Editing Carlson symmetric form (section)

==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.