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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, there are two types of Euler integral: # The ''Euler
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
of the first kind'' is the beta function \mathrm(z_1,z_2) = \int_0^1t^(1-t)^\,dt = \frac # The ''Euler integral of the second kind'' is the
gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers excep ...
\Gamma(z) = \int_0^\infty t^\,\mathrm e^\,dt For
positive integer In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
s and , the two integrals can be expressed in terms of
factorial In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \ ...
s and
binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
s: \Beta(n,m) = \frac = \frac = \left( \frac + \frac \right) \frac \Gamma(n) = (n-1)!


See also

*
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...
*
List of topics named after Leonhard Euler 200px, Leonhard Euler (1707–1783) In mathematics and physics, many topics are named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler inclu ...


References


External links and references


Wolfram MathWorld on the Euler Integral
* NIST Digital Library of Mathematical Function
dlmf.nist.gov/5.2.1
relation 5.2.1 an
dlmf.nist.gov/5.12
relation 5.12.1 Gamma and related functions {{sia, mathematics