Complex sine
= Complex sine
{title2=$\sin z=(e^{iz}-e^{-iz})/(2i)$}
The <sine> function extends to an <entire function> on the <complex plane> by the displayed exponential formula. Substituting $z=x+iy$ and grouping the real and imaginary terms gives $\sin(x+iy)=\sin x\cosh y+i\cos x\sinh y$, for real $x,y$. Thus its values grow exponentially in imaginary directions, even though the real sine is bounded.