In
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
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 i ...
of the first kind'' is the
beta function
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral
: \Beta(z_1,z_2) = \int_0^1 t^( ...
# 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 except ...
For
positive integers 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) \t ...
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:
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 in ma ...
*
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 includ ...
References
External links and references
Wolfram MathWorld on the Euler Integral* NIST Digital Library of Mathematical Function
dlmf.nist.gov/5.2.1relation 5.2.1 an
dlmf.nist.gov/5.12relation 5.12.1
Gamma and related functions
{{sia, mathematics