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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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