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QUESTION

Show that the functions

\begin{itemize}
\item[i)] $y^3-3x^2 y$ and

\item[ii)] ${y\over (x^2+y^2)}$

\end{itemize}

are harmonic in some region of the plane. In each case find a
conjugate \indent harmonic function and identify the corresponding
analytic function.


ANSWER

The Cauchy inequalities imply that if $z_0$ is a complex number,
and if $M_R$ is the maximum value of $|f(z)|$ on the circle centre
$z_0$, radius $R$, then $|f^{''}(z_0)|\le 2M_R/R^2$. As $f$ is
entire, (analytic thoughout $\bf C)$, $R$ can be as large as we
please, so that $f^{''}(w)=0$, for all $w\in {\bf C}.$ Thus
$f^{'}(w)=a$, a complex constant. Therefore $$\int_0^zf^{'}(w)
dw=az+b$$
 (b a complex constant) and hence
$$f(z)=az+k$$ ($k$ a complex constant). As $|f(z)|\le A|z|$, by
putting $z=0$, we get $f(0)=0$, so that $k=0$ and $f(z)=az$.


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