Euler integral

For the Euler–Poisson integral, see Gaussian integral.

In mathematics, there are two types of Euler integral:

1. The Euler integral of the first kind is the beta function $$\mathrm{B}(z_1,z_2)=\underset{0}{\overset{1}{\int }}{t}^{z_1-1}(1-t{)}^{z_2-1}dt=\frac{\mathrm{\Gamma }(z_1)\mathrm{\Gamma }(z_2)}{\mathrm{\Gamma }(z_1+z_2)}$$
2. The Euler integral of the second kind is the gamma function $$\mathrm{\Gamma }(z)=\underset{0}{\overset{\mathrm{\infty }}{\int }}{t}^{z-1}{\mathrm{e}}^{-t}dt$$

For positive integers m and n, the two integrals can be expressed in terms of factorials and binomial coefficients: $$\mathrm{B}(n,m)=\frac{(n-1)!(m-1)!}{(n+m-1)!}=\frac{n+m}{nm(\genfrac{}{}{0ex}{}{n+m}{n})}=(\frac{1}{n}+\frac{1}{m})\frac{1}{(\genfrac{}{}{0ex}{}{n+m}{n})}$$ $$\mathrm{\Gamma }(n)=(n-1)!$$

See also

• Leonhard Euler
• List of topics named after Leonhard Euler

External links and references

• Wolfram MathWorld on the Euler Integral
• NIST Digital Library of Mathematical Functions dlmf.nist.gov/5.2.1 relation 5.2.1 and dlmf.nist.gov/5.12 relation 5.12.1