Difference between revisions of "Gamma"
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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|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|Harry Bateman|next=Gamma(z) as integral of a power of log(1/t) for Re(z) greater than 0}}: §1.1 (1) | * {{BookReference|Higher Transcendental Functions Volume I|1953|Harry Bateman|next=Gamma(z) as integral of a power of log(1/t) for Re(z) greater than 0}}: §1.1 (1) | ||
Revision as of 16:16, 21 June 2016
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 $$\Gamma(x)=\displaystyle\int_0^{\infty} \xi^{x-1}e^{-\xi} \mathrm{d}\xi.$$ The analytic continuation of $\Gamma$ leads to a meromorphic function with poles at the negative integers.
Graph of $\Gamma$.
Domain coloring of $\Gamma$.
$ (1948).
Plot of $\Gamma$ and $\dfrac{1}{\Gamma}$ from Abramowitz&Stegun.
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Contents
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
Value of Gamma(1)
Factorial property of gamma
Gamma at positive integers
Relationship between Hurwitz zeta and gamma function
Gamma-Sine Relation
Bohr-Mollerup theorem
Videos
Gamma Function (playlist)
The Gamma Function: intro (5)
Gamma Integral Function - Introduction
Gamma function
Mod-04 Lec-09 Analytic continuation and the gamma function (Part I)
gamma function - Part 1
Beta Function, Gamma Function and their Properties
What's the Gamma Function?
euler gamma function
Thermodynamics 19 a : Gamma Function 1/2
The Gamma Function: why 0!=1 (5)
Gamma Function Of One-Half: Part 1
Gamma Function Of One-Half: Part 2
Gamma function at 1/2
Contour Integral Definition of the Gamma Function
See Also
Loggamma
Polygamma
Reciprocal gamma
References
- 1895: Johann Heinrich Graf: Einleitung in die Theorie der Gammafunktion und der Euler'schen Integrale ... (previous) ... (next): $\S 3. (15_a)$
- 1920: Edmund Taylor Whittaker and George Neville Watson: A course of modern analysis ... (previous) ... (next): $\S 12 \cdot 1$
- 1953: Harry Bateman: Higher Transcendental Functions Volume I ... (next): §1.1 (1)
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 6.1.1
The sine product formula and the gamma function
Leonhard Euler's Integral: A Historical Profile of the Gamma Function
