Euler's Integral definition of Euler's Integral in the Free Online Encyclopedia.

Euler’s Integral

The integral

is known as Euler’s integral of the first kind or the beta function. It was studied by L. Euler in 1730 and 1731 and had been previously considered by I. Newton and J. Wallis.

The integral

is known as Euler’s integral of the second kind or the gamma function. It was considered by Euler in 1729 and 1730 in a form equivalent to equation (2); equation (2) itself was given by Euler in 1781.

The name “Euler’s integral” was introduced by A. Legendre. Such integrals make it possible to generalize to the case of continuously varying arguments the binomial coefficients and the factorial n!: if a and b are natural numbers, then

Integrals (1) and (2) converge absolutely if a and b are positive; they do not exist if a and b are negative. The following relations hold:

The second relation expresses the beta function in terms of the gamma function. There are a number of relations between Euler’s integrals, for different values of the argument, that generalize the corresponding relations between the binomial coefficients. Euler’s integrals also may be considered for complex values of the arguments a and b. Euler’s integrals are encountered in many problems of the theory of special functions; many definite integrals that do not have an elementary expression reduce to Euler’s integrals.

The name “Euler’s integral” is also sometimes applied to the integral

which expresses the hypergeometric function.

REFERENCES

Fikhtengol’ts, G. M. Kurs differentsial’nogo i integral’nogo ischisleniia, 7th ed., vol. 2. Moscow, 1969.
Artin, E. Vvedenie v teoriiu gamma-funktsii. Moscow-Leningrad, 1934. (Translated from German.)
Whittaker, E. T., and G. N. Watson. Kurs sovremennogo analiza, 2nd ed., part 2. Moscow, 1963. (Translated from English.)

Mentioned in	?		Encyclopedia browser	?		Full browser	?
No references found	Eulenspiegel, Till Euler Euler Angles Euler Characteristic Euler characteristic of a topological space X Euler diagram Euler equation Euler equations of motion Euler force Euler formula for long columns Euler method Euler Number Euler number 1 Euler number 2 Euler Numbers Euler Phi-Function Euler Substitutions Euler transformation Euler's Constant Euler's criterion Euler's Equation Euler's Equations Euler's equations of motion Euler's expansion Euler's Formula Euler's Integral Euler's numbers Euler's phi function Euler's spiral Euler's theorem Euler, Leonhard Euler, Ulf Svante Von Euler-Chelpin Euler-Chelpin, Hans Karl August Simon Von Euler-Fourier Formulas Euler-Lagrange equation Euler-Maclaurin formula Euler-Maclaurin Summation Formula Euler-Rodrigues parameter Eulerian coordinates Eulerian correlation Eulerian description Eulerian equation Eulerian graph Eulerian nutation Eulerian path Eulerian wind Euless Euli, Sandro EuLisp eulittoral	Euler totient theorem Euler tour Euler trail Euler transformation Euler Venn diagram Euler walk Euler wenn diagram Euler zeta function Euler's angle Euler's Box Euler's Box Euler's circles Euler's constant Euler's criterion Euler's equation Euler's equation of motion Euler's equations Euler's equations of motion Euler's expansion Euler's formula Euler's formula for polyhedra Euler's formula in complex analysis Euler's forward method Euler's generalization Euler's homogeneous function theorem Euler's Integral Euler's method Euler's numbers Euler's phi function Euler's polyhedron formula Euler's polyhedron theorem Euler's quadratic residue theorem Euler's spiral Euler's summation formula Euler's theorem Euler's theorem on homogeneous functions Euler's totient function Euler's totient rule Euler's totient theorem Euler, Leonhard Euler, Leonhard Euler, Leonhard Euler, Ulf Svante von Euler, Ulf Svante von Euler, Ulf Svante von Euler-Chelpin Euler-Chelpin Euler-Chelpin, Hans Karl August Simon von Euler-Chelpin, Hans Karl August Simon von Euler-Fermat theorem Euler-Fourier Formulas Euler-Lagrange Derivative Euler-Lagrange differential equation Euler-Lagrange differential equations Euler-Lagrange equation Euler-Lagrange equations

Mentioned in

Encyclopedia browser

Full browser

No references found

Euler characteristic of a topological space X

Euler diagram

Euler equation

Euler equations of motion