Gamma

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 █