## Elliptic Integral of the Second Kind

Let the Modulus satisfy . (This may also be written in terms of the Parameter or Modular Angle .) The incomplete elliptic integral of 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
 (3) (4)

so the elliptic integral can also be written as
 (5)

The complete elliptic integral of the second kind, illustrated above as a function of the Parameter , is defined by

 (6) (7) (8) (9)

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 are also 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.