Difference between revisions of "Gamma"

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[[Value of Gamma(1)]]<br />
{{:Gamma at positive integers}}
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[[Factorial property of gamma]]<br />
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[[Gamma at positive integers]]<br />
  
 
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Revision as of 09:24, 4 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.

Properties

Gamma(z) as integral of a power of log(1/t) for Re(z) greater than 0 Value of Gamma(1)
Factorial property of gamma
Gamma at positive integers

Proposition: The following formula holds: $$\Gamma(x)=2\displaystyle\int_0^{\infty} e^{-t^2}t^{2x-1}dt.$$

Proof:

Proposition: The following formula holds: $$\displaystyle\int_0^{\frac{\pi}{2}} \cos^{2x-1}(\theta)\sin^{2y-1}(\theta) d\theta = \dfrac{\Gamma(x)\Gamma(y)}{2\Gamma(x+y)}.$$

Proof:

Theorem: The following formula holds for $\mathrm{Re}(z)>0$: $$\Gamma(z)=\displaystyle\int_0^1 \log \left( \dfrac{1}{t} \right)^{z-1} dt,$$ where $\Gamma$ denotes the gamma function and $\log$ denotes the logarithm.

Proof:

Proposition: The following formula holds: $$\Gamma(z)=\lim_{k \rightarrow \infty} \dfrac{k!k^z}{z(z+1)\ldots(z+k)}.$$

Proof:

Gamma function Weierstrass product

Theorem

The following formula holds: $$\Gamma(s)\zeta(s,a) = \displaystyle\int_0^{\infty} \dfrac{x^{s-1}e^{-ax}}{1-e^{-x}} \mathrm{d}x,$$ where $\Gamma$ denotes the gamma function and $\zeta$ denotes the Hurwitz zeta function.

Proof

References

Proposition: $\Gamma \left( \dfrac{1}{2} \right) = \sqrt{\pi}$.

Proof:

Corollary: $\displaystyle\int_0^{\infty} e^{-t^2} dt = \dfrac{1}{2}\sqrt{\pi}$.

Theorem (Convexity): The gamma function is logarithmically convex.

Proof:

Theorem (Legendre Duplication Formula): $$\Gamma(2x)=\dfrac{2^{2x-1}}{\sqrt{\pi}} \Gamma(x)\Gamma \left( x +\dfrac{1}{2} \right).$$

Proof:

Proposition: If $z=0,-1,-2,\ldots$ then $\Gamma(z)=\infty$.

Proof:

  1. REDIRECT Gamma(z)Gamma(1-z)=pi/sin(pi z)

Theorem

The gamma function is the unique function $f$ such that $f(1)=1$, $f(x+1)=xf(x)$ for $x>0$, and $f$ is logarithmically convex.

Proof

References

Theorem: The following formula holds: $$\displaystyle\lim_{t \rightarrow \infty} \dfrac{\Gamma(t+\alpha)}{\Gamma(t)t^{\alpha}}=\displaystyle\lim_{t \rightarrow \infty} \dfrac{\Gamma(t)t^{\alpha}}{\Gamma(t+\alpha)}=1.$$

Proof: proof goes here █

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

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