Carlson symmetric form information

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:^[1]

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

R_{J}(x,y,z,p)={\tfrac {3}{2}}\int _{0}^{\infty }{\frac {dt}{(t+p){\sqrt {(t+x)(t+y)(t+z)}}}}

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

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

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)}}}}

Since $R_{C}$ and $R_{D}$ are special cases of $R_{F}$ and $R_{J}$ , all elliptic integrals can ultimately be evaluated in terms of just $R_{F}$ , $R_{J}$ , and $R_{G}$ .

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 $R_{F}(x,y,z)$ 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.

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

^ F.W.J. Olver, D.W. Lozier, R.F. Boisvert, and C.W. Clark, editors, 2010, NIST Handbook of Mathematical Functions (Cambridge University Press), Section 19.16, "Symmetic Integrals". Retrieved 2024-04-16..

[1] F.W.J. Olver, D.W. Lozier, R.F. Boisvert, and C.W. Clark, editors, 2010, NIST Handbook of Mathematical Functions (Cambridge University Press), Section 19.16, "Symmetic Integrals". Retrieved 2024-04-16..

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