Gamma

The gamma function is the function defined by the integral (initially for positive values of $x$) by the formula $$\Gamma(x)=\displaystyle\int_0^{\infty} \xi^{x-1}e^{-\xi} d\xi.$$ The analytic continuation of $\Gamma$ leads to a meromorphic function with poles at the negative integers.

• Gamma.png

  Graph of $\Gamma$ on $[-7,7] \subset \mathbb{R}$.

• Complex gamma.jpg

  Domain coloring of analytic continuation of $\Gamma$.

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  $ (1948).

Recurrence

Proposition: $\Gamma(1)=1$

Proof: Compute directly $$\begin{array}{ll} \Gamma(1) &= \displaystyle\int_0^{\infty} \xi^{1-1} e^{-\xi} d\xi \\ &= \displaystyle\int_0^{\infty} e^{-\xi} d\xi \\ &= \left[ -e^{-\xi} \right]_{0}^{\infty} \\ &= 1.█ \end{array}$$

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

Proof: Use integration by parts to compute $$\begin{array}{ll} \Gamma(x+1) &= \displaystyle\int_0^{\infty} \xi^x e^{-\xi} d\xi \\ &= -\xi^x e^{-\xi}|_0^{\infty} \displaystyle\int_0^{\infty} x \xi^{x-1} e^{-\xi} d\xi \\ &= x\Gamma(x).█ \end{array}$$

Corollary: If $x \in \mathbb{N}$, then $\Gamma(x+1)=x!$, where $x!$ denotes the factorial.

Other formulas

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

Proof: █

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.

Proof: █

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: █

$\Gamma(z)$ at half integers

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

Proof: █

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

Other properties

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: 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: █

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: █

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