Conformal map projecting a line to a sine wave

Question

I'm looking for an analytic complex function that will map a straight line on to a sine wave. Are there any known examples?

To be more specific, let

$f(x+iy) = u(x,y) + iv(x,y)$.

I want to find a $u$ and $v$ such that for some $y_0$ and $k$

$v(x,y_0) = v_0 + \sin ku(x,y_0) $

Answer 1

Try: $f(z) = z + i \sin(k z)$.