find holomorphic function $f(z)$ for $u (x,y) = \cosh(ax)\sin(by)$

Question

Question: [For which positive real numbers $a$ and $b$ is

$u(x,y) = \cosh(ax)\sin(by)$

harmonic? When $a$ and $b$ satisfy this condition find a holomorphic function $f(z)$ such that $\Re f = u$]

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I got $a=b$ for the condition so that

$u(x,y) = \cosh(ax)\sin(ay)$ and $v(x,y) = -\sinh(ax)\cos(ay) + k$

$f(x,y) = \cosh(ax)\sin(ay) + i(-\sinh(ax)\cos(ay) + k)$

but I cannot form them in terms of $z$

Answer 1

It is $\sin(-iaz)+ik=-\sin(iaz)+ik$ where $z=x+iy$.

Answer 2

Hint: $\sin(x+ iy) = \sin x \cos(iy) + \cos x\sin (iy) = \sin x \cosh y + i\cos x\sinh y.$