释义 |
Elliptic Integral of the Second KindLet the Modulus satisfy . (This may also be written in terms ofthe Parameter or Modular Angle .) The incomplete elliptic integralof the second kind is then defined as
 | (1) |
A generalization replacing with gives
 | (2) |
To place the elliptic integral of the second kind in a slightly different form, let
so the elliptic integral can also be written as
The complete elliptic integral of the second kind, illustrated above as a function of the Parameter ,is defined by
where is the Hypergeometric Function and is a Jacobi Elliptic Function. The complete elliptic integral of the second kind satisfies the Legendre Relation
 | (10) |
where and are complete Elliptic Integrals of the First and second kinds,and and are the complementary integrals. The Derivative is
 | (11) |
(Whittaker and Watson 1990, p. 521). If is a singular value (i.e.,
 | (12) |
where is the Elliptic Lambda Function), and and the Elliptic Alpha Function arealso known, then
 | (13) |
See also Elliptic Integral of the First Kind, Elliptic Integral of the Third Kind, Elliptic Integral Singular Value References
Abramowitz, M. and Stegun, C. A. (Eds.). ``Elliptic Integrals.'' Ch. 17 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 587-607, 1972.Spanier, J. and Oldham, K. B. ``The Complete Elliptic Integrals and '' and ``The Incomplete Elliptic Integrals and .'' Chs. 61 and 62 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 609-633, 1987. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990. |