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:^[1] $${\displaystyle R_{F}(x,y,z)={\tfrac {1}{2}}\int _{0}^{\infty }{\frac {dt}{\sqrt {(t+x)(t+y)(t+z)}}}}$$ $${\displaystyle R_{J}(x,y,z,p)={\tfrac {3}{2}}\int _{0}^{\infty }{\frac {dt}{(t+p){\sqrt {(t+x)(t+y)(t+z)}}}}}$$ $${\displaystyle R_{G}(x,y,z)={\tfrac {1}{4}}\int _{0}^{\infty }{\frac {1}{\sqrt {(t+x)(t+y)(t+z)}}}{\biggl (}{\frac {x}{t+x}}+{\frac {y}{t+y}}+{\frac {z}{t+z}}{\biggr )}t\,dt}$$ $${\displaystyle R_{C}(x,y)=R_{F}(x,y,y)={\tfrac {1}{2}}\int _{0}^{\infty }{\frac {dt}{(t+y){\sqrt {(t+x)}}}}}$$ $${\displaystyle R_{D}(x,y,z)=R_{J}(x,y,z,z)={\tfrac {3}{2}}\int _{0}^{\infty }{\frac {dt}{(t+z)\,{\sqrt {(t+x)(t+y)(t+z)}}}}}$$

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

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 ${\displaystyle R_{F}(x,y,z)}$ is the same for any permutation of its arguments, and the value of ${\displaystyle R_{J}(x,y,z,p)}$ is the same for any permutation of its first three arguments.

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