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}

Read more about this topic:  Carlson Symmetric Form

Famous quotes containing the words special and/or cases:

    “He’s leaving Germany by special request of the Nazi government. First he sends a dispatch about Danzig and how 10,000 German tourists are pouring into the city every day with butterfly nets in their hands and submachine guns in their knapsacks. They warn him right then. What does he do next? Goes to a reception at von Ribbentropf’s and keeps yelling for gefilte fish!”
    —Billy Wilder (b. 1906)

    “And in cases where profound conviction has been wrought, the eloquent man is he who is no beautiful speaker, but who is inwardly drunk with a certain belief. It agitates and tears him, and perhaps almost bereaves him of the power of articulation.”
    —Ralph Waldo Emerson (1803–1882)