Difference between revisions of "Gamma"

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The gamma function is the function defined by the integral (initially for positive values of $x$) by the formula
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__NOTOC__
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$$\Gamma(x)=\displaystyle\int_0^{\infty} x^{t-1}e^{-x} dx.$$
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The gamma function $\Gamma \colon \mathbb{C} \setminus \{0,-1,-2,\ldots\} \rightarrow \mathbb{C}$ is the function initially defined for $\mathrm{Re}(z)>0$ by the integral by the formula
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$$\Gamma(z)=\displaystyle\int_0^{\infty} \xi^{z-1}e^{-\xi} \mathrm{d}\xi.$$
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The [[analytic continuation]] of $\Gamma$ leads to a [[meromorphic function]] with [[pole | poles]] at the negative integers.
  
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=Recurrence=
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<div align="center">
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<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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<gallery>
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<strong>Proposition:</strong> $\Gamma(1)=1$
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File:Gammaplot.png|Graph of $\Gamma$.
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<div class="mw-collapsible-content">
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File:Complexgammaplot.png|[[Domain coloring]] of $\Gamma$.
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<strong>Proof:</strong> Compute directly
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File:Absvalgamma.png|Plot of $z=|\Gamma(x+iy)|$ (1948).
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$$\begin{array}{ll}
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File:Gamma and reciprocal gamma (abramowitzandstegun).png|Plot of $\Gamma$ and [[Reciprocal gamma|$\dfrac{1}{\Gamma}$]] from Abramowitz&Stegun.
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\Gamma(1) &= \displaystyle\int_0^{\infty} e^{-t}t^{1-1} dt \\
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</gallery>
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&= \displaystyle\int_0^{\infty} e^{-t} dt \\
 
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&= \left[ -e^{-t} \right]_{-\infty}^{\infty} \\
 
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&= 0.
 
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\end{array}$$ █
 
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</div>
 
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</div>
 
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<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
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<strong>Theorem:</strong> $\Gamma(x+1)=x\Gamma(x); x>0$
 
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<div class="mw-collapsible-content">
 
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<strong>Proof:</strong> █
 
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</div>
 
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</div>
 
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<div class="toccolours" style="width:800px">
 
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<strong>Corollary:</strong> If $x \in \mathbb{N}$, then $\Gamma(x+1)=x!$, where $x!$ denotes the [[factorial]].
 
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</div>
 
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=Other formulas=
 
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<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
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<strong>Proposition:</strong> The following formula holds
 
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$$\Gamma(x)=\lim_{k \rightarrow \infty} \dfrac{k!k^z}{z(z+1)\ldots(z+k)}.$$
 
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<div class="mw-collapsible-content">
 
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<strong>Proof:</strong> █
 
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</div>
 
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</div>
 
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<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
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<strong>Proposition:</strong> The following formula holds
 
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$$\dfrac{1}{\Gamma(z)} = ze^{\gamma z} \displaystyle\prod_{k=1}^{\infty} \left( 1 + \dfrac{z}{k}\right)e^{-\frac{z}{k}},$$
 
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where $\gamma$ is the [[Euler-Mascheroni constant]].
 
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<div class="mw-collapsible-content">
 
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<strong>Proof:</strong> █
 
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</div>
 
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</div>
 
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<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
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<strong>Proposition:</strong> The following formula holds
 
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$$\Gamma(x)=2\displaystyle\int_0^{\infty} e^{-t^2}t^{2x-1}dt.$$
 
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<div class="mw-collapsible-content">
 
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<strong>Proof:</strong> █
 
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</div>
 
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</div>
 
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<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
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<strong>Proposition:</strong> The following formula holds
 
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$$\displaystyle\int_0^{\frac{\pi}{2}} \cos^{2x-1}(\theta)\sin^{2y-1}(\theta) d\theta = \dfrac{\Gamma(x)\Gamma(y)}{2\Gamma(x+y)}.$$
 
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<div class="mw-collapsible-content">
 
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<strong>Proof:</strong> █
 
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</div>
 
 
</div>
 
</div>
  
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=$\Gamma(z)$ at half integers=
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=Properties=
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<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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[[Gamma(z) as integral of a power of log(1/t) for Re(z) greater than 0]]<br />
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<strong>Proposition:</strong> $\Gamma \left( \dfrac{1}{2} \right) = \sqrt{\pi}$.
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[[Gamma function written as a limit of a factorial, exponential, and a rising factorial]]<br />
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<div class="mw-collapsible-content">
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[[Gamma function written as infinite product]]<br />
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<strong>Proof:</strong> █
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[[Gamma(1)=1]]<br />
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</div>
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[[Gamma(z+1)=zGamma(z)]]<br />
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</div>
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[[Gamma(n+1)=n!]]<br />
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[[Relationship between Hurwitz zeta and gamma function]]<br />
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[[Gamma(z)Gamma(1-z)=pi/sin(pi z)]]<br />
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[[Bohr-Mollerup theorem]]<br />
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[[Gamma'(z)/Gamma(z)=-gamma-1/z+Sum z/(k(z+k))]]<br />
  
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<div class="toccolours" style="width:800px">
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=Videos=
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<strong>Corollary:</strong> $\displaystyle\int_0^{\infty} e^{-t^2} dt = \dfrac{1}{2}\sqrt{\pi}$.
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[https://www.youtube.com/watch?v=Yor2Q584o6A What's the Gamma Function? (16 September 2008)]<br />
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</div>
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[https://www.youtube.com/watch?v=AIPgWKnBFgc Gamma Function Of One-Half: Part 1 (10 August 2010)]<br />
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[https://www.youtube.com/watch?v=KAn0jKFkmrI Gamma Function Of One-Half: Part 2 (10 August 2010)]<br />
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<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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[https://www.youtube.com/watch?v=vSekqP29PjA Gamma Integral Function - Introduction (5 December 2011)]<br />
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<strong>Theorem (Legendre Duplication Formula):</strong>
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[https://www.youtube.com/watch?v=Uos_gY--5GI gamma function - Part 1 (9 February 2012)]<br />
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$$\Gamma(2x)=\dfrac{2^{2x-1}}{\sqrt{\pi}} \Gamma(x)\Gamma \left( x +\dfrac{1}{2} \right).$$
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[https://www.youtube.com/watch?v=2iBNo4j3vRo&list=PL3E4136E122545FBE&index=1 Gamma Function (playlist) (26 February 2012)]<br />
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<div class="mw-collapsible-content">
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[https://www.youtube.com/watch?v=yu9k2iPta-k Gamma function (20 October 2012)]<br />
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<strong>Proof:</strong> █
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[https://www.youtube.com/watch?v=Vc8dIykQRhY Beta Function, Gamma Function and their Properties (17 August 2013)]<br />
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</div>
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[https://www.youtube.com/watch?v=6kSe2PnDEvM Thermodynamics 19 a : Gamma Function 1/2 (31 August 2013)]<br />
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</div>
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[https://www.youtube.com/watch?v=O45LOf2NfNI euler gamma function (14 September 2013)]<br />
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[https://www.youtube.com/watch?v=XZIVrkkYBRI The Gamma Function: intro (5) (13 February 2014)]<br />
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<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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[https://www.youtube.com/watch?v=Q1rHWCP_40s The Gamma Function: why 0!=1 (5) (13 February 2014)]<br />
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<strong>Proposition:</strong>  If $z=0,-1,-2,\ldots$ then $\Gamma(z)=\infty$.
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[https://www.youtube.com/watch?v=C3PmT6oNEew Mod-04 Lec-09 Analytic continuation and the gamma function (Part I) (3 June 2014)]<br />
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[https://www.youtube.com/watch?v=XAoe4th0F1k Gamma function at 1/2 (3 January 2015)]<br />
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<div class="mw-collapsible-content">
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[https://www.youtube.com/watch?v=l7LoSBv6o2k Contour Integral Definition of the Gamma Function (18 January 2015)]<br />
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<strong>Proof:</strong> proof goes here █
 
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</div>
 
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</div>
 
  
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<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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=External links=
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<strong>Theorem:</strong> The following relationship between $\Gamma$ and the [[Sine | $\sin$]] function holds:
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[http://ocw.mit.edu/courses/mathematics/18-104-seminar-in-analysis-applications-to-number-theory-fall-2006/projects/chan.pdf The sine product formula and the gamma function]<br />
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$$\Gamma(x)\Gamma(1-x) = \dfrac{\pi}{\sin(\pi x)}.$$
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[http://www.jstor.org/discover/10.2307/2309786?sid=21105065140641&uid=3739256&uid=2129&uid=70&uid=3739744&uid=4&uid=2 Leonhard Euler's Integral: A Historical Profile of the Gamma Function]<br />
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<div class="mw-collapsible-content">
 
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<strong>Proof:</strong> proof goes here █
 
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</div>
 
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</div>
 
  
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<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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=See Also=
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<strong>Proposition:</strong> $\Gamma(x) = \displaystyle\lim_{n \rightarrow \infty} \dfrac{n^x n!}{x(x+1)\ldots(x_n)}$
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[[Loggamma]]<br />
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<div class="mw-collapsible-content">
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[[Polygamma]]<br />
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<strong>Proof:</strong> proof goes here █
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[[Reciprocal gamma]] <br />
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</div>
 
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</div>
 
  
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<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
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<strong>Proposition:</strong> $\Gamma(x)\Gamma(1-x)=\dfrac{\pi}{\sin(\pi x)}$
 
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<div class="mw-collapsible-content">
 
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<strong>Proof:</strong> proof goes here █
 
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</div>
 
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</div>
 
  
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<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
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<strong>Bohr-Mollerup Theorem:</strong> The gamma function is the unique function $f$ such that
 
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*$f(1)=1$
 
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*$f(x+1)=xf(x)$ for $x>0$
 
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*$f$ is [[logarithmically convex]].
 
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<div class="mw-collapsible-content">
 
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<strong>Proof:</strong> proof goes here █
 
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</div>
 
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</div>
 
  
 
=References=
 
=References=
−
[http://www.plouffe.fr/simon/math/Artin%20E.%20The%20Gamma%20Function%20%281931%29%2823s%29.pdf The Gamma Function by Emil Artin]
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* {{BookReference|Einleitung in die Theorie der Gammafunktion und der Euler'schen Integrale|1895|Johann Heinrich Graf|prev=findme|next=findme}}: $\S 3 (15_a)$
 +
* {{BookReference|A course of modern analysis|1920|Edmund Taylor Whittaker|author2=George Neville Watson|edpage=Third edition|prev=findme|next=Euler-Mascheroni constant}}: $\S 12 \cdot 1$
 +
* {{BookReference|Higher Transcendental Functions Volume I|1953|Arthur Erdélyi|author2=Wilhelm Magnus|author3=Fritz Oberhettinger|author4=Francesco G. Tricomi|next=Gamma(z) as integral of a power of log(1/t) for Re(z) greater than 0}}: $\S 1.1 (1)$
 +
* {{BookReference|Special Functions of Mathematical Physics and Chemistry|1956|Ian N. Sneddon|prev=findme|next=Beta}}: $\S 5 (5.1)$
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* {{BookReference|Special Functions|1960|Earl David Rainville|prev=findme|next=findme}}: $15.(1)$
 +
* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=findme|next=Gauss' formula for gamma function}}: $6.1.1$
 +
* {{BookReference|Special Functions for Scientists and Engineers|1968|W.W. Bell|prev=findme|next=Beta}}: $(2.1)$
 +
* {{BookReference|The Special Functions And Their Approximations, Volume I|1969|Yudell L. Luke|prev=findme|next=findme}} $2.1 (1)$
 +
* {{BookReference|Hypergeometric Orthogonal Polynomials and Their q-Analogues|2010|Roelof Koekoek|author2=Peter A. Lesky|author3=René F. Swarttouw|prev=findme|next=Gamma(z+1)=zGamma(z)}}: $(1.2.1)$
 +
* {{BookReference|Special functions, a graduate text|2010|Richard Beals|author2=Roderick Wong|prev=findme|next=Gamma(z+1)=zGamma(z)}}: $(2.1.1)$
 +
[[Category:SpecialFunction]]

Latest revision as of 18:12, 16 June 2018

The gamma function $\Gamma \colon \mathbb{C} \setminus \{0,-1,-2,\ldots\} \rightarrow \mathbb{C}$ is the function initially defined for $\mathrm{Re}(z)>0$ by the integral by the formula $$\Gamma(z)=\displaystyle\int_0^{\infty} \xi^{z-1}e^{-\xi} \mathrm{d}\xi.$$ The analytic continuation of $\Gamma$ leads to a meromorphic function with poles at the negative integers.

Properties

Gamma(z) as integral of a power of log(1/t) for Re(z) greater than 0
Gamma function written as a limit of a factorial, exponential, and a rising factorial
Gamma function written as infinite product
Gamma(1)=1
Gamma(z+1)=zGamma(z)
Gamma(n+1)=n!
Relationship between Hurwitz zeta and gamma function
Gamma(z)Gamma(1-z)=pi/sin(pi z)
Bohr-Mollerup theorem
Gamma'(z)/Gamma(z)=-gamma-1/z+Sum z/(k(z+k))

Videos

What's the Gamma Function? (16 September 2008)
Gamma Function Of One-Half: Part 1 (10 August 2010)
Gamma Function Of One-Half: Part 2 (10 August 2010)
Gamma Integral Function - Introduction (5 December 2011)
gamma function - Part 1 (9 February 2012)
Gamma Function (playlist) (26 February 2012)
Gamma function (20 October 2012)
Beta Function, Gamma Function and their Properties (17 August 2013)
Thermodynamics 19 a : Gamma Function 1/2 (31 August 2013)
euler gamma function (14 September 2013)
The Gamma Function: intro (5) (13 February 2014)
The Gamma Function: why 0!=1 (5) (13 February 2014)
Mod-04 Lec-09 Analytic continuation and the gamma function (Part I) (3 June 2014)
Gamma function at 1/2 (3 January 2015)
Contour Integral Definition of the Gamma Function (18 January 2015)

External links

The sine product formula and the gamma function
Leonhard Euler's Integral: A Historical Profile of the Gamma Function

See Also

Loggamma
Polygamma
Reciprocal gamma


References

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