Editing Gamma function (section)

== Log-gamma function ==

[[File:LogGamma Analytic Function.png|thumb|The analytic function {{math|logΓ(''z'')}}]]

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 <code>lgamma</code> or <code>lngamma</code> in programming environments or <code>gammaln</code> 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>{{cite web |title=Log Gamma Function |url=http://mathworld.wolfram.com/LogGammaFunction.html |website=Wolfram MathWorld |access-date=3 January 2019}}</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|alt=Logarithmic gamma function in the complex plane from −2 − 2i to 2 + 2i with colors|thumb|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 {{mvar|z}} from the neighbouring strip. In particular, starting with a good approximation for a&nbsp;{{mvar|z}} with large real part one may go step by step down to the desired&nbsp;{{mvar|z}}. Following an indication of [[Carl Friedrich Gauss]], Rocktaeschel (1922) proposed for {{math|logΓ(''z'')}} an approximation for large {{math|Re(''z'')}}:
<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 {{math|logΓ(''z'')}} for {{mvar|z}} with a smaller {{math|Re(''z'')}} 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 {{math|logΓ(''z'')}} and {{math|Γ(''z'')}}, 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 {{math|{{abs|''z''}} → ∞}} at constant {{math|{{abs|arg(''z'')}} < π}}.  (See sequences {{OEIS link|A001163}} and {{OEIS link|A001164}} in the [[On-Line Encyclopedia of Integer Sequences|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 {{math|{{abs|''z''}} → ∞}} at constant {{math|{{abs|arg(''z'')}} < π}}.  (See sequences {{OEIS link|A046968}} and {{OEIS link|A046969}} in the [[On-Line Encyclopedia of Integer Sequences|OEIS]].)

The coefficients of the terms with {{math|''k'' > 1}} of {{math|''z''<sup>1−''k''</sup>}} in the last expansion are simply
<math display="block">\frac{B_k}{k(k-1)}</math>
where the {{math|''B<sub>k</sub>''}} are the [[Bernoulli numbers]].

The gamma function also has Stirling Series (derived by [[Charles Hermite]] in 1900) equal to<ref>{{cite web |title=Leonhard Euler's Integral: An Historical Profile of the Gamma Function |url=https://www.maa.org/sites/default/files/pdf/upload_library/22/Chauvenet/Davis.pdf |archive-url=https://web.archive.org/web/20140912213629/http://www.maa.org/sites/default/files/pdf/upload_library/22/Chauvenet/Davis.pdf |archive-date=2014-09-12 |url-status=live  |access-date=11 April 2022}}</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 function|convex]] on the positive real axis. Another characterisation is given by the [[Wielandt theorem]].

The gamma function is the unique function that simultaneously satisfies
# <math>\Gamma(1) = 1</math>,
# <math>\Gamma(z+1) = z \Gamma(z)</math> for all complex numbers <math>z</math> except the non-positive integers, and,
# for integer {{mvar|n}}, <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 {{math|logΓ}} 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 {{math|''ζ''(''k'')}} denoting the [[Riemann zeta function]] at {{mvar|k}}.

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 {{math|1=''z'' = 1}} to obtain an integral for {{math|''γ''}}, we can replace the {{math|''γ''}} 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 {{mvar|z}}. 
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">{{cite journal |last=Blagouchine |first=Iaroslav V. |year=2015 |title=A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations |journal=Journal of Number Theory |volume=148 |pages=537–592 |arxiv=1401.3724 |doi=10.1016/j.jnt.2014.08.009}}</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 {{math|''G''}}-function]]<ref name="Alexejewsky">{{cite journal|first=W. P. |last=Alexejewsky |title=Über eine Classe von Funktionen, die der Gammafunktion analog sind |trans-title=On a class of functions analogous to the gamma function |journal=Leipzig Weidmannsche Buchhandlung |volume=46 |date=1894 |pages=268–275}}</ref><ref name="Barnes">{{cite journal|first=E. W. |last=Barnes |title=The theory of the ''G''-function |journal=Quart. J. Math. |volume=31 |date=1899 |pages=264–314}}</ref> (see [[Barnes G-function|Barnes {{math|''G''}}-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 {{math|Re(''z'') > −1}}.

It can also be written in terms of the [[Hurwitz zeta function]]:<ref name="Adamchik">{{cite journal|first=Victor S. |last=Adamchik |title=Polygamma functions of negative order |journal=J. Comput. Appl. Math. |volume=100 |issue=2 |date=1998 |pages=191–199 |doi=10.1016/S0377-0427(98)00192-7|doi-access=free }}</ref><ref name="Gosper">{{cite journal|first=R. W. |last=Gosper |title=<math>\textstyle \int_{n/4}^{m/6} \log F(z) \,dz</math> in special functions, ''q''-series and related topics |journal=J. Am. Math. Soc. |volume=14 |date=1997}}</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">{{cite journal|first1=Olivier |last1=Espinosa|first2=Victor H. |last2=Moll|title= On Some Integrals Involving the Hurwitz Zeta Function: Part 1|journal=The Ramanujan Journal |volume=6 |date=2002 |issue=2|pages=159–188 |doi=10.1023/A:1015706300169|s2cid=128246166}}</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">{{cite journal|first1=David H. |last1=Bailey|first2=David |last2=Borwein|first3=Jonathan M.|last3=Borwein|title= On Eulerian log-gamma integrals and Tornheim-Witten zeta functions|journal=The Ramanujan Journal |volume=36 |date=2015 |issue=1–2|pages=43–68 |doi=10.1007/s11139-012-9427-1|s2cid=7335291}}</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">{{cite journal|first1=T. |last1=Amdeberhan|first2=Mark W.|last2=Coffey|first3=Olivier|last3=Espinosa|first4=Christoph|last4=Koutschan|first5=Dante V.|last5=Manna|first6=Victor H.|last6=Moll|title= Integrals of powers of loggamma|journal=Proc. Amer. Math. Soc.|volume=139|issue=2 |date=2011 |pages=535–545 |doi=10.1090/S0002-9939-2010-10589-0|doi-access=free}}</ref>
<math display="block">
\lim_{n\to\infty} \frac{L_n}{n!}=1.
</math>