CVR18

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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  {e^{{tz}}}{(z+2)^{2}}}\,

This has a pole at z=-2\, of multiplicity k=2\,

Use the formula {\mathrm  {Res}}_{{z=z_{0}}}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=-2}}=\lim _{{z\to -2}}{\frac  {d}{dz}}\left[(z+2)^{2}f(z)\right]=\lim _{{z\to -2}}{\frac  {d}{dz}}\left[e^{{tz}}\right]=\lim _{{z\to -2}}te^{{tz}}=-2e^{{-2t}},

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=-2}}+{\mathrm  {Res}}_{{z=\infty }}=-2e^{{-2t}}+{\mathrm  {Res}}_{{z=\infty }}=0\,

so {\mathrm  {Res}}_{{z=\infty }}=2e^{{-2t}}\,


Main Page : Complex Variables : Residues