Gamma

The gamma function $\Gamma \colon \mathbb{C} \setminus \{0,-1,-2,\ldots\} \rightarrow \mathbb{C}$ is the function initially defined for $x>0$ 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.

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

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

• 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): $\S 1.1 (1)$
• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $6.1.1$