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QUESTION


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\item[(a)]
Let $f(z)=(z+1)/(z-1)$. Find the Taylor series for $f(z)$ that is
valid in the disc $|z|<1$.\medskip

\item[(b)]
find a Laurent series for $f(z)$ that is valid in the annular
domain $1<|z|<\infty$.

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ANSWER


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\item[(a)]
Write $f(z)=-(1+z)(1-z)^{-1}$. Thus we have
$$f(z)=-(1+z)(1+z+z^2+\cdots  )=-(1+2z+2z^2+2z^3+\cdots).$$ This
is valid if $|z|<1$.

\item[(b)]
Now write $f(z)=(1+{1\over z})(1-{1\over z})^{-1}$ Thus we have
$$f(z)=(1+{1\over
z})(1+z^{-1}+z^{-2}+\cdots)=(1+2z^{-1}+2z^{-2}+\cdots).$$ This is
valid if $|z|>1$.

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