Carlson Symmetric Form - Negative Arguments

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, or the fourth argument, p, of 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

and

\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}

where

which must be greater than zero for 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.

Read more about this topic:  Carlson Symmetric Form

Famous quotes containing the words negative and/or arguments:

    “There is no reason why parents who work hard at a job to support a family, who nurture children during the hours at home, and who have searched for and selected the best [daycare] arrangement possible for their children need to feel anxious and guilty. It almost seems as if our culture wants parents to experience these negative feelings.”
    —Gwen Morgan (20th century)

    “There is no assurance of the great fact in question [namely, immortality]. All the arguments are mere probabilities, analogies, fancies, whims. We believe, or disbelieve, or are in doubt according to our own make-up—to accidents, to education, to environment. For myself, I do not reach either faith or belief ... that I—the conscious person talking to you—will meet you in the world beyond—you being yourself a conscious person—the same person now reading what I say.”
    —Rutherford Birchard Hayes (1822–1893)