Gaussian function

In mathematics, there are two types of Euler integral:

1. Euler integral of the first kind: the Beta function

$\mathrm {\mathrm {B} } (x,y)=\int _{0}^{1}t^{x-1}(1-t)^{y-1}\,dt={\frac {\Gamma (x)\Gamma (y)}{\Gamma (x+y)}}$

2. Euler integral of the second kind: the Gamma function

$\Gamma (z)=\int _{0}^{\infty }t^{z-1}\,e^{-t}\,dt$

For positive integers m and n

$\mathrm {\mathrm {B} } (n,m)={(n-1)!(m-1)! \over (n+m-1)!}={n+m \over nm{n+m \choose n}}$

$\Gamma (n)=(n-1)!\,$

See also

• Euler integral (thermodynamics)
• Leonhard Euler
• List of topics named after Leonhard Euler

References

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