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Complete Elliptic Integral of the First Kind

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

K(k)=F(1/2pi,k)
(1)
=pi/2sum_(n=0)^(infty)[((2n-1)!!)/((2n)!!)]^2k^(2n)
(2)
=1/2pi_2F_1(1/2,1/2;1;k^2)
(3)

where F(phi,k) is the incomplete elliptic integral of the first kind and _2F_1(a,b;c;x) is the hypergeometric function.

It is implemented in the Wolfram Language as EllipticK[m], where m=k^2 is the parameter.

It satisfies the identity

pi/(2sqrt(1-k^2))P_(-1/2)((1+k^2)/(1-k^2))=1/(sqrt(1-k^2))K(sqrt((k^2)/(k^2-1))),
(4)

where P_n(x) is a Legendre polynomial. This simplifies to

pi/(2sqrt(1-k^2))P_(-1/2)((1+k^2)/(1-k^2))=K(k)
(5)

for all complex values of k except possibly for real k with |k|>1.

In addition, K(k) satisfies the identity

[K(sqrt(1/2(1-sqrt((1-2k^2)^2))))]^2 =(pi^2)/4sum_(n=0)^infty[((2n-1)!!)/((2n)!!)]^3(2kk^')^(2n),
(6)

where k^'=sqrt(1-k^2) is the complementary modulus. Amazingly, this reduces to the beautiful form

[K(k)]^2=(pi^2)/4sum_(n=0)^infty[((2n-1)!!)/((2n)!!)]^3(2kk^')^(2n)
(7)

for 0<k<=1/sqrt(2) (Watson 1908, 1939).

K(k) can be computed in closed form for special values of k=k_n, where k_n is a called an elliptic integral singular value. Other special values include

K(-iinfty)=0
(8)
K(-infty)=0
(9)
K(0)=1/2pi
(10)
K(infty)=0
(11)
K(iinfty)=0.
(12)

K(ik) satisfies

K(ik)=1/(sqrt(k^2+1))K(sqrt((k^2)/(k^2+1)))
(13)

possibly modulo issues of sqrt(k^2), which can be derived from equation 17.4.17 in Abramowitz and Stegun (1972, p. 593).

K(k) is related to the Jacobi elliptic functions through

K(k)=1/2pitheta_3^2(q),
(14)

where the nome is defined by

q=e^(-piK^'(k)/K(k)),
(15)

with K^'(k)=K(k^'), where k^'=sqrt(1-k^2) is the complementary modulus.

K(k) satisfies the Legendre relation

E(k)K^'(k)+E^'(k)K(k)-K(k)K^'(k)=1/2pi,
(16)

where K(k) and E(k) are complete elliptic integrals of the first and second kinds, respectively, and K^'(k) and E^'(k) are the complementary integrals. The modulus k is often suppressed for conciseness, so that K(k) and E(k) are often simply written K and E, respectively.

The integral K(k^') for complementary modulus is given by

K(k^')=int_0^1(dt)/(sqrt((1-t^2)(1-k^('2)t^2)))
(17)

(Whittaker and Watson 1990, p. 501), and

(dK)/(dk)=(E(k))/(k(1-k^2))-(K(k))/k
(18)
d/(dk)(kk^('2)(dK)/(dk))=kK(k)
(19)

(Whittaker and Watson 1990, p. 521), so

E(k)=k(1-k^2)[(dK)/(dk)+(K(k))/k]
(20)
=(1-k^2)[k(dK)/(dk)+K(k)]
(21)

(cf. Whittaker and Watson 1990, p. 521).

EllipticKODE

The solution to the differential equation

d/(dk)[k(1-k^2)(dy)/(dk)]-ky=0
(22)

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

y=C_1K(k)+C_2K^'(k),
(23)

where the two solutions are illustrated above and K^'(k)=K(sqrt(1-k^2)).

Definite integrals of K(k) include

int_0^1K(k)dk=2K
(24)
int_0^1K(sqrt(k))dk=2
(25)
int_0^1K(k^(1/4))dk=(20)/9
(26)
int_0^1(K(k^(1/4)))/(k^(1/4))dk=4,
(27)

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

See also

Complete Elliptic Integral of the Third Kind, Complete Elliptic Integral of the Second Kind, Elliptic Integral of the First Kind, Elliptic Integral Singular Value

Related Wolfram sites

http://functions.wolfram.com/EllipticIntegrals/EllipticK/

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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.

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Complete Elliptic Integral of the First Kind

Cite this as:

Weisstein, Eric W. "Complete Elliptic Integral of the First Kind." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CompleteEllipticIntegraloftheFirstKind.html

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