Proposition: $\Gamma(1)=1$

Theorem: $\Gamma(x+1)=x\Gamma(x); x>0$

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

Proposition: The following formula holds $$\dfrac{1}{\Gamma(z)} = ze^{\gamma z} \displaystyle\prod_{k=1}^{\infty} \left( 1 + \dfrac{z}{k}\right)e^{-\frac{z}{k}},$$ where $\gamma$ is the Euler-Mascheroni constant.

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

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

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

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

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

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

Theorem: The following relationship between $\Gamma$ and the $\sin$ function holds: $$\Gamma(x)\Gamma(1-x) = \dfrac{\pi}{\sin(\pi x)}.$$

Proposition: $\Gamma(x) = \displaystyle\lim_{n \rightarrow \infty} \dfrac{n^x n!}{x(x+1)\ldots(x_n)}$

Proposition: $\Gamma(x)\Gamma(1-x)=\dfrac{\pi}{\sin(\pi x)}$

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.