It could be simple if I was allowed to use partial Derivatives. Without calculating partial derivative I don't know how to prove that it's harmonic. Can someone please amswer this question?
2026-04-06 09:41:43.1775468503
Without calculating the partial derivatives, explain why sin x cosh y and cos x sinh y are harmonic functions in C.
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Note that, if $x,y\in\Bbb R$,\begin{align}\sin(x+yi)&=\sin(x)\cos(yi)+\cos(x)\sin(yi)\\&=\sin(x)\cosh(y)+\cos(x)\sinh(y)i\end{align}and therefore your functions are the real part and the imaginary part of a holomorphic function. Therefore, they are harmonic.