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Carlson Elliptic Integrals

The Carlson elliptic integrals, also known as the Carlson symmetric forms, are a standard set of canonical elliptic integrals which provide a convenient alternative to Legendre's elliptic integrals of the first, second, and third kind. Carlson and Legendre elliptic integrals may be converted to each other.

The Carlson elliptic integrals are defined as

R_C(x,y)=R_F(x,y,y)
(1)
=1/2int_0^infty(dt)/((t+y)sqrt(t+x))
(2)
R_D(x,y,z)=R_J(x,y,z,z)
(3)
=3/2int_0^infty(dt)/(sqrt(t+x)sqrt(t+y)(t+z)^(3/2))
(4)
R_E(x,y)=1/piint_0^infty(x/(t+x)+y/(t+y))(sqrt(t)dt)/(sqrt(t+x)sqrt(t+y))
(5)
R_F(x,y,z)=1/2int_0^infty(dt)/(sqrt(t+x)sqrt(t+y)sqrt(t+z))
(6)
R_G(x,y,z)=1/4int_0^infty((xt)/(t+x)+(yt)/(t+y)+(zt)/(t+z))(dt)/(sqrt(t+x)sqrt(t+y)sqrt(t+z))
(7)
R_J(x,y,z,p)=3/2int_0^infty(dt)/((t+p)sqrt(t+x)sqrt(t+y)sqrt(t+z))
(8)
R_K(x,y)=1/piint_0^infty(dt)/(sqrt(t)sqrt(t+x)sqrt(t+y))
(9)
R_M(x,y,p)=2/piint_0^infty(dt)/((t+p)sqrt(t+x)sqrt(t+y)).
(10)

They are implemented in the Wolfram Language as CarlsonRC[x, y], CarlsonRD[x, y, z], CarlsonRE[x, y], CarlsonRF[x, y, z], CarlsonRG[x, y, z], CarlsonRJ[x, y, z, rho], CarlsonRK[x, y], and CarlsonRM[x, y, rho].

For 0<=phi<=2pi and 0<=k^2sin^2phi<=1, the incomplete elliptic integrals of the first, second, and third kind are related to the Carlson elliptic integrals by

F(phi,k)=sinphiR_F(cos^2phi,1-k^2sin^2phi,1)
(11)
E(phi,k)=sinphiR_F(cos^2phi,1-k^2sin^2phi,1)-1/3k^2sin^3phiR_D(cos^2phi,1-k^2sin^2phi,1)
(12)
Pi(phi,n,k)=sinphiR_F(cos^2phi,1-k^2sin^2phi,1)+1/3nsin^3phiR_J(cos^2phi,1-k^2sin^2phi,1,1-nsin^2phi).
(13)

Expressing the complete Legendre-Jacobi integrals in terms of the incomplete Carlson integrals by plugging phi=pi/2 into the above gives

K(k)=R_F(0,1-k^2,1)
(14)
E(k)=R_F(0,1-k^2,1)-1/3k^2R_D(0,1-k^2,1)
(15)
Pi(n,k)=R_F(0,1-k^2,1)+1/3nR_J(0,1-k^2,1,1-n)
(16)

(Press and Teukolsky 1990) and

K(k)=1/2piR_K(1,1-k^2)
(17)
E(k)=1/2piR_E(1,1-k^2)
(18)
Pi(n,k)=1/2piR_K(1,1-k^2)+1/4npiR_M(1,1-k^2,1-n).
(19)

The functions also satisfy the following homogeneity properties:

R_F(kappax,kappay,kappaz)=kappa^(-1/2)R_F(x,y,z)
(20)
R_J(kappax,kappay,kappaz,kappap)=kappa^(-3/2)R_J(x,y,z,p)
(21)

(Press and Teukolsky 1990).

Special values include

R_D(0,2,1)=(3pi)/L
(22)
R_F(0,1,2)=L/2
(23)
R_K(1,2)=L/pi,
(24)

where L is the lemniscate constant.

See also

Elliptic Integral

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References

Carlson, B. C. Special Functions of Applied Mathematics. New York: Academic Press, 1977.Carlson, B. C. "Elliptic Integrals of the First Kind." SIAM J. Math. Anal. 8, 231-242, 1977.Carlson, B. C. "A Table of Elliptic Integrals of the Second Kind." Math. Comput. 49, 595-606, 1987.Carlson, B. C. "A Table of Elliptic Integrals of the Third Kind." Math. Comput. 51, 267-280, 1988.Carlson, B. C. "Numerical Computation of Real or Complex Elliptic Integrals." Numer. Algorithms 10, 13-26, 1995.Carlson, B. C. "Elliptic Integrals." Ch. 19 in Digital Library of Mathematical Functions. 2020-12-15. https://dlmf.nist.gov/19.Press, W. H. and Teukolsky, S. A. "Elliptic Integrals." Computers in Physics 4, 92-98, 1990.

Cite this as:

Weisstein, Eric W. "Carlson Elliptic Integrals." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CarlsonEllipticIntegrals.html

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