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QUESTION Prove directly that $\sin x \cosh y$ is a harmonic
function.
ANSWER Let $u(x,y)=\sin x \cosh y$. We know by question 3, that
this is harmonic as it is the real part of the analytic function
$\sin z$. This also follows easily by definition. For
$u_{xx}+u_{yy}=-\cos x \cosh y+\sin x \cosh y=0$.
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