Elliptic Integral of the First Kind

Let the elliptic modulus  satisfy , and the Jacobi amplitude be given by  with . The incomplete elliptic integral of the first kind is then defined as

(1)

The elliptic integral of the first kind is implemented in the Wolfram Language as EllipticF[phi, m] (note the use of the parameter  instead of the modulus ).

Letting

			(2)
			(3)
			(4)

Equation (1) can be written as

			(5)
			(6)

Letting

			(7)
			(8)

then the integral can also be written as

(9)

where  is the complementary elliptic modulus.

The inverse function of  is given by the Jacobi amplitude

(10)

The integral

(11)

which arises in computing the period of a pendulum, is also an elliptic integral of the first kind. Use

			(12)
			(13)

to write

			(14)
			(15)
			(16)

so

(17)

Now let

(18)

so the angle  is transformed to

(19)

which ranges from 0 to  as  varies from 0 to . Taking the differential gives

(20)

or

(21)

Plugging this in gives

			(22)
			(23)
			(24)

so

			(25)
			(26)

Making the slightly different substitution , so  leads to an equivalent, but more complicated expression involving an incomplete elliptic integral of the first kind,

			(27)
			(28)

Therefore, the identity

(29)

holds over at least some region of the complex plane. The region of applicability is , which is shown above.

The elliptic integral of the first kind satisfies

(30)

Special values of  include

			(31)
			(32)

where  is known as the complete elliptic integral of the first kind.

REFERENCES:

Abramowitz, M. and Stegun, I. 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.

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000.

Spanier, J. and Oldham, K. B. "The Complete Elliptic Integrals  and " and "The Incomplete Elliptic Integrals and ." Chs. 61-62 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 609-633, 1987.

Tölke, F. "Parameterfunktionen." Ch. 3 in Praktische Funktionenlehre, zweiter Band: Theta-Funktionen und spezielle Weierstraßsche Funktionen. Berlin: Springer-Verlag, pp. 83-115, 1966.

Tölke, F. "Umkehrfunktionen der Jacobischen elliptischen Funktionen und elliptische Normalintegrale erster Gattung. Elliptische Amplitudenfunktionen sowie Legendresche F- und E-Funktion. Elliptische Normalintegrale zweiter Gattung. Jacobische Zeta- und Heumansche Lambda-Funktionen," and "Normalintegrale dritter Gattung. Legendresche -Funktion. Zurückführung des allgemeinen elliptischen Integrals auf Normalintegrale erster, zweiter, und dritter Gattung." Chs. 6-7 in Praktische Funktionenlehre, dritter Band: Jacobische elliptische Funktionen, Legendresche elliptische Normalintegrale und spezielle Weierstraßsche Zeta- und Sigma Funktionen. Berlin: Springer-Verlag, pp. 58-144, 1967.

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

Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 122, 1997.