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QUESTION


Find the residue at $z=0$ of each of the functions (a) ${1\over
z^3}$,\ \  (b) ${1\over z+z^2}$, \ \ (c) ${z-\sin z\over z}$,\ \
(d) $\sinh z\over z^4(1-z^2)$.




ANSWER


The residue at $z=0$ is the coefficient of ${1\over z}$ in the
Laurent expansion about $z=0$. (a) $0$,(b) there is a simple pole
at $z=0$so its residue is $\lim_{z\rightarrow 0} {z\over
z+z^2}=1$, (c) This function has a removable singularity at $z=0$
so that the residue at $z-0$ is $0$. (d) Using Taylor expansions
about $z=0$ we get ${1\over
z^4}(z+z^3/3!+\cdots)((1+z^2+z^4+\cdots)$. The coefficient of
${1\over z}$ is 7/6.




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