# Tag Archives: shifted

## Contour method for shifted logarithm branch

Prove  $a,b,c,d >0$ $$\int^\infty_0 \frac{\log(a^2+b^2x^2)}{c^2+d^2x^2}\,dx = \frac{\pi}{cd} \log \frac{ad+bc}{d}$$ Consider the function $$f(z) = \frac{\log(a-ibz)}{c^2+d^2z^2}$$ We need the logarithm with the branch cut $y<-\frac{a}{b} , x =0$ . Note that this corresponds to \log(a+ibz) = \log\sqrt{(a+y)^2+b^2x^2}+i\theta … Continue reading