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QUESTION
Find the real and imaginary parts of $\sin(1+i)$.
ANSWER
$f(z)$ analytic implies that $\partial u/\partial x=\partial
v/\partial y$.
$\overline {f(z)}$ analytic implies that $\partial u/\partial
x=-\partial v/\partial y$. Thus $\partial u/\partial x=\partial
v/\partial y=0$, and also
$\partial u/\partial y=\partial v/\partial x=0$. Thus $u$ and $v$
are constants, (See Theorem 3.4) and so $f$ is constant. Now
suppose that $f$ is constant and that $|f|$ is constant. Then
$|f^2|=f\overline f$ is constant. We deduce that $\overline f$ is
constant so by the first part $f$ is constant.
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