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.

:

Other properties

Homogeneity

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 $lambda=sqrt\{xy\}+sqrt\{yz\}+sqrt\{zx\}$.

:

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

* [http://arxiv.org/abs/math/9310223v1 B. C. Carlson, John L. Gustafson 'Asymptotic approximations for symmetric elliptic integrals' 1993 arXiv]

* [http://arxiv.org/abs/math/9409227v1 B. C. Carlson 'Numerical Computation of Real Or Complex Elliptic Integrals' 1994 arXiv]