The sine function can take a complex argument. e.g. sin(x + iy)


But does it get used that way in any field? Either practical (e.g. electrical engineering) or in other fields of math? Naturally, I am not interested in trivial examples where the real or imaginary part of the argument is always zero.


I've asked elsewhere and the best anyone has come up with is that it can be a 2D solution of the Laplace equation. Anything more substantial? Any other interesting properties? Anyone stumbled upon it in deep in some analysis somewhere? Or is it really never used?



Are there any uses for complex sine? [sin z]
It turns up in a functional equation for the Gamma function, $\Gamma(s)\Gamma(1-s)={\pi\over\sin\pi s}$. From there one goes on to the functional equation for the Riemann zeta-function, $$\zeta(s)=2(2\pi)^{s-1}\Gamma(1-s)\sin(\pi s/2)\zeta(1-s)$$ The million dollar question is where are the (complex) zeros of $\zeta(s)$? 


