Carlson Symmetric Form - Special Cases

Special Cases

When any two, or all three of the arguments of are the same, then a substitution of renders the integrand rational. The integral can then be expressed in terms of elementary transcendental functions.

R_{C}(x,y) = R_{F}(x,y,y) = \frac{1}{2} \int _{0}^{\infty}\frac{1}{\sqrt{t + x} (t + y)} dt = \int _{\sqrt{x}}^{\infty}\frac{1}{u^{2} - x + y} du = \begin{cases} \frac{\arccos \sqrt{\frac{x}{y}}}{\sqrt{y - x}}, & x < y \\ \frac{1}{\sqrt{y}}, & x = y \\ \frac{\mathrm{arccosh} \sqrt{\frac{x}{y}}}{\sqrt{x - y}}, & x > y \\ \end{cases}

Similarly, when at least two of the first three arguments of are the same,

R_{J}(x,y,y,p) = 3 \int _{\sqrt{x}}^{\infty}\frac{1}{(u^{2} - x + y) (u^{2} - x + p)} du = \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^{\frac{3}{2}}}, &y = p = x \\ \end{cases}

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