Difference between revisions of "Gamma"

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[[Gamma function written as infinite product]]<br />
 
[[Gamma function written as infinite product]]<br />
 
[[Gamma(1)=1]]<br />
 
[[Gamma(1)=1]]<br />
[[Factorial property of gamma]]<br />
+
[[Gamma(z+1)=zGamma(z)]]<br />
[[Gamma at positive integers]]<br />
+
[[Gamma(n+1)=n!]]<br />
 
[[Relationship between Hurwitz zeta and gamma function]]<br />
 
[[Relationship between Hurwitz zeta and gamma function]]<br />
[[Gamma-Sine Relation]]<br />
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[[Gamma(z)Gamma(1-z)=pi/sin(pi z)]]<br />
 
[[Bohr-Mollerup theorem]]<br />
 
[[Bohr-Mollerup theorem]]<br />
 
[[Gamma'(z)/Gamma(z)=-gamma-1/z+Sum z/(k(z+k))]]<br />
 
[[Gamma'(z)/Gamma(z)=-gamma-1/z+Sum z/(k(z+k))]]<br />
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[[Polygamma]]<br />
 
[[Polygamma]]<br />
 
[[Reciprocal gamma]] <br />
 
[[Reciprocal gamma]] <br />
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 +
  
 
=References=
 
=References=
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* {{BookReference|Special Functions for Scientists and Engineers|1968|W.W. Bell|prev=findme|next=Beta}}: $(2.1)$
 
* {{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|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=findme}}: $(1.2.1)$
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* {{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)}}: $(1.2.1)$
 
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* {{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]]
 
[[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