Editing Carlson symmetric form (section)

{{Short description|Set of elliptic integrals}}
In [[mathematics]], the '''Carlson symmetric forms of [[elliptic integral]]s''' are a small canonical set of elliptic integrals to which all others may be reduced. They are a modern alternative to the [[Legendre form]]s. The Legendre forms may be expressed in terms of the Carlson forms and vice versa.

The Carlson elliptic integrals are:<ref>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, {{cite web
|url=https://dlmf.nist.gov/19.16 |title=Symmetic Integrals
|access-date=2024-04-16 }}.</ref>
<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).