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QUESTION
For which values of $z$ is $\tanh z$ {\it not} analytic? What is
the largest circle (centre at tha origin) for which the Taylor
series about $z=0$ for $\tanh z$ converges to $\tanh z$?. Find the
first two non-zero terms ofthis series.
ANSWER
$\tanh z=\sinh z/\cosh z$ is not analytic if and only if $\cosh
z=0$. This is when $e^z+e^-z=0$ or when $e^{2z}=-1=e^{i\pi}$. Thus
a point closest to the origin where we do not get convergence is
$z=i\pi/2$, so we get convergence if $|z|<\pi/2$. We compute the
Taylor series in usual way as in question 1.
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