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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 R C {\displaystyle R_{C}} and R D {\displaystyle R_{D}} are special cases of R F {\displaystyle R_{F}} and R J {\displaystyle R_{J}} , all elliptic integrals can ultimately be evaluated in terms of just R F {\displaystyle R_{F}} and R J {\displaystyle R_{J}}.
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 {\displaystyle R_{F}} is the same for any permutation of its arguments, and the value of R J {\displaystyle R_{J}} is the same for any permutation of its first three arguments.