Carlson symmetric forms

Builds on Elliptic integral Improper integral

Definition

Let $x, y, z$ be nonnegative real numbers, at most one of which is $0$ , and let $p > 0$ . The Carlson symmetric forms are improper integrals. They are $R_F(x, y, z) = \frac{1}{2} \int_0^\infty \frac{dt}{\sqrt{(t + x)(t + y)(t + z)}},$ $R_J(x, y, z, p) = \frac{3}{2} \int_0^\infty \frac{dt}{(t + p)\sqrt{(t + x)(t + y)(t + z)}},$ together with the special cases $R_C(x, y) = R_F(x, y, y), \qquad R_D(x, y, z) = R_J(x, y, z, z),$ where for $R_C$ one takes $x \ge 0$ and $y > 0$ , and for $R_D$ one takes $z > 0$ .

Remarks

The integrals converge: near $t = \infty$ the integrand of $R_F$ behaves like $t^{-3/2}$ , and near $t = 0$ at most one factor vanishes, which gives at worst $t^{-1/2}$ . $R_F$ is symmetric in $x, y, z$ , $R_J$ in $x, y, z$ , and $R_D$ in $x, y$ . Example: $R_F(x, x, x) = \frac12\int_0^\infty (t + x)^{-3/2}\,dt = x^{-1/2}$ for $x > 0$ . Non-example: $R_F(0, 0, 1)$ is not defined, since two arguments are $0$ and the integrand $1/(t\sqrt{t + 1})$ is not integrable at $t = 0$ . With $k' = \sqrt{1 - k^2}$ and $0 \le k < 1$ , the complete elliptic integrals are $K(k) = R_F(0, k'^2, 1), \qquad E(k) = R_F(0, k'^2, 1) - \frac{k^2}{3} R_D(0, k'^2, 1).$ $R_C$ is elementary; for $0 \le x < y$ it equals $\arccos(\sqrt{x/y}) / \sqrt{y - x}$ .

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Tag	`HNGT8S7JAG`
URI	`m2://HNGT8S7JAG`
Kind	definition
Writing	W0 stub
MSC	33E05, 33F05
Defines	Carlson symmetric forms
Uses	Elliptic integral, Improper integral

Carlson symmetric forms

Definition

Remarks

Reading path

Further reading

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