Carlson symmetric form

Carlson symmetric form

In mathematics, the Carlson symmetric forms of elliptic integrals, R_C, R_D, R_F and R_J are defined by

:R_C(x,y) := frac{1}{2} int_0^infty (t+x)^{-1/2} (t+y)^{-1},dt

:R_D(x,y,z) := frac{3}{2} int_0^infty (t+x)^{-1/2} (t+y)^{-1/2} (t+z)^{-3/2},dt

:R_F(x,y,z) := frac{1}{2} int_0^infty (t+x)^{-1/2} (t+y)^{-1/2} (t+z)^{-1/2},dt

:R_J(x,y,z,p) := frac{3}{2} int_0^infty (t+x)^{-1/2} (t+y)^{-1/2} (t+z)^{-1/2} (t+p)^{-1},dt

Note that R_C is a special case of R_F and R_D is a special case of R_J;

:R_Cleft(x,y ight)=R_Fleft(x,y,y ight)

:R_Dleft(x,y,z ight)=R_Jleft(x,y,z,z ight).

The term "symmetric" refers to the fact that these functions are unchanged by the exchange of certain of their arguments. The value of R_F(x,y,z) is the same for any permutation of its arguments, and the value of R_J(x,y,z,p) is the same for any permutation of its first three arguments.

Relations concerning to Legendre form of elliptic integrals

Incomplete elliptic integrals

Incomplete elliptic integrals can be calculated easily using Carlson symmetric forms:

:F(phi,k)=sinphi R_Fleft(cos^2phi,1-k^2sin^2phi,1 ight)

:E(phi,k)=sinphi R_Fleft(cos^2phi,1-k^2sin^2phi,1 ight) -frac{1}{3}k^2sin^3phi R_Dleft(cos^2phi,1-k^2sin^2phi,1 ight)

:Pi(phi,n,k)=sinphi R_Fleft(cos^2phi,1-k^2sin^2phi,1 ight)+frac{1}{3}nsin^3phi R_Jleft(cos^2phi,1-k^2sin^2phi,1,1-nsin^2phi ight)

Complete elliptic integrals

Complete elliptic integrals can be calculated by substituting phi=frac{pi}{2}:

:K(k)=R_Fleft(0,1-k^2,1 ight)

:E(k)=R_Fleft(0,1-k^2,1 ight)-frac{1}{3}k^2 R_Dleft(0,1-k^2,1 ight)

:Pi(n,k)=R_Fleft(0,1-k^2,1 ight)+frac{1}{3}n R_J left(0,1-k^2,1,1-n ight)

pecial cases

When any two, or all three of the arguments of R_F are the same, then a substitution of sqrt{t + x} = u renders the integrand rational. The integral can then be expressed in terms of elementary transcendental functions.

:R_{C}(x,y) = R_{F}(x,y,y) = frac{1}{2} int _{0}^{infty}frac{1}{sqrt{t + x} (t + y)} dt =int _{sqrt{x^{infty}frac{1}{u^{2} - x + y} du =egin{cases} frac{arccosleft( sqrt{x/y} ight)}{sqrt{y - x, & x < y \ frac{1}{sqrt{y, & x = y \ frac{mathrm{arccosh}left( sqrt{x/y} ight)}{sqrt{x - y, & x > y \ end{cases}

Other properties


By substituting in the integral definitions t = kappa u for any constant kappa, it is found that

:R_Fleft(kappa x,kappa y,kappa z ight)=kappa^{-1/2}R_F(x,y,z)

:R_Jleft(kappa x,kappa y,kappa z,kappa p ight)=kappa^{-3/2}R_J(x,y,z,p)

Duplication theorem

:R_F(x,y,z)=2R_F(x+lambda,y+lambda,z+lambda)=R_Fleft(frac{x+lambda}{4},frac{y+lambda}{4},frac{z+lambda}{4} ight),

where lambda=sqrt{xy}+sqrt{yz}+sqrt{zx}.

:egin{align}R_{J}(x,y,z,p) & = 2 R_{J}(x + lambda,y + lambda,z + lambda,p + lambda) + 6 R_{C}(d^{2},d^{2} + (p - x) (p - y) (p - z)) \ & = frac{1}{4} R_{J}left( frac{x + lambda}{4},frac{y + lambda}{4},frac{z + lambda}{4},frac{p + lambda}{4} ight) + 6 R_{C}(d^{2},d^{2} + (p - x) (p - y) (p - z)) end{align}

where d = (sqrt{p} + sqrt{x}) (sqrt{p} + sqrt{y}) (sqrt{p} + sqrt{z}) and lambda = sqrt{x y} + sqrt{y z} + sqrt{z x}

Numerical evaluation

The duplication theorem can be used for a fast and robust evaluation of the Carlson symmetric form of elliptic integralsand therefore also for the evaluation of Legendre-form of elliptic integrals. Let us calculate R_F(x,y,z):first, define x_0=x, y_0=y and z_0=z. Then iterate the series

:lambda_n=sqrt{x_ny_n}+sqrt{y_nz_n}+sqrt{z_nx_n},:x_{n+1}=frac{x_n+lambda_n}{4}, y_{n+1}=frac{y_n+lambda_n}{4}, z_{n+1}=frac{z_n+lambda_n}{4}until the desired precision is reached: if x, y and z are non-negative, all of the series will converge quickly to a given value, say, mu. Therefore,

:R_Fleft(x,y,z ight)=R_Fleft(mu,mu,mu ight)=mu^{-1/2}.

Evaluating R_C(x,y) is much the same due to the relation

:R_Cleft(x,y ight)=R_Fleft(x,y,y ight).

External links

* [ B. C. Carlson, John L. Gustafson 'Asymptotic approximations for symmetric elliptic integrals' 1993 arXiv]

* [ B. C. Carlson 'Numerical Computation of Real Or Complex Elliptic Integrals' 1994 arXiv]

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