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:

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

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