In Landau's book on quantum mechanics an expression of the form
$$ (x-a)^{-1/4} $$
is analytically continued around the upper half-plane along the semicircle from $0$ to $\pi$. The result stated is
$$(a-x)^{-1/4}e^{-i\pi/4}$$
How does one derive this result? Landau writes $(x-a)=\rho e^{i\phi}$ and lets $\phi$ go from $0$ to $\pi$. In this manner
$$(x-a)^{-1/4}\to \rho^{-1/4}e^{-i\phi /4}$$
but why is this not simply
$$(x-a)^{-1/4}e^{-i\pi/4}$$
at the end of the semicircle?