Special Functions

Gamma Function

\[ \Gamma (z)=\int_{0}^{\infty} t^{z-1}e^{-t} dt \]

• \(\Gamma(n + 1) = n \Gamma(n)\)
• \(\Gamma(n) = (n-1)!\), for every positive integer
• \(\Gamma(1/2) = \sqrt{\pi}\)

Beta function (Euler integral of first kind)

\[ \mathrm {B} (z_{1},z_{2}) = \int_{0}^{1}t^{z_{1}-1}(1-t)^{z_{2}-1}dt \]

• symmetric \(\mathrm {B} (z_{1},z_{2}) = \mathrm {B} (z_{2},z_{1})\)
• relationship to gamma $$ \mathrm {B} (z_{1},z_{2}) = \frac{\Gamma(z_1)\Gamma(z_2)}{\Gamma(z_1+z_2)} $$
• \[ \mathrm {B} (z_{1},z_{2}) = \int_{0}^{\infty} \frac{t^{z_1 - 1}} {(1+x)^{z_1 + z_2}} dt \]
• \[ \mathrm {B} (z_{1},z_{2}) = 2 \int_{0}^{\pi/2} \sin^{2z_1 - 1}(t) \cos^{2z_2 - 1}(t) dt \]