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\begin{document}

{\bf Question}

Show that the functions (a) $x^2-y^2+2y$ and (b) $\sin x\cosh y$
are harmonic in any finite region of the $z$-plane where $z=x+iy$.

\medskip

{\bf Answer}

\begin{description}
\item[(a)]
If $\phi=x^2-y^2+2y$

$\ds\frac{\pl ^2 \phi}{\pl x^2}=2,\ \ds\frac{\pl^2\phi}{\pl
y^2}=-2 \Rightarrow \bigtriangledown ^2 phi=0$

$\Rightarrow \phi$ harmonic in {\bf{R}}.

\item[(b)]
If $\phi=\sin x \cosh y$

$\ds\frac{\pl^2\phi}{\pl x^2}=-\sin x \cosh y,\
\ds\frac{\pl^2\phi}{\pl y^2}=\sin x\cosh y \Rightarrow
\bigtriangledown ^2 \phi=0$

$\Rightarrow \phi$ harmonic in {\bf{R}}.
\end{description}
\end{document}