In 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 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^ ...

\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 except th ...

\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 ...

\Beta(n,m) = \frac = \frac = \left( \frac + \frac \right) \frac

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

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

See also

References

External links and references