Editing Carlson symmetric form (section)

= \int _{\sqrt{x}}^{\infty} \frac{du}{u^{2} - x + y} =
\begin{cases}
  \frac{\arccos \sqrt{{x}/{y}}}{\sqrt{y - x}},  & x < y \\
  \frac{1}{\sqrt{y}}, & x = y \\
  \frac{\operatorname{arcosh} \sqrt{{x}/{y}}}{\sqrt{x - y}},  & x > y \\
\end{cases}</math>

Similarly, when at least two of the first three arguments of <math>R_J</math> are the same,

:<math>R_{J}(x,y,y,p)
= 3 \int _{\sqrt{x}}^{\infty} \frac{du}{(u^{2} - x + y) (u^{2} - x + p)}
= \begin{cases}
  \frac{3}{p - y} (R_{C}(x,y) - R_{C}(x,p)),  & y \ne p \\
  \frac{3}{2 (y - x)} \left( R_{C}(x,y) - \frac{1}{y} \sqrt{x}\right),  & y = p \ne x \\
  \frac{1}{y^{{3}/{2}}},  &y = p = x \\
\end{cases}</math>

==Properties==

===Homogeneity===

By substituting in the integral definitions <math>t = \kappa u</math> for any constant <math>\kappa</math>, it is found that

:<math>R_F\left(\kappa x,\kappa y,\kappa z\right)=\kappa^{-1/2}R_F(x,y,z)</math>

:<math>R_J\left(\kappa x,\kappa y,\kappa z,\kappa p\right)=\kappa^{-3/2}R_J(x,y,z,p)</math>

===Duplication theorem===

:<math>R_F(x,y,z)=2R_F(x+\lambda,y+\lambda,z+\lambda)=
R_F\left(\frac{x+\lambda}{4},\frac{y+\lambda}{4},\frac{z+\lambda}{4}\right),</math>

where <math>\lambda=\sqrt{x}\sqrt{y}+\sqrt{y}\sqrt{z}+\sqrt{z}\sqrt{x}</math>.

:<math>\begin{align}R_{J}(x,y,z,p) & = 2 R_{J}(x + \lambda,y + \lambda,z + \lambda,p + \lambda) + 6 R_{C}(d^{2},d^{2} + (p - x) (p - y) (p - z)) \\
 & = \frac{1}{4} R_{J}\left( \frac{x + \lambda}{4},\frac{y + \lambda}{4},\frac{z + \lambda}{4},\frac{p + \lambda}{4}\right) + 6 R_{C}(d^{2},d^{2} + (p - x) (p - y) (p - z)) \end{align}</math><ref name="Carlson94"/>

where <math>d = (\sqrt{p} + \sqrt{x}) (\sqrt{p} + \sqrt{y}) (\sqrt{p} + \sqrt{z})</math> and <math>\lambda =\sqrt{x}\sqrt{y}+\sqrt{y}\sqrt{z}+\sqrt{z}\sqrt{x}</math>

==Series Expansion==

In obtaining a [[Taylor series]] expansion for <math>R_{F}</math> or <math>R_{J}</math> it proves convenient to expand about the mean value of the several arguments. So for <math>R_{F}</math>, letting the mean value of the arguments be <math>A = (x + y + z)/3</math>, and using homogeneity, define <math>\Delta x</math>, <math>\Delta y</math> and <math>\Delta z</math> by

:<math>\begin{align}R_{F}(x,y,z) & = R_{F}(A (1 - \Delta x),A (1 - \Delta y),A (1 - \Delta z)) \\
 & = \frac{1}{\sqrt{A}} R_{F}(1 - \Delta x,1 - \Delta y,1 - \Delta z) \end{align}</math>

that is <math>\Delta x = 1 - x/A</math> etc. The differences <math>\Delta x</math>, <math>\Delta y</math> and <math>\Delta z</math> are defined with this sign (such that they are ''subtracted''), in order to be in agreement with Carlson's papers. Since <math>R_{F}(x,y,z)</math> is symmetric under permutation of <math>x</math>, <math>y</math> and <math>z</math>, it is also symmetric in the quantities <math>\Delta x</math>, <math>\Delta y</math> and <math>\Delta z</math>. It follows that both the integrand of <math>R_{F}</math> and its integral can be expressed as functions of the [[elementary symmetric polynomial]]s in <math>\Delta x</math>, <math>\Delta y</math> and <math>\Delta z</math> which are

:<math>E_{1} = \Delta x + \Delta y + \Delta z = 0</math>

:<math>E_{2} = \Delta x \Delta y + \Delta y \Delta z + \Delta z \Delta x</math>

:<math>E_{3} = \Delta x \Delta y \Delta z</math>

Expressing the integrand in terms of these polynomials, performing a multidimensional Taylor expansion and integrating term-by-term...

:<math>\begin{align}R_{F}(x,y,z) & = \frac{1}{2 \sqrt{A}} \int _{0}^{\infty}\frac{1}{\sqrt{(t + 1)^{3} - (t + 1)^{2} E_{1} + (t + 1) E_{2} - E_{3}}} dt \\
 & = \frac{1}{2 \sqrt{A}} \int _{0}^{\infty}\left( \frac{1}{(t + 1)^{\frac{3}{2}}} - \frac{E_{2}}{2 (t + 1)^{\frac{7}{2}}} + \frac{E_{3}}{2 (t + 1)^{\frac{9}{2}}} + \frac{3 E_{2}^{2}}{8 (t + 1)^{\frac{11}{2}}} - \frac{3 E_{2} E_{3}}{4 (t + 1)^{\frac{13}{2}}} + O(E_{1}) + O(\Delta^{6})\right) dt \\
 & = \frac{1}{\sqrt{A}} \left( 1 - \frac{1}{10} E_{2} + \frac{1}{14} E_{3} + \frac{1}{24} E_{2}^{2} - \frac{3}{44} E_{2} E_{3} + O(E_{1}) + O(\Delta^{6})\right) \end{align}</math>

The advantage of expanding about the mean value of the arguments is now apparent; it reduces <math>E_{1}</math> identically to zero, and so eliminates all terms involving <math>E_{1}</math> - which otherwise would be the most numerous.

An ascending series for <math>R_{J}</math> may be found in a similar way. There is a slight difficulty because <math>R_{J}</math> is not fully symmetric; its dependence on its fourth argument, <math>p</math>, is different from its dependence on <math>x</math>, <math>y</math> and <math>z</math>. This is overcome by treating <math>R_{J}</math> as a fully symmetric function of ''five'' arguments, two of which happen to have the same value <math>p</math>. The mean value of the arguments is therefore taken to be

:<math>A = \frac{x + y + z + 2 p}{5}</math>

and the differences <math>\Delta x</math>, <math>\Delta y</math> <math>\Delta z</math> and <math>\Delta p</math> defined by

:<math>\begin{align}R_{J}(x,y,z,p) & = R_{J}(A (1 - \Delta x),A (1 - \Delta y),A (1 - \Delta z),A (1 - \Delta p)) \\
 & = \frac{1}{A^{\frac{3}{2}}} R_{J}(1 - \Delta x,1 - \Delta y,1 - \Delta z,1 - \Delta p) \end{align}</math>

The [[elementary symmetric polynomial]]s in <math>\Delta x</math>, <math>\Delta y</math>, <math>\Delta z</math>, <math>\Delta p</math> and (again) <math>\Delta p</math> are in full

:<math>E_{1} = \Delta x + \Delta y + \Delta z + 2 \Delta p = 0</math>

:<math>E_{2} = \Delta x \Delta y + \Delta y \Delta z + 2 \Delta z \Delta p + \Delta p^{2} + 2 \Delta p \Delta x + \Delta x \Delta z + 2 \Delta y \Delta p</math>

:<math>E_{3} = \Delta z \Delta p^{2} + \Delta x \Delta p^{2} + 2 \Delta x \Delta y \Delta p + \Delta x \Delta y \Delta z + 2 \Delta y \Delta z \Delta p + \Delta y \Delta p^{2} + 2 \Delta x \Delta z \Delta p</math>

:<math>E_{4} = \Delta y \Delta z \Delta p^{2} + \Delta x \Delta z \Delta p^{2} + \Delta x \Delta y \Delta p^{2} + 2 \Delta x \Delta y \Delta z \Delta p</math>

:<math>E_{5} = \Delta x \Delta y \Delta z \Delta p^{2}</math>

However, it is possible to simplify the formulae for <math>E_{2}</math>, <math>E_{3}</math> and <math>E_{4}</math> using the fact that <math>E_{1} = 0</math>. Expressing the integrand in terms of these polynomials, performing a multidimensional Taylor expansion and integrating term-by-term as before...

:<math>\begin{align}R_{J}(x,y,z,p) & = \frac{3}{2 A^{\frac{3}{2}}} \int _{0}^{\infty}\frac{1}{\sqrt{(t + 1)^{5} - (t + 1)^{4} E_{1} + (t + 1)^{3} E_{2} - (t + 1)^{2} E_{3} + (t + 1) E_{4} - E_{5}}} dt \\
 & = \frac{3}{2 A^{\frac{3}{2}}} \int _{0}^{\infty}\left( \frac{1}{(t + 1)^{\frac{5}{2}}} - \frac{E_{2}}{2 (t + 1)^{\frac{9}{2}}} + \frac{E_{3}}{2 (t + 1)^{\frac{11}{2}}} + \frac{3 E_{2}^{2} - 4 E_{4}}{8 (t + 1)^{\frac{13}{2}}} + \frac{2 E_{5} - 3 E_{2} E_{3}}{4 (t + 1)^{\frac{15}{2}}} + O(E_{1}) + O(\Delta^{6})\right) dt \\
 & = \frac{1}{A^{\frac{3}{2}}} \left( 1 - \frac{3}{14} E_{2} + \frac{1}{6} E_{3} + \frac{9}{88} E_{2}^{2} - \frac{3}{22} E_{4} - \frac{9}{52} E_{2} E_{3} + \frac{3}{26} E_{5} + O(E_{1}) + O(\Delta^{6})\right) \end{align}</math>

As with <math>R_{J}</math>, by expanding about the mean value of the arguments, more than half the terms (those involving <math>E_{1}</math>) are eliminated.

==Negative arguments==

In general, the arguments x, y, z of Carlson's integrals may not be real and negative, as this would place a [[branch point]] on the path of integration, making the integral ambiguous. However, if the second argument of <math>R_C</math>, or the fourth argument, p, of <math>R_J</math> is negative, then this results in a [[simple pole]] on the path of integration. In these cases the [[Cauchy principal value]] (finite part) of the integrals may be of interest; these are

:<math>\mathrm{p.v.}\; R_C(x, -y) = \sqrt{\frac{x}{x + y}}\,R_C(x + y, y),</math>

and

:<math>\begin{align}\mathrm{p.v.}\; R_{J}(x,y,z,-p) & = \frac{(q - y) R_{J}(x,y,z,q) - 3 R_{F}(x,y,z) + 3 \sqrt{y} R_{C}(x z,- p q)}{y + p} \\
 & = \frac{(q - y) R_{J}(x,y,z,q) - 3 R_{F}(x,y,z) + 3 \sqrt{\frac{x y z}{x z + p q}} R_{C}(x z + p q,p q)}{y + p} \end{align}</math>
where

:<math>q = y + \frac{(z - y) (y - x)}{y + p}.</math>

which must be greater than zero for <math>R_{J}(x,y,z,q)</math> to be evaluated. This may be arranged by permuting x, y and z so that the value of y is between that of x and z.

==Numerical evaluation==

The duplication theorem can be used for a fast and robust evaluation of the Carlson symmetric form of elliptic integrals
and therefore also for the evaluation of Legendre-form of elliptic integrals. Let us calculate <math>R_F(x,y,z)</math>:
first, define <math>x_0=x</math>, <math>y_0=y</math> and <math>z_0=z</math>. Then iterate the series

:<math>\lambda_n=\sqrt{x_n}\sqrt{y_n}+\sqrt{y_n}\sqrt{z_n}+\sqrt{z_n}\sqrt{x_n},</math>
:<math>x_{n+1}=\frac{x_n+\lambda_n}{4}, y_{n+1}=\frac{y_n+\lambda_n}{4}, z_{n+1}=\frac{z_n+\lambda_n}{4}</math>
until the desired precision is reached: if <math>x</math>, <math>y</math> and <math>z</math> are non-negative, all of the series will converge quickly to a given value, say, <math>\mu</math>. Therefore,

:<math>R_F\left(x,y,z\right)=R_F\left(\mu,\mu,\mu\right)=\mu^{-1/2}.</math>

Evaluating <math>R_C(x,y)</math> is much the same due to the relation

:<math>R_C\left(x,y\right)=R_F\left(x,y,y\right).</math>

==References and External links==
<references>
<ref name="Carlson94">{{cite journal
| last   = Carlson
| first  = Bille C.
| arxiv = math/9409227v1
| title  = Numerical computation of real or complex elliptic integrals
| journal = Numerical Algorithms
| date   = 1994
| volume = 10
| pages = 13–26
| doi = 10.1007/BF02198293
}}</ref>
</references>
*[https://arxiv.org/abs/math/9310223v1 B. C. Carlson, John L. Gustafson 'Asymptotic approximations for symmetric elliptic integrals' 1993 arXiv]
*[https://arxiv.org/abs/math/9409227v1 B. C. Carlson 'Numerical Computation of Real Or Complex Elliptic Integrals' 1994 arXiv]
*[http://dlmf.nist.gov/19#PT3 B. C. Carlson 'Elliptic Integrals:Symmetric Integrals' in Chap. 19 of ''Digital Library of Mathematical Functions''. Release date 2010-05-07. National Institute of Standards and Technology.]
*[http://dlmf.nist.gov/about/bio/BCCarlson 'Profile: Bille C. Carlson' in ''Digital Library of Mathematical Functions''. National Institute of Standards and Technology.]
*{{Citation | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press | publication-place=New York | isbn=978-0-521-88068-8 | chapter=Section 6.12. Elliptic Integrals and Jacobian Elliptic Functions | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=309 | access-date=2011-08-10 | archive-date=2011-08-11 | archive-url=https://web.archive.org/web/20110811154417/http://apps.nrbook.com/empanel/index.html#pg=309 | url-status=dead }}
*[[Fortran]] code from [[SLATEC]] for evaluating [http://www.netlib.org/slatec/src/drf.f RF],  [http://www.netlib.org/slatec/src/drj.f RJ], [http://www.netlib.org/slatec/src/drc.f RC],  [http://www.netlib.org/slatec/src/drd.f RD],

[[Category:Elliptic functions]]