Are there any uses for complex sine? [sin z] 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?
 A: Euler discovered an infinite product expansion for the sine function,
$$
\sin z=z\prod_{k\geq 1}\left(1-\frac{z^2}{k^2\pi^2}\right)
$$
by analogy with the factorization of a polynomial with known zeroes (i.e. roots) into linear terms. In order for this to be true, it is crucial to know all zeros, real as well as complex. As a consequence of the product formula, Euler evaluated zeta function at even integers in terms of Bernoulli numbers. 
Euler gave an interesting proof of the product formula based on the idea with zeros, where it is shown that the products of $n$ factors approximate $\sin z$ as $n\to\infty.$ I've read some recent papers making this proof entirely rigorous (e.g. cited in Varadarajan's Bull of AMS article). The first step is to write
$$
\sin z=\frac{1}{2i}(e^{iz}-e^{-iz})=
\frac{1}{2i}\lim_{n\to\infty}\left(\left(1+\frac{iz}{n}\right)^n-\left(1-\frac{iz}{n}\right)^n\right).
$$
A standard proof of the product formula in complex analysis textbooks (due to Weierstrass?) also crucially relies on the fact that $\sin z$ is a complex analytic function, rather than merely real analytic.
A: 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 $\zeta(s)$? 
