# Elliptic integral

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Related to Elliptic integrals: elliptic integral of the second kind, elliptic integral of the third kind, elliptic integral of the first kind

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Also found in: Encyclopedia.

Related to Elliptic integrals: elliptic integral of the second kind, elliptic integral of the third kind, elliptic integral of the first kind

k is the complete elliptic integral of the second kind of modulus k and additional modulus of complete elliptic integrals, k' = [square root of 1 - [k.sup.2]] [11].

Using quaternion calculus of variations and optimal control theory, Arthurs and Walsh [2] obtained a system of differential equations that can be integrated analytically in terms of elliptic integrals.

Cody, "Chebyshev approximations for the complete elliptic integrals," Mathematics of Computation, vol.

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.

An axisymmetric collocation boundary element method with the polynomial approximation of the complete elliptic integrals is a well-known technique for solution of the potential problems.

The oblate spheroidal solutions that result if [DELTA][rho] > 0 were obtained in terms of elliptic integrals by Chandrasekhar [7].

Fortunately, the transformation for calculation of parallel-plate capacitance can be expressed by a linear combination of elliptic integrals.

Here, K, K' are the complete elliptic integrals of the first kind:

The ellipsoidal inner integral for each surface area can be determined analytically (which involves using elliptic integrals).

In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse.

Note that all functions used in the analysis (e.g., elliptic integrals of first kind) are available in most mathematical software packages.

The hypergeometric function and complete elliptic integrals. Given complex numbers a, b, and c with c [not equal to] 0,-1,-2, ..., the Gaussian hypergeometric function is the analytic continuation to the slit plane C \ [1,[infinity]) of the series

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