# elliptic integral of the second kind

## elliptic integral of the second kind

[ə¦lip·tik ¦int·ə·grəl əvthə ¦sek·ənd ‚kīnd]
(mathematics)
Any elliptic integral which approaches infinity as one of the limits of integration y approaches infinity, or which is infinite for some value of y, but which has no logarithmic singularities in y.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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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]] .
where E([THETA] | m) is the elliptic integral of the second kind, am(x | m) is the Jacobi elliptic function amplitude, and E(m) is the complete elliptic integral of the second kind.
A calculation of [partial derivative][u.sup.*.sub.ax]/[partial derivative]n, with the use of the relation for K'(m) in terms of the complete elliptic integral of the second kind E(m), gives
The complete elliptic integral of the second kind E is proportional to the circumference of the ellipse C:
where [omega] = (1/2) [square root of -A([[gamma].sub.1] - [[gamma].sub.3])], k = [square root of ([[gamma].sub.2] - [[gamma].sub.3])/([[gamma].sub.1] - [[gamma].sub.3])], sn(*, k) is the Jacobian elliptic function with the modulus k, E(am([u.sub.1],k),k) is the normal elliptic integral of the second kind, and am([u.sub.1], k) reads amplitude [u.sub.1] (see ).
Complete elliptic integral of the second kind is defined as The complete elliptic integral of the second kind E is proportional to the circumference of the ellipse C

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