Solution
= Solution
Computing the <Laplacian> gives
$$
\Delta\!\left(x\cosh y\sin x\right)=2\cos x\cosh y
$$
and
$$
\Delta\!\left(Ay\sinh y\cos x\right)=2A\cos x\cosh y.
$$
Thus <Laplace's equation> holds exactly when
$$
\boxed{A=-1}.
$$
For $z=x+iy$,
$$
\sin z=\sin x\cosh y+i\cos x\sinh y,
$$
so
$$
\operatorname{Re}(z\sin z)
=x\sin x\cosh y-y\cos x\sinh y
=\phi(x,y).
$$
One suitable <analytic function> is therefore
$$
\boxed{f(z)=z\sin z}.
$$