Carlson symmetric form

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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 $R J$ ;

$R_C\left(x,y\right)=R_F\left(x,y,y\right)$

$R_D\left(x,y,z\right)=R_J\left(x,y,z,z\right).$

1 Relations concerning to Legendre form of elliptic integrals
- 1.1 Incomplete elliptic integrals
- 1.2 Complete elliptic integrals
2 Other properties
3 Numerical evaluation
4 External links

[edit] Relations concerning to Legendre form of elliptic integrals

[edit] Incomplete elliptic integrals

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

$F(\phi,k)=\sin\phi R_F\left(\cos^2\phi,1-k^2\sin^2\phi,1\right)$

$E(\phi,k)=\sin\phi R_F\left(\cos^2\phi,1-k^2\sin^2\phi,1\right) -\frac{1}{3}k^2\sin^3\phi R_D\left(\cos^2\phi,1-k^2\sin^2\phi,1\right)$

$\Pi(\phi,n,k)=\sin\phi R_F\left(\cos^2\phi,1-k^2\sin^2\phi,1\right)+ \frac{1}{3}n\sin^3\phi R_J\left(\cos^2\phi,1-k^2\sin^2\phi,1,1-n\sin^2\phi\right)$

[edit] Complete elliptic integrals

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

$K(k)=R_F\left(0,1-k^2,1\right)$

$E(k)=R_F\left(0,1-k^2,1\right)-\frac{1}{3}k^2 R_D\left(0,1-k^2,1\right)$

$\Pi(n,k)=R_F\left(0,1-k^2,1\right)+\frac{1}{3}n R_J \left(0,1-k^2,1,1-n\right)$

[edit] Other properties

Duplication theorem:

$R_F(x,y,z)=2R_F(x+\lambda,y+\lambda,z+\lambda)= R_F\left(\frac{x+\lambda}{4},\frac{y+\lambda}{4},\frac{z+\lambda}{4}\right),$

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

Homogeneity:

$R_F\left(\lambda x,\lambda y,\lambda z\right)=\lambda^{-1/2}R_F(x,y,z)$

$R_J\left(\lambda x,\lambda y,\lambda z,\lambda p\right)=\lambda^{-3/2}R_J(x,y,z,p)$

[edit] Numerical evaluation

The duplication theorem can be used for a fast and robust evaluation of the Carlson symmetric form of elliptic integrals and 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, $μ$ . Therefore,

$R_F\left(x,y,z\right)=R_F\left(\mu,\mu,\mu\right)=\mu^{-1/2}.$