I am playing with a differential equation whose solutions are conical functions (Legendre functions with a complex order) with a pure imaginary argument $ P_{i\nu -1/2}(i \sinh \xi) $ or $Q_{i\nu -1/2}(i \sinh \xi)$. Playing around by plotting them using Mathemica suggests that $$ Q_{i\nu-1/2}(i\sinh \xi)= -i\frac{\pi }2 P_{i\nu-1/2}(i\sinh \xi). $$ Is this a correct result with the usual definitions of these functions?
I worry because Mathematica has an ideosyncratic definition of the associated conical functions $P^k_{i\nu -1/2}(x)$, which are usually defined to be real functions of $x$ when $x$ is real, but Mathematica makes them purely imaginary when $k$ is an odd integer and seems to have weird definitions when $k$ is not an integer.
