!&=&+2\\\Gamma \left({\tfrac {7}{2}}\right)&=&{\tfrac {15}{8}}{\sqrt {\pi }}&\approx &+3.32335\,09704\,47842\,55118\\\Gamma (4)&=&3!&=&+6\end{array}}}$

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):

${\displaystyle {\frac {1}{\Gamma (-3)}}={\frac {1}{\Gamma (-2)}}={\frac {1}{\Gamma (-1)}}={\frac {1}{\Gamma (0)}}=0.}$