The variable  (also denoted ) used in elliptic functions and elliptic integrals is called the amplitude (or Jacobi amplitude). It can be defined by

			(1)
			(2)

where  is a Jacobi elliptic function with elliptic modulus. As is common with Jacobi elliptic functions, the modulus  is often suppressed for conciseness. The Jacobi amplitude is the inverse function of the elliptic integral of the first kind. The amplitude function is implemented in the Wolfram Language as JacobiAmplitude[u, m], where  is the parameter.

It is related to the elliptic integral of the first kind  by

(3)

(Abramowitz and Stegun 1972, p. 589).

The derivative of the Jacobi amplitude is given by

(4)

or using the notation ,

(5)

The amplitude function has the special values

			(6)
			(7)

where  is a complete elliptic integral of the first kind. In addition, it obeys the identities

			(8)
			(9)
			(10)
			(11)
			(12)
			(13)

which serve as definitions for the Jacobi elliptic functions.

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 589-590, 1972.

Fischer, G. (Ed.). Plate 132 in Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Bildband.Braunschweig, Germany: Vieweg, p. 129, 1986.

Jacobi, C. G. J. J. für Math. 18, 12 and 20, 1838.

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, p. 494, 1990.