Finding the harmonic conjugate of $u(x,y)=\sinh(x) \sin(y)$ I know this is already a harmonic function but I am having trouble finding its harmonic conjugate.  My instructor did this: 
$v_{x}=\cosh x \sin y \implies v(x,y)=\sin y \sinh(x)+g_{1}(x)$
$v_{y}=\sinh x \cosh y \implies v(x,y)=\sinh x \sinh y+g_{2}(x)$
I guess I'm supposed to find what $v(x,y)$ equals and eliminate one of the $g$'s?  I don't know.  
Can someone help?

It is easy to show that the Laplace equations is satisfied, $\nabla^2u=0$. Therefore, we can go about find the harmonic conjugate. By the Cauchy-Reimann equations, we have
\begin{alignat}{2}
u_x &=\cosh(x)\sin(y) &&{}= v_y\\
u_y &=\sinh(x)\cos(y) &&{}=-v_x
\end{alignat}
We can integrate $u_x = v_y$ by $y$ so 
$$
v(x,y) = \int\cosh(x)\sin(y)dy = -\cosh(x)\cos(y) + g(x)\tag{1}
$$
Now, $-v_x = u_y$ so 
$$
-v_x = \sinh(x)\cos(y) - g'(x) = \sinh(x)\cos(y)\tag{2}
$$
Can you take it from here? Next, you will be solving for $g(x)$ in equation $(2)$ and then plugging it into equation $(1)$ to identify the harmonic conjugate $v(x,y)$.
