Difference between revisions of "Gamma"
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Revision as of 05:55, 23 September 2014
The gamma function is the function defined by the integral (initially for positive values of $x$) $$\Gamma(x)=\displaystyle\int_0^{\infty} x^{t-1}e^{-x} dx.$$
Properties
Proposition: $\Gamma(1)=1$
Proof: Compute directly $$\begin{array}{ll} \Gamma(1) &= \displaystyle\int_0^{\infty} e^{-t}t^{1-1} dt \\ &= \displaystyle\int_0^{\infty} e^{-t} dt \\ &= \left[ -e^{-t} \right]_{-\infty}^{\infty} \\ &= 0. \end{array}$$ █
Theorem: $\Gamma(x+1)=x\Gamma(x); x>0$
Proof: proof goes here █
Theorem: If $x \in \mathbb{N}$, then $\Gamma(x+1)=x!$.
Proof: proof goes here █
Theorem (Legendre Duplication Formula): $$\Gamma(2x)=\dfrac{2^{2x-1}}{\sqrt{\pi}} \Gamma(x)\Gamma \left( x +\dfrac{1}{2} \right).$$
Proof: proof goes here █
Proposition: If $z=0,-1,-2,\ldots$ then $\Gamma(z)=\infty$.
Proof: proof goes here █
Theorem: The following relationship between $\Gamma$ and the $\sin$ function holds: $$\Gamma(x)\Gamma(1-x) = \dfrac{\pi}{\sin(\pi x)}.$$
Proof: proof goes here █
Proposition: $\Gamma(x) = \displaystyle\lim_{n \rightarrow \infty} \dfrac{n^x n!}{x(x+1)\ldots(x_n)}$
Proof: proof goes here █
Proposition: $\Gamma(x)\Gamma(1-x)=\dfrac{\pi}{\sin(\pi x)}$
Proof: proof goes here █
Bohr-Mollerup Theorem: The gamma function is the unique function $f$ such that
- $f(1)=1$
- $f(x+1)=xf(x)$ for $x>0$
- $f$ is logarithmically convex.
Proof: proof goes here █
