Carlson symmetric form

In mathematics, the Carlson symmetric forms of elliptic integrals are a small canonical set of elliptic integrals to which all others may be reduced. They are a modern alternative to the Legendre forms. The Legendre forms may be expressed in terms of the Carlson forms and vice versa.

The Carlson elliptic integrals are:

<math display="block">R_F(x,y,z) = \tfrac{1}{2}\int_0^\infty \frac{dt}{\sqrt{(t+x)(t+y)(t+z)</math>

<math display="block">R_J(x,y,z,p) = \tfrac{3}{2}\int_0^\infty \frac{dt}{(t+p)\sqrt{(t+x)(t+y)(t+z)</math>

<math display="block">R_G(x,y,z) = \tfrac{1}{4}\int_0^\infty\frac{1}{\sqrt{(t+x)(t+y)(t+z)

\biggl(\frac{x}{t+x} + \frac{y}{t+y} + \frac{z}{t+z} \biggr) t\,dt </math>

<math display="block">R_C(x,y) = R_F(x,y,y) = \tfrac{1}{2} \int_0^\infty \frac{dt}{(t+y)\sqrt{(t+x)</math>

<math display="block">R_D(x,y,z) = R_J(x,y,z,z) = \tfrac{3}{2} \int_0^\infty \frac{dt}{ (t+z) \,\sqrt{(t+x)(t+y)(t+z)</math>

Since <math>R_C</math> and <math>R_D</math> are special cases of <math>R_F</math> and <math>R_J</math>, all elliptic integrals can ultimately be evaluated in terms of just <math>R_F</math>, <math>R_J</math>, and <math>R_G</math>.

The term symmetric refers to the fact that in contrast to the Legendre forms, these functions are unchanged by the exchange of certain subsets of their arguments. The value of <math>R_F(x,y,z)</math> is the same for any permutation of its arguments, and the value of <math>R_J(x,y,z,p)</math> is the same for any permutation of its first three arguments.

The Carlson elliptic integrals are named after Bille C. Carlson (1924–2013).

Relation to the Legendre forms

Incomplete elliptic integrals

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

:<math>\begin{align}

F(\phi,k)&=\sin\phi R_F\left(\cos^2\phi,1-k^2\sin^2\phi,1\right), \\[5mu]

E(\phi,k)&=\sin\phi R_F\left(\cos^2\phi,1-k^2\sin^2\phi,1\right) - \tfrac{1}{3}k^2\sin^3\phi R_D\left(\cos^2\phi,1-k^2\sin^2\phi,1\right), \\[5mu]

\Pi(\phi,n,k)&=\sin\phi R_F\left(\cos^2\phi,1-k^2\sin^2\phi,1\right)+

\tfrac{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).

\end{align}</math>

(Note: the above are only valid for <math>\textstyle -\tfrac\pi2\le\phi\le\frac\pi2</math> and <math>0\le k^2\sin^2\phi\le1</math>)

Complete elliptic integrals

Complete elliptic integrals can be calculated by substituting <math>\phi=\tfrac{\pi}{2}</math>:

:<math>\begin{align}

K(k) &= R_F\left(0,1-k^2,1\right), \\[5mu]

E(k) &= R_F\left(0,1-k^2,1\right)-\tfrac{1}{3}k^2 R_D\left(0,1-k^2,1\right), \\[5mu]

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

\end{align}</math>

Special cases

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

:<math>\begin{align} R_{C}(x,y)

&= R_{F}(x,y,y)

= \frac{1}{2} \int _{0}^{\infty} \frac{dt}{\sqrt{t + x} (t + y)}

= \int _{\sqrt{x^{\infty} \frac{du}{u^{2} - x + y} \\[5mu]

&=

\begin{cases}

\dfrac{\arccos \sqrt}{\sqrt{y - x, & x < y \\[3mu]

\dfrac{1}{\sqrt{y, & x = y \\[3mu]

\dfrac{\operatorname{arcosh} \sqrt}{\sqrt{x - y, & x > y

\end{cases}

\end{align}</math>

Similarly, when at least two of the first three arguments of <math>R_J</math> are the same,

:<math>\begin{align}

R_{J}(x,y,y,p)

&= 3 \int _{\sqrt{x^{\infty} \frac{du}{(u^{2} - x + y) (u^{2} - x + p)} \\[5mu]

&= \begin{cases}

\dfrac{3}{p - y} (R_{C}(x,y) - R_{C}(x,p)), & y \ne p \\[3mu]

\dfrac{3}{2 (y - x)} \left( R_{C}(x,y) - \dfrac{1}{y} \sqrt{x}\right), & y = p \ne x \\[3mu]

\dfrac{1}{y^}, &y = p = x

\end{cases}

\end{align}</math>

Properties

Homogeneity

By substituting in the integral definitions <math>t = \kappa u</math> for any constant <math>\kappa</math>, it is found that

:<math>\begin{align}

R_F\left(\kappa x,\kappa y,\kappa z\right) &= \kappa^{-1/2}R_F(x,y,z), \\[5mu]

R_J\left(\kappa x,\kappa y,\kappa z,\kappa p\right) &= \kappa^{-3/2}R_J(x,y,z,p).

\end{align}</math>

Duplication theorem

:<math>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),</math>

where <math>\lambda=\sqrt{\vphantom{ty} x}\sqrt{\vphantom{ty} y}+\sqrt{\vphantom{ty} y}\sqrt{\vphantom{ty} z}+\sqrt{\vphantom{ty} z}\sqrt{\vphantom{ty} x}</math>.

:<math>\begin{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)) \\[5mu]

& = \frac{1}{4} R_{J}\left( \frac{x + \lambda}{4},\frac{y + \lambda}{4},\frac{z + \lambda}{4},\frac{p + \lambda}{4}\right) + 6 R_{C}(d^{2},d^{2} + (p - x) (p - y) (p - z)) \end{align}</math>

where <math>d = \bigl(\sqrt{\vphantom{ty} p} + \sqrt{\vphantom{ty} x}\bigr) \bigl(\sqrt{\vphantom{ty} p} + \sqrt{\vphantom{ty} y}\bigr) \bigl(\sqrt{\vphantom{ty} p} + \sqrt{\vphantom{ty} z}\bigr)</math> and <math>\lambda =\sqrt{\vphantom{ty} x}\sqrt{\vphantom{ty} y}+\sqrt{\vphantom{ty} y}\sqrt{\vphantom{ty} z}+\sqrt{\vphantom{ty} z}\sqrt{\vphantom{ty} x}</math>.

</references>

• 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
• B. C. Carlson 'Elliptic Integrals:Symmetric Integrals' in Chap. 19 of Digital Library of Mathematical Functions. Release date 2010-05-07. National Institute of Standards and Technology.
• 'Profile: Bille C. Carlson' in Digital Library of Mathematical Functions. National Institute of Standards and Technology.
• Fortran code from SLATEC for evaluating RF, RJ, RC, RD,