Conjugate of complex Harmonic function?

Question

$$ \begin{array}{l}{\text { Let } f=u(x, y)+i v(x y), \text { where } u=\sin (x) \sinh (y)} \\ {\text { find the conjugate of } u(x, y) \text { if it is harmonic. }}\end{array} $$

I am pretty lost on how to go about this. My understanding is to check if $f$ is harmonic the second derivatives of $u$ and $v$ should add to 0.

1. How to verify $u(x,y) is harmonic?

2. How to find the conjugate of $u(x,y)$

Answer 1

If $u$ is harmonic, then $$u_{xx} + u_{yy} = 0$$

Then, find $u_x(z,0)$ and $u_y(z,0)$ by replacing $z = x$ and $y=0$ in the partial derivatives.

Next, $$f(x) = \int [u_x(z,0) - iu_y(z,0)]dz + c_1 + ic_2$$

(Milne-Thomson theorem)

Then substitute $z=x+iy$ and isolate $u = \sin(x)\sinh(y) + c_1$.

The remaining function will be the conjugate of $u$.