Complex sine ($\sin z=(e^{iz}-e^{-iz})/(2i)$)

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.