= Upper-half-plane contour for square-root logarithmic integrals
{title2=$I=\pi/\sqrt2,\quad J=\pi^2/(2\sqrt2)$}
Set $I=\int_0^\infty x^{1/2}/(1+x^2)\,dx$ and $J=\int_0^\infty x^{1/2}\log x/(1+x^2)\,dx$. In an indented upper semicircle use the <principal complex logarithm> and $z^{1/2}=e^{\log z/2}$. The negative real boundary contributes $iJ-\pi I$, the positive boundary contributes $J$, and both circular arcs vanish. The pole at $i$ has <residue> $\pi e^{i\pi/4}/4$. Thus $(1+i)J-\pi I=(\pi^2/(2\sqrt2))(-1+i)$; comparing real and imaginary parts yields the displayed integrals.
Back to article page
