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Find the residues of f(z)\, at all its isolated singular points and at infinity (if infinity is not a limit point of singular points), where f(z)\, is given by

f(z)={\frac  {\sin 2z}{(z+1)^{3}}}\,

This has poles at z=-1\, of multiplicity k=3\,.

Use the formula {\mathrm  {Res}}_{{z=z_{0}^{k}}}f(z)={\frac  {1}{(k-1)!}}\lim _{{z\to z_{0}}}{\frac  {d^{{k-1}}}{{dz}^{{k-1}}}}\left[(z-z_{0})^{k}f(z)\right]\,

{\mathrm  {Res}}_{{z=-1}}=\lim _{{z\to -1}}{\frac  {d^{2}}{{dz}^{2}}}(z+1)^{3}f(z)={\frac  {1}{2}}\lim _{{z\to -1}}{\frac  {d^{2}}{{dz}^{2}}}\sin 2z\,

={\frac  {1}{2}}\lim _{{z\to -1}}(-4)\sin 2z=-2\sin(-2)=2\sin 2\,

If f(z)\, is analytic except at isolated singular points, then the sum of all the residues of f(z)\, equals 0.

So {\mathrm  {Res}}_{{z=-1}}+{\mathrm  {Res}}_{{z=\infty }}=2\sin 2+{\mathrm  {Res}}_{{z=\infty }}=0\, and {\mathrm  {Res}}_{{z=\infty }}=-2\sin 2\,

Main Page : Complex Variables : Residues