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QUESTION
If $f(z)=u(x,y)+iv(x,y)$ and $\overline{f(z)}=u(x,y)-iv(x,y)$ are
both analytic in a region $D$ show that $f$ is constant in $D$.
(Hint: Cauchy-Riemann equations). Also show that if $f$ is
analytic and $|f|$ is constant then $f$ is constant.
ANSWER
(i) Either do this directly using the Cauchy-Riemann equations, or
note that $(x+iy)^3=x^3-3xy^2+i(3x^2y-y^3)$. Thus $y^3-3xy$ is the
imaginary part of $-z^3$ and so $y^3-3x^2y$ is harmonic, its
conjugate harmonic function is $3xy^2-x^3$ and the corresponding
analytic function is $-z^3$.
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(ii) $1/z=1/(x+iy)$=$(x-iy)/(x^2+y^2)$. Thus $y/(x^2+y^2)$ is the
imaginary part of $-1/z$ and is thus harmonic in any region not
containing the origin. Thus $y/(x^2+y^2)$ is the imaginary part of
$-1/z$, so that $y/(x^2+y^2)$ is harmonic, its conjugate harmonic
function is $-x/(x^2+y^2)$ and $-1/z$ is the corresponding
analytic function.
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