# Elliptic integral

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 (Math.) See Integral. one of an important class of integrals, occurring in the higher mathematics; - so called because one of the integrals expresses the length of an arc of an ellipse.

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1,2] are the outer and inner parts of the total contour, respectively; K(k) is the complete elliptic integral of the first kind of module k ;
In terms of the Gauss hypergeometric function, the complete elliptic integral of the first kind can be expressed as
j] is the complete elliptic integral of the first kind, and cd is Jacobi elliptic function.
0]) and [alpha] = 4K([beta])/[lambda], and K([beta]) is the complete elliptic integral of first kind.
In the case of real spectra, [phi] = [pi] / 2 and L[[pi] / 2, k] is a complete elliptic integral of the form
Complications formerly encountered in numerical computation of Legendre's complete elliptic integral of the third kind were avoided by defining and tabulating Heuman's Lambda function (for circular cases) and a modification of Jacobi's Zeta function (for hyperbolic cases).
Complete elliptic integral of the first kind is defined as Elliptic Integrals are said to be complete when the amplitude
Among the topics are complete elliptic integrals, the Riemann zeta function, some automatic proofs, the error function, hyper-geometric functions, Bessel-K functions, polylogarithm functions, evaluation by series, the exponential integral, confluent hyper-geometric and Whittaker functions, and the evaluation of entries in Gradshteyn and Ryzhik employing the method of brackets.
where E(w) and K(w) are respectively complete elliptic integrals of the first and second kind, defined according to the convention, (7)

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