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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 {\frac  {\sin z}{z^{2}+4}}\,

The poles are at z=\pm 2i\,.

\lim _{{z\to 2i}}{\frac  {\sin z}{z+2i}}={\frac  {\sin 2i}{4i}}={\frac  {\sinh 2}{4}}\,

\lim _{{z\to -2i}}{\frac  {\sin z}{z-2i}}={\frac  {\sin -2i}{-4i}}={\frac  {\sinh 2}{4}}\,

Main Page : Complex Variables : Residues