Gamma function

In mathematics, the gamma function (represented by Γ, capital Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function <math>\Gamma(z)</math> is defined for all complex numbers <math>z</math> except non-positive integers, and for every positive integer <math>z=n</math>, <math display="block">\Gamma(n) = (n-1)!\,.</math>The gamma function can be defined via a convergent improper integral for complex numbers with positive real part:

<math display="block"> \Gamma(z) = \int_0^\infty t^{z-1} e^{-t}\text{ d}t, \ \qquad \Re(z) > 0\,.</math>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.

The gamma function has no zeros, so the reciprocal gamma function Template:Math is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential function:

<math display="block"> \Gamma(z) = \mathcal M \{e^{-x} \} (z)\,.</math>

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

File:Generalized factorial function more infos.svg

<math>\Gamma(x+1)</math> interpolates the factorial function to non-integer values.

The gamma function can be seen as a solution to the interpolation problem of finding a smooth curve <math>y=f(x)</math> that connects the points of the factorial sequence: <math>(x,y) = (n, n!) </math> for all positive integer values of <math>n</math>. The simple formula for the factorial, Template:Math is only valid when Template:Mvar is a positive integer, and no elementary function has this property, but a good solution is the gamma function <math>f(x) = \Gamma(x+1) </math>.<ref name="Davis" />

The gamma function is not only smooth but analytic (except at the non-positive integers), 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 <math>k\sin(m\pi x)</math> for an integer <math>m</math>.<ref name="Davis" /> Such a function is known as a pseudogamma function, the most famous being the Hadamard function.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

File:Gamma plus sin pi z.svg

The gamma function, Template:Math in blue, plotted along with Template:Math in green. Notice the intersection at positive integers. Both are valid extensions of the factorials to a meromorphic function on the complex plane.

A more restrictive requirement is the functional equation which interpolates the shifted factorial <math>f(n) = (n{-}1)! </math> :<ref>Template:Cite book Extract of page 28</ref><ref>Template:Cite book Expression G.2 on page 293</ref> <math display="block">f(x+1) = x f(x)\ \text{ for all } x>0, \qquad f(1) = 1.</math>

But this still does not give a unique solution, since it allows for multiplication by any periodic function <math>g(x)</math> with <math>g(x) = g(x+1)</math> and <math>g(0)=1</math>, such as <math>g(x) = e^{k\sin(m\pi x)}</math>.

One way to resolve the ambiguity is the Bohr–Mollerup theorem, which shows that <math>f(x) = \Gamma(x)</math> is the unique interpolating function for the factorial, defined over the positive reals, which is logarithmically convex,<ref name="Kingman1961">Template:Cite journal</ref> meaning that <math>y = \log f(x) </math> is convex.<ref>{{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web |_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:Bohr-MollerupTheorem%7CBohr-MollerupTheorem.html}} |title = Bohr–Mollerup Theorem |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}</ref>

Definition

Main definition

The notation <math>\Gamma (z)</math> is due to Legendre.<ref name="Davis" /> If the real part of the complex number Template:Mvar is strictly positive (<math>\Re (z) > 0</math>), then the integral <math display="block"> \Gamma(z) = \int_0^\infty t^{z-1} e^{-t}\, dt</math> converges absolutely, and is known as the Euler integral of the second kind. (Euler's integral of the first kind is the beta function.<ref name="Davis" />) Using integration by parts, one sees that:

File:Plot of gamma function in complex plane in 3D with color and legend and 1000 plot points created with Mathematica.svg

Absolute value (vertical) and argument (color) of the gamma function on the complex plane

<math display="block">\begin{align}

  \Gamma(z+1) & = \int_0^\infty t^{z} e^{-t} \, dt \\

&= \Bigl[-t^z e^{-t}\Bigr]_0^\infty + \int_0^\infty z t^{z-1} e^{-t}\, dt \\ &= \lim_{t\to \infty}\left(-t^z e^{-t}\right) - \left(-0^z e^{-0}\right) + z\int_0^\infty t^{z-1} e^{-t}\, dt. \end{align}</math>

Recognizing that <math>-t^z e^{-t}\to 0</math> as <math>t\to \infty,</math> <math display="block">\begin{align}

  \Gamma(z+1) & = z\int_0^\infty t^{z-1} e^{-t}\, dt \\
&= z\Gamma(z).

\end{align}</math>

Then Template:Nowrap can be calculated as: <math display="block">\begin{align}

  \Gamma(1) & = \int_0^\infty t^{1-1} e^{-t}\,dt \\
  & = \int_0^\infty  e^{-t} \, dt \\
  & = 1.

\end{align}</math>

Thus we can show that <math>\Gamma(n) = (n-1)!</math> for any positive integer Template:Mvar by induction. Specifically, the base case is that <math>\Gamma(1) = 1 = 0!</math>, and the induction step is that <math>\Gamma(n+1) = n\Gamma(n) = n(n-1)! = n!.</math>

The identity <math display="inline">\Gamma(z) = \frac {\Gamma(z + 1)} {z}</math> can be used (or, yielding the same result, analytic continuation can be used) to uniquely extend the integral formulation for <math>\Gamma (z)</math> to a meromorphic function defined for all complex numbers Template:Mvar, except integers less than or equal to zero.<ref name="Davis" /> It is this extended version that is commonly referred to as the gamma function.<ref name="Davis" />

Alternative definitions

There are many equivalent definitions.

Euler's definition as an infinite product

For a fixed integer <math>m</math>, as the integer <math>n</math> increases, we have that<ref>Template:Cite journal</ref> <math display="block">\lim_{n \to \infty} \frac{n! \, \left(n+1\right)^m}{(n+m)!} = 1\,.</math>

If <math>m</math> 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, let us assume that this equation continues to hold when <math>m</math> is replaced by an arbitrary complex number <math>z</math>, in order to define the Gamma function for non-integers:

<math display="block">\lim_{n \to \infty} \frac{n! \, \left(n+1\right)^z}{(n+z)!} = 1\,.</math> Multiplying both sides by <math>(z-1)!</math> gives <math display="block">\begin{align} (z-1)!

     &= \frac{1}{z} \lim_{n \to \infty} n!\frac{z!}{(n+z)!} (n+1)^z \\[8pt]
     &= \frac{1}{z} \lim_{n \to \infty} (1 \cdot2\cdots n)\frac{1}{(1+z) \cdots (n+z)} \left(\frac{2}{1} \cdot \frac{3}{2} \cdots \frac{n+1}{n}\right)^z \\[8pt]
     &= \frac{1}{z} \prod_{n=1}^\infty \left[ \frac{1}{1+\frac{z}{n}} \left(1 + \frac{1}{n}\right)^z \right].

\end{align}</math>This infinite product, which is due to Euler,<ref>Template:Cite journal</ref> converges for all complex numbers <math>z</math> except the non-positive integers, which fail because of a division by zero. In fact, the above assumption produces a unique definition of <math>\Gamma(z)</math> as Template:Tmath.

Intuitively, this formula indicates that <math>\Gamma(z)</math> is approximately the result of computing <math>\Gamma(n+1)=n!</math> for some large integer <math>n</math>, multiplying by <math>(n+1)^z</math> to approximate <math>\Gamma(n+z+1)</math>, and then using the relationship <math>\Gamma(x+1) = x \Gamma(x)</math> backwards <math>n+1</math> times to get an approximation for <math>\Gamma(z)</math>; and furthermore that this approximation becomes exact as <math>n</math> increases to infinity.

The infinite product for the reciprocal <math display="block">\frac{1}{\Gamma(z)} = z \prod_{n=1}^\infty \left[ \left(1+\frac{z}{n}\right) / {\left(1 + \frac{1}{n}\right)^z} \right]</math> is an entire function, converging for every complex number Template:Mvar.

Weierstrass's definition

The definition for the gamma function due to Weierstrass is also valid for all complex numbers <math>z</math> except non-positive integers: <math display="block">\Gamma(z) = \frac{e^{-\gamma z}} z \prod_{n=1}^\infty \left(1 + \frac z n \right)^{-1} e^{z/n},</math> where <math>\gamma \approx 0.577216</math> is the Euler–Mascheroni constant.<ref name="Davis" /> This is the Hadamard product of <math>1/\Gamma(z)</math> in a rewritten form.

Template:Collapse top Equivalence of the integral definition and Weierstrass definition

By the integral definition, the relation <math>\Gamma (z+1)=z\Gamma (z)</math> and Hadamard factorization theorem, <math display="block>\frac{1}{\Gamma (z)}=ze^{c_1 z+c_2}\prod_{n=1}^\infty e^{-\frac{z}{n}}\left(1+\frac{z}{n}\right)</math> for some constants <math>c_1,c_2</math> since <math>1/\Gamma</math> is an entire function of order <math>1</math>. Since <math>z\Gamma (z)\to 1</math> as <math>z\to 0</math>, <math>c_2=0</math> (or an integer multiple of <math>2\pi i</math>) and since <math>\Gamma (1)=1</math>, <math display="block">\begin{align}e^{-c_1} &=\prod_{n=1}^\infty e^{-\frac{1}{n}}\left(1+\frac{1}{n}\right)\\ &=\exp\left(\lim_{N\to\infty}\sum_{n=1}^N \left(\log\left(1+\frac{1}{n}\right)-\frac{1}{n}\right)\right)\\ &=\exp\left(\lim_{N\to\infty}\left(\log (N+1)-\sum_{n=1}^N \frac{1}{n}\right)\right).\end{align}</math>

where <math>c_1=\gamma+2\pi i k</math> for some integer <math>k</math>. Since <math>\Gamma (z)\in\mathbb{R}</math> for <math>z\in\mathbb{R}\setminus\mathbb{Z}_0^-</math>, we have <math>k=0</math> and <math display="block>\frac{1}{\Gamma (z)}=ze^{\gamma z}\prod_{n=1}^\infty e^{-\frac{z}{n}}\left(1+\frac{z}{n}\right)</math>

Equivalence of the Weierstrass definition and Euler definition

<math display="block">\begin{align}\Gamma (z)&=\frac{e^{-\gamma z}}{z}\prod_{n=1}^{\infty}\left(1+\frac{z}{n}\right)^{-1}e^{z/n}\\ &=\frac1z\lim_{n\to\infty}e^{z\left(\log (n+1)-1-\frac{1}{2}-\frac{1}{3}-\cdots-\frac{1}{n}\right)}\frac{e^{z\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}\right)}}{\left(1+z\right)\left(1+\frac{z}{2}\right)\cdots\left(1+\frac{z}{n}\right)}\\ &=\frac1z\lim_{n\to\infty}\frac{1}{\left(1+z\right)\left(1+\frac{z}{2}\right)\cdots\left(1+\frac{z}{n}\right)}e^{z\log\left(n+1\right)}\\ &=\lim_{n\to\infty}\frac{n!(n+1)^z}{z(z+1)\cdots (z+n)},\quad z\in\mathbb{C}\setminus\mathbb{Z}_0^-\end{align}</math> Template:Collapse bottom

Properties

General

Besides the fundamental property discussed above: <math display="block">\Gamma(z+1) = z\ \Gamma(z)</math> other important functional equations for the gamma function are Euler's reflection formula <math display="block">\Gamma(1-z) \Gamma(z) = \frac{\pi}{\sin \pi z}, \qquad z \not\in \Z</math> which implies <math display="block">\Gamma(z - n) = (-1)^{n-1} \; \frac{\Gamma(-z) \Gamma(1+z)}{\Gamma(n+1-z)}, \qquad n \in \Z</math> and the Legendre duplication formula <math display="block">\Gamma(z) \Gamma\left(z + \tfrac12\right) = 2^{1-2z} \; \sqrt{\pi} \; \Gamma(2z).</math>

Template:Collapse top Proof 1

With Euler's infinite product <math display=block>\Gamma(z) = \frac1z \prod_{n=1}^{\infty} \frac{(1+1/n)^z}{1 + z/n}</math> compute <math display=block>\frac{1}{\Gamma(1-z)\Gamma(z)} = \frac{1}{(-z)\Gamma(-z)\Gamma(z)} = z \prod_{n=1}^{\infty} \frac{(1-z/n)(1+z/n)}{(1+1/n)^{-z}(1+1/n)^{z}} = z \prod_{n=1}^{\infty} \left(1 - \frac{z^2}{n^2}\right) = \frac{\sin \pi z}{\pi}\,,</math> where the last equality is a known result. A similar derivation begins with Weierstrass's definition.

Proof 2

First prove that <math display="block">I=\int_{-\infty}^\infty \frac{e^{ax}}{1+e^x}\, dx=\int_0^\infty \frac{v^{a-1}}{1+v}\, dv=\frac{\pi}{\sin\pi a},\quad a\in (0,1).</math> Consider the positively oriented rectangular contour <math>C_R</math> with vertices at <math>R</math>, <math>-R</math>, <math>R+2\pi i</math> and <math>-R+2\pi i</math> where <math>R\in\mathbb{R}^+</math>. Then by the residue theorem, <math display="block">\int_{C_R}\frac{e^{az}}{1+e^z}\, dz=-2\pi ie^{a\pi i}.</math> Let <math display="block">I_R=\int_{-R}^R \frac{e^{ax}}{1+e^x}\, dx</math> and let <math>I_R'</math> be the analogous integral over the top side of the rectangle. Then <math>I_R\to I</math> as <math>R\to\infty</math> and <math>I_R'=-e^{2\pi i a}I_R</math>. If <math>A_R</math> denotes the right vertical side of the rectangle, then <math display="block">\left|\int_{A_R} \frac{e^{az}}{1+e^z}\, dz\right|\le \int_0^{2\pi}\left|\frac{e^{a(R+it)}}{1+e^{R+it}}\right|\, dt\le Ce^{(a-1)R}</math> for some constant <math>C</math> and since <math>a<1</math>, the integral tends to <math>0</math> as <math>R\to\infty</math>. Analogously, the integral over the left vertical side of the rectangle tends to <math>0</math> as <math>R\to\infty</math>. Therefore <math display="block">I-e^{2\pi ia}I=-2\pi ie^{a\pi i},</math> from which <math display="block">I=\frac{\pi}{\sin \pi a},\quad a\in (0,1).</math> Then <math display="block">\Gamma (1-z)=\int_0^\infty e^{-u}u^{-z}\, du=t\int_0^\infty e^{-vt}(vt)^{-z}\, dv,\quad t>0</math> and <math display="block">\begin{align}\Gamma (z)\Gamma (1-z)&=\int_0^\infty\int_0^\infty e^{-t(1+v)}v^{-z}\, dv\, dt\\ &=\int_0^\infty \frac{v^{-z}}{1+v}\, dv\\&=\frac{\pi}{\sin \pi (1-z)}\\&=\frac{\pi}{\sin \pi z},\quad z\in (0,1).\end{align}</math> Proving the reflection formula for all <math>z\in (0,1)</math> proves it for all <math>z\in\mathbb{C}\setminus\mathbb{Z}</math> by analytic continuation. Template:Collapse bottom

The beta function can be represented as <math display="block">\Beta (z_1,z_2)=\frac{\Gamma (z_1)\Gamma (z_2)}{\Gamma (z_1+z_2)}=\int_0^1 t^{z_1-1}(1-t)^{z_2-1} \, dt.</math>

Setting <math>z_1=z_2=z</math> yields <math display="block">\frac{\Gamma^2(z)}{\Gamma (2z)}=\int_0^1 t^{z-1}(1-t)^{z-1} \, dt.</math>

After the substitution <math>t=\frac{1+u}{2}</math>: <math display="block">\frac{\Gamma^2(z)}{\Gamma (2z)}=\frac{1}{2^{2z-1}}\int_{-1}^1 \left(1-u^{2}\right)^{z-1} \, du.</math>

The function <math>(1-u^2)^{z-1}</math> is even, hence <math display="block">2^{2z-1}\Gamma^2(z)=2\Gamma (2z)\int_0^1 (1-u^2)^{z-1} \, du.</math>

Now <math display="block">\Beta \left(\frac{1}{2},z\right)=\int_0^1 t^{\frac{1}{2}-1}(1-t)^{z-1} \, dt, \quad t=s^2.</math>

Then <math display="block">\Beta \left(\frac{1}{2},z\right)=2\int_0^1 (1-s^2)^{z-1} \, ds = 2\int_0^1 (1-u^2)^{z-1} \, du.</math>

This implies <math display="block">2^{2z-1}\Gamma^2(z)=\Gamma (2z)\Beta \left(\frac{1}{2},z\right).</math>

Since <math display="block">\Beta \left(\frac{1}{2},z\right)=\frac{\Gamma \left(\frac{1}{2}\right)\Gamma (z)}{\Gamma \left(z+\frac{1}{2}\right)}, \quad \Gamma \left(\frac{1}{2}\right)=\sqrt{\pi},</math> the Legendre duplication formula follows: <math display="block">\Gamma (z)\Gamma \left(z+\frac{1}{2}\right)=2^{1-2z}\sqrt{\pi} \; \Gamma (2z).</math>

The duplication formula is a special case of the multiplication theorem (see <ref name="ReferenceA">Template:Dlmf</ref> Eq. 5.5.6): <math display="block">\prod_{k=0}^{m-1}\Gamma\left(z + \frac{k}{m}\right) = (2 \pi)^{\frac{m-1}{2}} \; m^{\frac12 - mz} \; \Gamma(mz).</math>

A simple but useful property, which can be seen from the limit definition, is: <math display="block">\overline{\Gamma(z)} = \Gamma(\overline{z}) \; \Rightarrow \; \Gamma(z)\Gamma(\overline{z}) \in \mathbb{R} .</math>

In particular, with Template:Math, this product is <math display="block">|\Gamma(a+bi)|^2 = |\Gamma(a)|^2 \prod_{k=0}^\infty \frac{1}{1+\frac{b^2}{(a+k)^2}}</math>

If the real part is an integer or a half-integer, this can be finitely expressed in closed form: <math display="block"> \begin{align} |\Gamma(bi)|^2 & = \frac{\pi}{b\sinh \pi b} \\[1ex] \left|\Gamma\left(\tfrac{1}{2}+bi\right)\right|^2 & = \frac{\pi}{\cosh \pi b} \\[1ex] \left|\Gamma\left(1+bi\right)\right|^2 & = \frac{\pi b}{\sinh \pi b} \\[1ex] \left|\Gamma\left(1+n+bi\right)\right|^2 & = \frac{\pi b}{\sinh \pi b} \prod_{k=1}^n \left(k^2 + b^2 \right), \quad n \in \N \\[1ex] \left|\Gamma\left(-n+bi\right)\right|^2 & = \frac{\pi}{b \sinh \pi b} \prod_{k=1}^n \left(k^2 + b^2 \right)^{-1}, \quad n \in \N \\[1ex] \left|\Gamma\left(\tfrac{1}{2} \pm n+bi\right)\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 \N \\[-1ex]& \end{align} </math>

First, consider the reflection formula applied to <math>z=bi</math>. <math display="block">\Gamma(bi)\Gamma(1-bi)=\frac{\pi}{\sin \pi bi}</math> Applying the recurrence relation to the second term: <math display="block">-bi \cdot \Gamma(bi)\Gamma(-bi)=\frac{\pi}{\sin \pi bi}</math> which with simple rearrangement gives <math display="block">\Gamma(bi)\Gamma(-bi)=\frac{\pi}{-bi\sin \pi bi}=\frac{\pi}{b\sinh \pi b}</math>

Second, consider the reflection formula applied to <math>z=\tfrac{1}{2}+bi</math>. <math display="block">\Gamma(\tfrac{1}{2}+bi)\Gamma\left(1-(\tfrac{1}{2}+bi)\right)=\Gamma(\tfrac{1}{2}+bi)\Gamma(\tfrac{1}{2}-bi)=\frac{\pi}{\sin \pi (\tfrac{1}{2}+bi)}=\frac{\pi}{\cos \pi bi}=\frac{\pi}{\cosh \pi b}</math>

Formulas for other values of <math>z</math> 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 <math display="block">\Gamma\left(\tfrac12\right)=\sqrt{\pi},</math> which can be found by setting <math display="inline">z = \frac{1}{2}</math> in the reflection formula, by using the relation to the beta function given below with <math display="inline">z_1 = z_2 = \frac{1}{2}</math>, or simply by making the substitution <math>t = u^2</math> in the integral definition of the gamma function, resulting in a Gaussian integral. In general, for non-negative integer values of <math>n</math> we have: <math display="block">\begin{align} \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{align}</math> where the double factorial <math>(2n-1)!! = (2n-1)(2n-3)\cdots(3)(1)</math>. See Particular values of the gamma function for calculated values.

It might be tempting to generalize the result that <math display="inline">\Gamma \left( \frac{1}{2} \right) = \sqrt\pi</math> by looking for a formula for other individual values <math>\Gamma(r)</math> where <math>r</math> 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 <math>\Gamma(r)</math> are not known to be expressible by themselves in terms of elementary functions. It has been proved that <math>\Gamma (n + r)</math> is a transcendental number and algebraically independent of <math>\pi</math> for any integer <math>n</math> and each of the fractions <math display="inline">r = \frac{1}{6}, \frac{1}{4}, \frac{1}{3}, \frac{2}{3}, \frac{3}{4}, \frac{5}{6}</math>.<ref>Template:Cite journal</ref> 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, Template:Math: <math display="block">\Gamma'(z)=\Gamma(z)\psi^{(0)}(z).</math> For a positive integer Template:Mvar the derivative of the gamma function can be calculated as follows:

File:Plot of gamma function in the complex plane from -2-i to 6+2i with colors created in Mathematica.svg

Colors showing the argument of the gamma function in the complex plane from Template:Math to Template:Math

<math display="block">\Gamma'(m+1) = m! \left( - \gamma + \sum_{k=1}^m\frac{1}{k} \right)= m! \left( - \gamma + H(m) \right)\,,</math> where H(m) is the mth harmonic number and Template:Math is the Euler–Mascheroni constant.

For <math>\Re(z) > 0</math> the <math>n</math>th derivative of the gamma function is: <math display="block">\frac{d^n}{dz^n}\Gamma(z) = \int_0^\infty t^{z-1} e^{-t} (\log t)^n \, dt.</math> (This can be derived by differentiating the integral form of the gamma function with respect to <math>z</math>, and using the technique of differentiation under the integral sign.)

Using the identity <math display="block">\Gamma^{(n)}(1)=(-1)^n B_n(\gamma, 1! \zeta(2), \ldots, (n-1)! \zeta(n))</math> where <math>\zeta(z)</math> is the Riemann zeta function, and <math>B_n</math> is the <math>n</math>-th Bell polynomial, we have in particular the Laurent series expansion of the gamma function <ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> <math display="block">\Gamma(z) = \frac1z - \gamma + \frac12\left(\gamma^2 + \frac{\pi^2}6\right)z - \frac16\left(\gamma^3 + \frac{\gamma\pi^2}2 + 2 \zeta(3)\right)z^2 + O(z^3).</math>

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 <math>x_1</math> and <math>x_2</math>, and for any <math>t \in [0, 1]</math>, <math display="block">\Gamma(tx_1 + (1 - t)x_2) \le \Gamma(x_1)^t\Gamma(x_2)^{1 - t}.</math>
• For any two positive real numbers <math>x_1</math> and <math>x_2</math>, and <math>x_2</math> > <math>x_1</math><math display="block"> \left(\frac{\Gamma(x_2)}{\Gamma(x_1)}\right)^{\frac{1}{x_2 - x_1}} > \exp\left(\frac{\Gamma'(x_1)}{\Gamma(x_1)}\right).</math>
• For any positive real number <math>x</math>, <math display="block"> \Gamma(x) \Gamma(x) > \Gamma'(x)^2.</math>

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

Logarithmic convexity and Jensen's inequality together imply, for any positive real numbers <math>x_1, \ldots, x_n</math> and <math>a_1, \ldots, a_n</math>, <math display="block">\Gamma\left(\frac{a_1x_1 + \cdots + a_nx_n}{a_1 + \cdots + a_n}\right) \le \bigl(\Gamma(x_1)^{a_1} \cdots \Gamma(x_n)^{a_n}\bigr)^{\frac{1}{a_1 + \cdots + a_n}}.</math>

There are also bounds on ratios of gamma functions. The best-known is Gautschi's inequality, which says that for any positive real number Template:Mvar and any Template:Math, <math display="block">x^{1 - s} < \frac{\Gamma(x + 1)}{\Gamma(x + s)} < \left(x + 1\right)^{1 - s}.</math>

Stirling's formula

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

File:Gamma cplot.svg

</math> is also displayed.

File:Gamma abs 3D.png

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

The behavior of <math>\Gamma(x)</math> for an increasing positive real variable is given by Stirling's formula <math display="block">\Gamma(x+1)\sim\sqrt{2\pi x}\left(\frac{x}{e}\right)^x,</math> where the symbol <math>\sim</math> means asymptotic convergence: the ratio of the two sides converges to 1 in the limit Template:Nowrap This growth is faster than exponential, <math>\exp(\beta x)</math>, for any fixed value of <math>\beta</math>.

Another useful limit for asymptotic approximations for <math>x \to + \infty</math> is: <math display="block"> {\Gamma(x+\alpha)}\sim{\Gamma(x)x^\alpha}, \qquad \alpha \in \Complex. </math>

When writing the error term as an infinite product, Stirling's formula can be used to define the gamma function: <ref>Template:Cite book</ref> <math display="block"> \Gamma(x) = \sqrt{\frac{2\pi}{x}} \left(\frac{x}{e}\right)^x \prod_{n=0}^{\infty} \left[\frac{1}{e}\left(1+\frac{1}{x+n}\right)^{x+n+\frac{1}{2}} \right]</math>

Extension to negative, non-integer values

Although the main definition of the gamma function—the Euler integral of the second kind—is only valid (on the real axis) for positive arguments, its domain can be extended with analytic continuation<ref>Template:Cite book</ref> to negative arguments by shifting the negative argument to positive values by using either the Euler's reflection formula, <math display="block"> \Gamma(-x) = \frac{1}{\Gamma(x+1)}\frac{\pi}{\sin\big(\pi(x+1)\big)}, </math> or the fundamental property, <math display="block"> \Gamma(-x):=\frac1{-x}\Gamma(-x+1) , </math> when <math>x\not\in\mathbb{Z}</math>. For example, <math display="block"> \Gamma\left(-\frac12\right)=-2\Gamma\left(\frac12\right) . </math>

Residues

The behavior for non-positive <math>z</math> is more intricate. Euler's integral does not converge for Template:Nowrap 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,<ref name="Davis" /> <math display="block">\Gamma(z)=\frac{\Gamma(z+n+1)}{z(z+1)\cdots(z+n)},</math> choosing <math>n</math> such that <math>z + n</math> is positive. The product in the denominator is zero when <math>z</math> equals any of the integers <math>0, -1, -2, \ldots</math>. 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.<ref name="Davis" />

For a function <math>f</math> of a complex variable <math>z</math>, at a simple pole <math>c</math>, the residue of <math>f</math> is given by: <math display="block">\operatorname{Res}(f,c)=\lim_{z\to c}(z-c)f(z).</math>

For the simple pole <math>z = -n</math>, the recurrence formula can be rewritten as: <math display="block">(z+n) \Gamma(z)=\frac{\Gamma(z+n+1)}{z(z+1)\cdots(z+n-1)}.</math> The numerator at <math>z = -n,</math> is <math display="block">\Gamma(z+n+1) = \Gamma(1) = 1</math> and the denominator <math display="block">z(z+1)\cdots(z+n-1) = -n(1-n)\cdots(n-1-n) = (-1)^n n!.</math> So the residues of the gamma function at those points are:<ref name="Mathworld">{{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web |_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:GammaFunction%7CGammaFunction.html}} |title = Gamma Function |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}</ref> <math display="block">\operatorname{Res}(\Gamma,-n)=\frac{(-1)^n}{n!}.</math>The gamma function is non-zero everywhere along the real line, although it comes arbitrarily close to zero as Template:Math. There is in fact no complex number <math>z</math> for which <math>\Gamma (z) = 0</math>, and hence the reciprocal gamma function <math display="inline">\frac {1}{\Gamma (z)}</math> is an entire function, with zeros at <math>z = 0, -1, -2, \ldots</math>.<ref name="Davis" />

Minima and maxima

On the real line, the gamma function has a local minimum at Template:Math<ref>Template:Cite OEIS</ref> where it attains the value Template:Math.<ref>Template:Cite OEIS</ref> The gamma function rises to either side of this minimum. The solution to Template:Math is Template:Math and the common value is Template:Math. The positive solution to Template:Math is Template:Math, the golden ratio, and the common value is Template:Math.<ref>Template:Cite OEIS</ref>

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 <math>z</math> and <math>z + n</math> is odd, and an even number if the number of poles is even.<ref name="Mathworld" /> The values at the local extrema of the gamma function along the real axis between the non-positive integers are:

Template:Math,

Template:Math,

Template:Math,

Template:Math,

Template:Math, 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 Template:Mvar is positive,<ref>Template:Cite book</ref> <math display="block">\Gamma (z)=\int_{-\infty}^\infty e^{zt-e^t}\, dt</math> and<ref>Whittaker and Watson, 12.2 example 1.</ref> <math display="block">\Gamma(z) = \int_0^1 \left(\log \frac{1}{t}\right)^{z-1}\,dt,</math> <math display="block">\Gamma(z) = 2c^z\int_{0}^{\infty}t^{2z-1}e^{-ct^{2}}\,dt \,,\; c>0</math> where the three integrals respectively follow from the substitutions <math>t=e^{-x}</math>, <math>t=-\log x</math> <ref>Template:Cite journal</ref> and <math>t=cx^2</math><ref>Template:Cite journal</ref> 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 <math>z=1/2,\; c=1</math> we get <math display="block"> \Gamma(1/2)=2\int_{0}^{\infty}e^{-t^{2}}\,dt=\sqrt{\pi} \;. </math>

Binet's first integral formula for the gamma function states that, when the real part of Template:Mvar is positive, then:<ref>Whittaker and Watson, 12.31.</ref> <math display="block">\operatorname{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.</math> The integral on the right-hand side may be interpreted as a Laplace transform. That is, <math display="block">\log\left(\Gamma(z)\left(\frac{e}{z}\right)^z\sqrt{\frac{z}{2\pi}}\right) = \mathcal{L}\left(\frac{1}{2t} - \frac{1}{t^2} + \frac{1}{t(e^t - 1)}\right)(z).</math>

Binet's second integral formula states that, again when the real part of Template:Mvar is positive, then:<ref>Whittaker and Watson, 12.32.</ref> <math display="block">\operatorname{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.</math>

Let Template:Math be a Hankel contour, meaning a path that begins and ends at the point Template:Math on the Riemann sphere, whose unit tangent vector converges to Template:Math at the start of the path and to Template:Math at the end, which has winding number 1 around Template:Math, and which does not cross Template:Closed-open. Fix a branch of <math>\log(-t)</math> by taking a branch cut along Template:Closed-open and by taking <math>\log(-t)</math> to be real when Template:Math is on the negative real axis. Assume Template:Mvar is not an integer. Then Hankel's formula for the gamma function is:<ref>Whittaker and Watson, 12.22.</ref> <math display="block">\Gamma(z) = -\frac{1}{2i\sin \pi z}\int_C (-t)^{z-1}e^{-t}\,dt,</math> where <math>(-t)^{z-1}</math> is interpreted as <math>\exp((z-1)\log(-t))</math>. The reflection formula leads to the closely related expression <math display="block">\frac{1}{\Gamma(z)} = \frac{i}{2\pi}\int_C (-t)^{-z}e^{-t}\,dt,</math> again valid whenever Template:Math is not an integer.

Continued fraction representation

The gamma function can also be represented by a sum of two continued fractions:<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> <math display="block">\begin{aligned}

   \Gamma (z) &= \cfrac{e^{-1}}{
       2 + 0 - z + 1\cfrac{z-1}{
           2 + 2 - z + 2\cfrac{z-2}{
               2 + 4 - z + 3\cfrac{z-3}{
                   2 + 6 - z + 4\cfrac{z-4}{
                       2 + 8 - z + 5\cfrac{z-5}{
                           2 + 10 - z + \ddots
                       }
                   }
               }
           }
       }
   } \\
   &+\ \cfrac{e^{-1}}{
       z + 0 - \cfrac{z+0}{
           z + 1 + \cfrac{1}{
               z + 2 - \cfrac{z+1}{
                   z + 3 + \cfrac{2}{
                       z + 4 - \cfrac{z+2}{
                           z + 5 + \cfrac{3}{
                               z + 6 - \ddots
                           }
                       }
                   }
               }
           }
       }
   }

\end{aligned}</math> where <math>z\in\mathbb{C}</math>.

Fourier series expansion

The logarithm of the gamma function has the following Fourier series expansion for <math>0 < z < 1:</math> <math display="block">\operatorname{log\Gamma}(z) = \left(\frac{1}{2} - z\right)(\gamma + \log 2) + (1 - z)\log\pi - \frac{1}{2}\log\sin(\pi z) + \frac{1}{\pi}\sum_{n=1}^\infty \frac{\log n}{n} \sin (2\pi n z),</math> which was for a long time attributed to Ernst Kummer, who derived it in 1847.<ref>Template:Cite book</ref><ref>Template:Cite book</ref> However, Iaroslav Blagouchine discovered that Carl Johan Malmsten first derived this series in 1842.<ref name="iaroslav_06">Template:Cite journal</ref><ref name="iaroslav_06bis">Template:Cite journal</ref>

Raabe's formula

In 1840 Joseph Ludwig Raabe proved that <math display="block">\int_a^{a+1}\log\Gamma(z)\, dz = \tfrac12\log2\pi + a\log a - a,\quad a>0.</math> In particular, if <math>a = 0</math> then <math display="block">\int_0^1\log\Gamma(z)\, dz = \tfrac12\log2\pi.</math>

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 <math>a \to \infty</math> gives the formula.

Pi function

An alternative notation introduced by Gauss is the <math>\Pi</math>-function, a shifted version of the gamma function: <math display="block">\Pi(z) = \Gamma(z+1) = z \Gamma(z) = \int_0^\infty e^{-t} t^z\, dt,</math> so that <math>\Pi(n) = n!</math> for every non-negative integer <math>n</math>.

Using the pi function, the reflection formula is: <math display="block">\Pi(z) \Pi(-z) = \frac{\pi z}{\sin( \pi z)} = \frac{1}{\operatorname{sinc}(z)}</math> using the normalized sinc function; while the multiplication theorem becomes: <math display="block">\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-\frac12} \Pi(z)\ .</math>

The shifted reciprocal gamma function is sometimes denoted <math display="inline">\pi(z) = \frac{1}{\Pi(z)}\ ,</math> an entire function.

The [[volume of an n-ball|volume of an Template:Math-ellipsoid]] with radii Template:Math can be expressed as <math display="block">V_n(r_1,\dotsc,r_n)=\frac{\pi^{\frac{n}{2}}}{\Pi\left(\frac{n}{2}\right)} \prod_{k=1}^n r_k.</math>

Relation to other functions

• In the first integral defining the gamma function, the limits of integration are fixed. The upper incomplete gamma function is obtained by allowing the lower limit of integration to vary:<math display="block">\Gamma(z,x) = \int_x^\infty t^{z-1} e^{-t} dt.</math>There is a similar lower incomplete gamma function.
• The gamma function is related to Euler's beta function by the formula <math display="block">\Beta(z_1,z_2) = \int_0^1 t^{z_1-1}(1-t)^{z_2-1}\,dt = \frac{\Gamma(z_1)\,\Gamma(z_2)}{\Gamma(z_1+z_2)}.</math>
• 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, <math>\zeta (z)</math>. <math display="block">\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).</math> It also appears in the following formula: <math display="block">\zeta(z) \Gamma(z) = \int_0^\infty \frac{u^{z}}{e^u - 1} \, \frac{du}{u},</math> which is valid only for <math>\Re (z) > 1</math>.Template:Pb The logarithm of the gamma function satisfies the following formula due to Lerch: <math display="block">\operatorname{log\Gamma}(z) = \zeta_H'(0,z) - \zeta'(0),</math> where <math>\zeta_H</math> is the Hurwitz zeta function, <math>\zeta</math> is the Riemann zeta function and the prime (Template:Math) denotes differentiation in the first variable.
• The gamma function is related to the stretched exponential function. For instance, the moments of that function are <math display="block">\langle\tau^n\rangle \equiv \int_0^\infty t^{n-1}\, e^{ - \left( \frac{t}{\tau} \right)^\beta} \, \mathrm{d}t = \frac{\tau^n}{\beta}\Gamma \left({n \over \beta }\right).</math>

Particular values

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Including up to the first 20 digits after the decimal point, some particular values of the gamma function are: <math display="block">\begin{array}{rcccl} \Gamma\left(-\tfrac{3}{2}\right) &=& \tfrac{4\sqrt{\pi}}{3} &\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{\sqrt{\pi}}{2} &\approx& +0.88622\,69254\,52758\,01364 \\ \Gamma(2) &=& 1! &=& +1 \\ \Gamma\left(\tfrac{5}{2}\right) &=& \tfrac{3\sqrt{\pi}}{4} &\approx& +1.32934\,03881\,79137\,02047 \\ \Gamma(3) &=& 2! &=& +2 \\ \Gamma\left(\tfrac{7}{2}\right) &=& \tfrac{15\sqrt{\pi}}{8} &\approx& +3.32335\,09704\,47842\,55118 \\ \Gamma(4) &=& 3! &=& +6 \end{array}</math> (These numbers can be found in the OEIS.<ref>Template:Cite OEIS</ref><ref>Template:Cite OEIS</ref><ref>Template:Cite OEIS</ref><ref>Template:Cite OEIS</ref><ref>Template:Cite OEIS</ref><ref>Template:Cite OEIS</ref> 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 Template:Math. The reciprocal gamma function is well defined and analytic at these values (and in the entire complex plane): <math display="block">\frac{1}{\Gamma(-3)} = \frac{1}{\Gamma(-2)} = \frac{1}{\Gamma(-1)} = \frac{1}{\Gamma(0)} = 0.</math>

Log-gamma function

File:LogGamma Analytic Function.png

The analytic function Template:Math

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<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> <math display="block">\operatorname{log\Gamma}(z) = - \gamma z - \log z + \sum_{k = 1}^\infty \left[ \frac z k - \log \left( 1 + \frac z k \right) \right].</math>

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 <math display="block"> \operatorname{log\Gamma}(z) = \operatorname{log\Gamma}(z+1) - \log z</math>

File:Plot of logarithmic gamma function in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg

Logarithmic gamma function in the complex plane from −2 − 2i to 2 + 2i with colors

is often used since it allows one to determine function values in one strip of width 1 in Template:Mvar from the neighbouring strip. In particular, starting with a good approximation for a Template:Mvar with large real part one may go step by step down to the desired Template:Mvar. Following an indication of Carl Friedrich Gauss, Rocktaeschel (1922) proposed for Template:Math an approximation for large Template:Math: <math display="block"> \operatorname{log\Gamma}(z) \approx (z - \tfrac{1}{2}) \log z - z + \tfrac{1}{2}\log(2\pi).</math>

This can be used to accurately approximate Template:Math for Template:Mvar with a smaller Template:Math via (P.E.Böhmer, 1939) <math display="block"> \operatorname{log\Gamma}(z-m) = \operatorname{log\Gamma}(z) - \sum_{k=1}^m \log(z-k).</math>

A more accurate approximation can be obtained by using more terms from the asymptotic expansions of Template:Math and Template:Math, which are based on Stirling's approximation. <math display="block">\Gamma(z)\sim z^{z - \frac12} e^{-z} \sqrt{2\pi} \left( 1 + \frac{1}{12z} + \frac{1}{288z^2} - \frac{139}{51\,840 z^3} - \frac{571}{2\,488\,320 z^4} \right) </math>

as Template:Math at constant Template:Math. (See sequences A001163 and A001164 in the OEIS.)

In a more "natural" presentation: <math display="block">\operatorname{log\Gamma}(z) = z \log z - z - \tfrac12 \log z + \tfrac12 \log 2\pi + \frac{1}{12z} - \frac{1}{360z^3} +\frac{1}{1260 z^5} +o\left(\frac1{z^5}\right)</math>

as Template:Math at constant Template:Math. (See sequences A046968 and A046969 in the OEIS.)

The coefficients of the terms with Template:Math of Template:Math in the last expansion are simply <math display="block">\frac{B_k}{k(k-1)}</math> where the Template:Math are the Bernoulli numbers.

The gamma function also has Stirling Series (derived by Charles Hermite in 1900) equal to<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> <math display="block">\operatorname{log\Gamma}(1+x)=\frac{x(x-1)}{2!} \log(2)+\frac{x(x-1)(x-2)}{3!} (\log(3)-2\log(2))+\cdots,\quad\Re (x)> 0.</math>

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. <math>\Gamma(1) = 1</math>,
2. <math>\Gamma(z+1) = z \Gamma(z)</math> for all complex numbers <math>z</math> except the non-positive integers, and,
3. for integer Template:Mvar, <math display="inline">\lim_{n \to \infty} \frac{\Gamma(n+z)}{\Gamma(n)\;n^z} = 1</math> for all complex numbers <math>z</math>.<ref name="Davis" />

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 Template:Math around 1: <math display="block">\operatorname{log\Gamma}(z+1)= -\gamma z +\sum_{k=2}^\infty \frac{\zeta(k)}{k} \, (-z)^k \qquad \forall\; |z| < 1</math> with Template:Math denoting the Riemann zeta function at Template:Mvar.

So, using the following property: <math display="block">\zeta(s) \Gamma(s) = \int_0^\infty \frac{t^s}{e^t-1} \, \frac{dt}{t}</math> an integral representation for the log-gamma function is: <math display="block">\operatorname{log\Gamma}(z+1)= -\gamma z + \int_0^\infty \frac{e^{-zt} - 1 + z t}{t \left(e^t - 1\right)} \, dt </math> or, setting Template:Math to obtain an integral for Template:Math, we can replace the Template:Math term with its integral and incorporate that into the above formula, to get: <math display="block">\operatorname{log\Gamma}(z+1)= \int_0^\infty \frac{e^{-zt} - ze^{-t} - 1 + z}{t \left(e^t -1\right)} \, dt\,. </math>

There also exist special formulas for the logarithm of the gamma function for rational Template:Mvar. For instance, if <math>k</math> and <math>n</math> are integers with <math>k<n</math> and <math>k\neq n/2 \,,</math> then<ref name="iaroslav_07">Template:Cite journal</ref> <math display="block"> \begin{align} \operatorname{log\Gamma} \left(\frac{k}{n}\right) = {} & \frac{\,(n-2k)\log2\pi\,}{2n} + \frac{1}{2}\left\{\,\log\pi-\log\sin\frac{\pi k}{n} \,\right\} + \frac{1}{\pi}\!\sum_{r=1}^{n-1}\frac{\,\gamma+\log r\,}{r}\cdot\sin\frac{\,2\pi r k\,}{n} \\ & {} - \frac{1}{2\pi}\sin\frac{2\pi k}{n}\cdot\!\int_0^\infty \!\!\frac{\,e^{-nx}\!\cdot\log x\,}{\,\cosh x -\cos( 2\pi k/n )\,}\,{\mathrm d}x. \end{align} </math>This formula is sometimes used for numerical computation, since the integrand decreases very quickly.

Integration over log-gamma

The integral <math display="block"> \int_0^z \operatorname{log\Gamma} (x) \, dx</math> can be expressed in terms of the [[Barnes G-function|Barnes Template:Math-function]]<ref name="Alexejewsky">Template:Cite journal</ref><ref name="Barnes">Template:Cite journal</ref> (see [[Barnes G-function|Barnes Template:Math-function]] for a proof): <math display="block">\int_0^z \operatorname{log\Gamma}(x) \, dx = \frac{z}{2} \log (2 \pi) + \frac{z(1-z)}{2} + z \operatorname{log\Gamma}(z) - \log G(z+1)</math> where Template:Math.

It can also be written in terms of the Hurwitz zeta function:<ref name="Adamchik">Template:Cite journal</ref><ref name="Gosper">Template:Cite journal</ref> <math display="block">\int_0^z \operatorname{log\Gamma}(x) \, dx = \frac{z}{2} \log(2 \pi) + \frac{z(1-z)}{2} - \zeta'(-1) + \zeta'(-1,z) .</math>

When <math>z=1</math> it follows that <math display="block"> \int_0^1 \operatorname{log\Gamma}(x) \, dx = \frac 1 2 \log(2\pi), </math> 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 <math>\operatorname{log\Gamma}</math>:<ref name="EspinosaMoll">Template:Cite journal</ref> <math display="block">\int_{0}^{1} \log ^{2} \Gamma(x) d x=\frac{\gamma^{2}}{12}+\frac{\pi^{2}}{48}+\frac{1}{3} \gamma L_{1}+\frac{4}{3} L_{1}^{2}-\left(\gamma+2 L_{1}\right) \frac{\zeta^{\prime}(2)}{\pi^{2}}+\frac{\zeta^{\prime \prime}(2)}{2 \pi^{2}},</math> where <math>L_1</math> is <math>\frac12\log(2\pi)</math>.

D. H. Bailey and his co-authors<ref name="Bailey">Template:Cite journal</ref> gave an evaluation for <math display="block">L_n:=\int_0^1 \log^n \Gamma(x) \, dx</math> when <math>n=1,2</math> in terms of the Tornheim–Witten zeta function and its derivatives.

In addition, it is also known that<ref name="ACEKNM">Template:Cite journal</ref> <math display="block"> \lim_{n\to\infty} \frac{L_n}{n!}=1. </math>

Approximations

File:Mplwp factorial gamma stirling.svg

Comparison of the gamma function (blue line) with the factorial (blue dots) and Stirling's approximation (red line)

Complex values of the gamma function can be approximated using Stirling's approximation or the Lanczos approximation, <math display="block">\Gamma(z) \sim \sqrt{2\pi}z^{z-1/2}e^{-z}\quad\hbox{as }z\to\infty\hbox{ in } \left|\arg(z)\right|<\pi.</math> This is precise in the sense that the ratio of the approximation to the true value approaches 1 in the limit as Template:Math goes to infinity.

The gamma function can be computed to fixed precision for <math>\operatorname{Re} (z) \in [1, 2]</math> by applying integration by parts to Euler's integral. For any positive number Template:Mvar the gamma function can be written <math display="block">\begin{align} \Gamma(z) &= \int_0^x e^{-t} t^z \, \frac{dt}{t} + \int_x^\infty e^{-t} t^z\, \frac{dt}{t} \\ &= x^z e^{-x} \sum_{n=0}^\infty \frac{x^n}{z(z+1) \cdots (z+n)} + \int_x^\infty e^{-t} t^z \, \frac{dt}{t}. \end{align}</math>

When Template:Math and <math>x \geq 1</math>, the absolute value of the last integral is smaller than <math>(x + 1)e^{-x}</math>. By choosing a large enough <math>x</math>, this last expression can be made smaller than <math>2^{-N}</math> for any desired value <math>N</math>. Thus, the gamma function can be evaluated to <math>N</math> bits of precision with the above series.

A fast algorithm for calculation of the Euler gamma function for any algebraic argument (including rational) was constructed by E.A. Karatsuba.<ref>E.A. Karatsuba, Fast evaluation of transcendental functions. Probl. Inf. Transm. Vol.27, No.4, pp. 339–360 (1991).</ref><ref>E.A. Karatsuba, On a new method for fast evaluation of transcendental functions. Russ. Math. Surv. Vol.46, No.2, pp. 246–247 (1991).</ref><ref>E.A. Karatsuba "Fast Algorithms and the FEE Method".</ref>

For arguments that are integer multiples of Template:Math, the gamma function can also be evaluated quickly using arithmetic–geometric mean iterations (see particular values of the gamma function).<ref>Template:Cite journal</ref>

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 <math>\Gamma(z)</math> is easily found. Therefore, it is worth investigating potential solutions. For the case that speed is more important than accuracy, published tables for <math>\Gamma(z)</math> 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 <math>\Gamma(z)</math> tables are usually published for argument values between 1 and 2, the property <math>\Gamma(z+1) = z\ \Gamma(z)</math> may be used to quickly and easily translate all real values <math>z <1 </math> and <math>z>2</math> into the range <math>1\leq z \leq 2</math>, such that only tabulated values of <math>z</math> between 1 and 2 need be used.<ref>Template:Cite journal</ref>

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 Template:Math is most difficult to avoid."<ref>Michon, G. P. "Trigonometry and Basic Functions Template:Webarchive". Numericana. Retrieved 5 May 2007.</ref>

Integration problems

The gamma function finds application in such diverse areas as quantum physics, astrophysics and fluid dynamics.<ref>Template:Cite book</ref> 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.<ref>Template:Cite book</ref>

The primary reason for the gamma function's usefulness in such contexts is the prevalence of expressions of the type <math>f(t)e^{-g(t)}</math> 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 Template:Math is a power function and Template:Math is a linear function, a simple change of variables <math>u:=a\cdot t</math> gives the evaluation

<math display="block">\int_0^\infty t^b e^{-at} \,dt = \frac{1}{a^b} \int_0^\infty u^b e^{-u} d\left(\frac{u}{a}\right) = \frac{\Gamma(b+1)}{a^{b+1}}.</math>

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 Template:Math 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 <math display="block">ae^{-\frac{(x-b)^2}{c^2}}</math> and integrals thereof, such as the error function. There are many interrelations between these functions and the gamma function; notably, the factor <math>\sqrt{\pi}</math> obtained by evaluating <math display="inline">\Gamma \left( \frac{1}{2} \right)</math> 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 Template:Math-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 Template:Mvar and Template:Mvar, with Template:Math, we can write <math display="block">(1 + z)^n = \sum_{k=0}^\infty \frac{\Gamma(n+1)}{k!\Gamma(n-k+1)} z^k,</math> which closely resembles the binomial coefficient when Template:Mvar is a non-negative integer, <math display="block">(1 + z)^n = \sum_{k=0}^n \frac{n!}{k!(n-k)!} z^k = \sum_{k=0}^n \binom{n}{k} z^k.</math>

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 Template:Mvar elements from a set of Template:Mvar elements; if Template:Math, there are of course no ways. If Template:Math, Template:Math 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 Template:Math and Template:Math are monic polynomials of degree Template:Mvar and Template:Mvar with respective roots Template:Math and Template:Math, we have <math display="block">\prod_{i=a}^b \frac{P(i)}{Q(i)} = \left( \prod_{j=1}^m \frac{\Gamma(b-p_j+1)}{\Gamma(a-p_j)} \right) \left( \prod_{k=1}^n \frac{\Gamma(a-q_k)}{\Gamma(b-q_k+1)} \right).</math>

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 Template:Math 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: <math display="block">\Gamma\left(\frac{s}{2}\right)\zeta(s)\pi^{-\frac{s}{2}} = \Gamma\left(\frac{1-s}{2}\right)\zeta(1-s)\pi^{-\frac{1-s}{2}}.</math>

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".<ref>Template:Cite book</ref> Another contender for that title might be <math display="block">\zeta(s) \; \Gamma(s) = \int_0^\infty \frac{t^s}{e^t-1} \, \frac{dt}{t}.</math>

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."<ref name="Davis">Template:Cite journal</ref>

18th century: Euler and Stirling

File:DanielBernoulliLettreAGoldbach-1729-10-06.jpg

Daniel Bernoulli's letter to Christian Goldbach, October 6, 1729

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<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> <math display="block">x!=\lim_{n\to\infty}\left(n+1+\frac{x}{2}\right)^{x-1}\prod_{k=1}^{n}\frac{k+1}{k+x}</math> which is well defined for real values of Template:Math 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 Template:Math other than the negative integers, <math display="block">n! = \prod_{k=1}^\infty \frac{\left(1+\frac{1}{k}\right)^n}{1+\frac{n}{k}}\,,</math> 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 <math display="block">n!=\int_0^1 (-\log s)^n\, ds\,,</math> which is valid when the real part of the complex number Template:Math is strictly greater than Template:Math (i.e., <math>\Re (n) > -1</math>). By the change of variables Template:Math, this becomes the familiar Euler integral. Euler published his results in the paper "De progressionibus transcendentibus seu quarum termini generales algebraice dari nequeunt" ("On transcendental progressions, that is, those whose general terms cannot be given algebraically"), submitted to the St. Petersburg Academy on 28 November 1729.<ref>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.</ref> 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 Template:Math, 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

File:Euler factorial paper.png

The first page of Euler's paper

Carl Friedrich Gauss rewrote Euler's product as <math display="block">\Gamma(z) = \lim_{m\to\infty}\frac{m^z m!}{z(z+1)(z+2)\cdots(z+m)}</math> 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.<ref name="Remmert">Template:Cite book</ref> 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, <math display="block">\Gamma(z) = \frac{e^{-\gamma z}}{z} \prod_{k=1}^\infty \left(1 + \frac{z}{k}\right)^{-1} e^\frac{z}{k},</math> where Template:Math is the Euler–Mascheroni constant. Weierstrass originally wrote his product as one for Template:Math, 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 Template:Math 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 "Template:Math-function"). The alternative "pi function" notation Template:Math 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 Template:Math instead of simply using "Template:Math". Consider that the notation for exponents, Template:Math, has been generalized from integers to complex numbers Template:Math without any change. Legendre's motivation for the normalization is not 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 Template:Math).<ref>Template:Cite journal</ref> 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 Template:Math against the multiplicative character Template:Math with respect to the Haar measure <math display="inline">\frac{dx}{x}</math> on the Lie group Template:Math. Thus this normalization makes it clearer that the gamma function is a continuous analogue of a Gauss sum.<ref>Template:Cite book Extract of page 205</ref>

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.<ref name="Knuth">Template:Cite book</ref> 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 Template:Mvar 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 review three centuries of work on the gamma function.<ref>Template:Cite journal</ref>

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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

File:Jahnke gamma function.png

A hand-drawn graph of the absolute value of the complex gamma function, from Tables of Higher Functions by Jahnke and Template:Ill.

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 Template:Ill, 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."<ref>Template:Cite news</ref>

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.<ref name=Davis />

File:Famous complex plot by Janhke and Emde (Tables of Functions with Formulas and Curves, 4th ed., Dover, 1945) gamma function from -4.5-2.5i to 4.5+2.5i.svg

Reproduction of a famous complex plot by Janhke and Emde (Tables of Functions with Formulas and Curves, 4th ed., Dover, 1945) of the gamma function from −4.5 − 2.5i to 4.5 + 2.5i

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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

See also

• Ascending factorial
• Cahen–Mellin integral
• Elliptic gamma function
• Lemniscate constant
• Pseudogamma function
• Hadamard's gamma function
• Inverse gamma function
• Lanczos approximation
• Multiple gamma function
• Multivariate gamma function
• [[p-adic gamma function|Template:Math-adic gamma function]]
• [[Pochhammer k-symbol|Pochhammer Template:Math-symbol]]
• [[q-gamma function|Template:Math-gamma function]]
• Ramanujan's master theorem
• Spouge's approximation
• Stirling's approximation

Notes

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Further reading

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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.
• Template:Springer
• Wolfram gamma function evaluator (arbitrary precision)
• Template:WolframFunctionsSite
• Volume of n-Spheres and the Gamma Function at MathPages