Euler integral explained

In mathematics, there are two types of Euler integral:^[1]

1. The Euler integral of the first kind is the beta function $$\backslash mathrm(z\_1,z\_2)\; =\; \backslash int\_0^1t^(1-t)^\backslash ,dt\; =\; \backslash frac$$
2. The Euler integral of the second kind is the gamma function^[2] $$\backslash Gamma(z)\; =\; \backslash int\_0^\backslash infty\; t^\backslash ,\backslash mathrm\; e^\backslash ,dt$$

For positive integers and, the two integrals can be expressed in terms of factorials and binomial coefficients:$$\backslash Beta(n,m)\; =\; \backslash frac\; =\; \backslash frac\; =\; \backslash left(\backslash frac\; +\; \backslash frac\; \backslash right)\; \backslash frac$$$$\backslash Gamma(n)\; =\; (n-1)!$$

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

Notes and References

1. Book: Jeffrey . Alan . Dai . Hui-Hui . Handbook of mathematical formulas and integrals . 2008 . Elsevier Academic Press . 978-0-12-374288-9 . 4th . Amsterdam . 180880679 . 234–235.
2. Book: Jahnke, Hans Niels . A history of analysis . 2003 . American mathematical society . 978-0-8218-2623-2 . History of mathematics . Providence (R.I.) . 116-117.