Difference between revisions of "Gamma"
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*$f(1)=1$ | *$f(1)=1$ | ||
*$f(x+1)=xf(x)$ for $x>0$ | *$f(x+1)=xf(x)$ for $x>0$ | ||
| − | *$f$ is [[logarithmically convex]] | + | *$f$ is [[logarithmically convex]]. |
<div class="mw-collapsible-content"> | <div class="mw-collapsible-content"> | ||
<strong>Proof:</strong> proof goes here █ | <strong>Proof:</strong> proof goes here █ | ||
</div> | </div> | ||
</div> | </div> | ||
Revision as of 11:01, 21 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
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 █
