Gamma function

Proof of factorial representation

The factorial definition of the gamma function can be derived using integration by parts. $${\displaystyle \int _{a}^{b}u\,dv\ =\ {\Bigl [}uv{\Bigr ]}_{a}^{b}-\int _{a}^{b}v\,du.}$$ Let ${\displaystyle u=t^{z}}$ so ${\displaystyle du=zt^{z-1}}$ and ${\displaystyle dv=e^{-t}}$ so ${\displaystyle v=-e^{-t}}$ $${\displaystyle \Gamma (z+1):=\int _{0}^{\infty }t^{z}e^{-t}\,dt}$$ so by applying integration by parts, $${\displaystyle {\begin{aligned}\Gamma (z+1)&={\Bigl [}-t^{z}e^{-t}{\Bigr ]}_{0}^{\infty }+\int _{0}^{\infty }z\,t^{z-1}e^{-t}\,dt=z\,\Gamma (z).\end{aligned}}}$$ as the first part vanishes (equals 0). Note that ${\displaystyle \Gamma (1)=1}$. Thus, ${\displaystyle \Gamma (z)=(z-1)!}$ for ${\displaystyle Re(z)>0}$, that is, real part of ${\displaystyle z}$ is positive.

Particular values

Some particular values of the gamma function are:

${\displaystyle {\begin{array}{lll}\Gamma (-3/2)&={\tfrac {4}{3}}{\sqrt {\pi }}&\approx 2.363271801207\\\Gamma (-1/2)&=-2{\sqrt {\pi }}&\approx -3.544907701811\\\Gamma (1/2)&={\sqrt {\pi }}&\approx 1.772453850905\\\Gamma (1)&=0!&=1\\\Gamma (3/2)&={\tfrac {1}{2}}{\sqrt {\pi }}&\approx 0.88622692545\\\Gamma (2)&=1!&=1\\\Gamma (5/2)&={\tfrac {3}{4}}{\sqrt {\pi }}&\approx 1.32934038818\\\Gamma (3)&=2!&=2\\\Gamma (7/2)&={\tfrac {15}{8}}{\sqrt {\pi }}&\approx 3.32335097045\\\Gamma (4)&=3!&=6\\\end{array}}}$

Pi function

Gauss introduced the Pi function. This is another way of denoting the gamma function. In terms of the gamma function, the Pi function is

${\displaystyle \Pi (z)=\Gamma (z+1)=z\;\Gamma (z)=\int _{0}^{\infty }e^{-t}t^{z+1}\,{\frac {{\rm {d}}t}{t}},}$

so that

${\displaystyle \Pi (n)=n!}$

for every non-negative integer n.

Analytic number theory

The gamma function is used to study the Riemann zeta function. A property of the Riemann zeta function is its functional equation:

${\displaystyle \Gamma \left({\frac {s}{2}}\right)\zeta (s)\pi ^{-s/2}=\Gamma \left({\frac {1-s}{2}}\right)\zeta (1-s)\pi ^{-(1-s)/2}.}$

Bernhard Riemann found a relation between these two functions. This was published in his 1859 paper "Über die Anzahl der Primzahlen unter einer gegebenen Grösse" ("On the Number of Prime Numbers less than a Given Quantity")

${\displaystyle \zeta (z)\;\Gamma (z)=\int _{0}^{\infty }{\frac {t^{z}}{e^{t}-1}}\;{\frac {dt}{t}}.}$