Complete Elliptic Integral of the First Kind

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

			(1)
			(2)
			(3)

where  is the incomplete elliptic integral of the first kind and  is the hypergeometric function.

It is implemented in the Wolfram Language as EllipticK[m], where  is the parameter.

It satisfies the identity

(4)

where  is a Legendre polynomial. This simplifies to

(5)

for all complex values of  except possibly for real  with .

In addition,  satisfies the identity

(6)

where  is the complementary modulus. Amazingly, this reduces to the beautiful form

(7)

for  (Watson 1908, 1939).

 can be computed in closed form for special values of , where  is a called an elliptic integral singular value. Other special values include

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

 satisfies

(13)

possibly modulo issues of , which can be derived from equation 17.4.17 in Abramowitz and Stegun (1972, p. 593).

 is related to the Jacobi elliptic functions through

(14)

where the nome is defined by

(15)

with , where  is the complementary modulus.

 satisfies the Legendre relation

(16)

where  and  are complete elliptic integrals of the first and second kinds, respectively, and  and are the complementary integrals. The modulus  is often suppressed for conciseness, so that  and  are often simply written  and , respectively.

The integral  for complementary modulus is given by

			(17)
			(18)

and  satisfies the differential equation

(19)

so

			(20)
			(21)

(Whittaker and Watson 1990, pp. 499 and 521).

The solution to the differential equation

(22)

(Zwillinger 1997, p. 122; Gradshteyn and Ryzhik 2000, p. 907) is

(23)

where the two solutions are illustrated above and .

Definite integrals of  include

			(24)
			(25)
			(26)
			(27)

where  (not to be confused with ) is Catalan's constant.

REFERENCES:

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

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

Watson G. N. "The Expansion of Products of Hypergeometric Functions." Quart. J. Pure Appl. Math. 39, 27-51, 1907.

Watson G. N. "A Series for the Square of the Hypergeometric Function." Quart. J. Pure Appl. Math. 40, 46-57, 1908.

Watson, G. N. "Three Triple Integrals." Quart. J. Math., Oxford Ser. 2 10, 266-276, 1939.

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.