Use the rational function formula to compute the residues at \(z=\pm i\) of \[ f(z) \da {1\over z^2 + 1} .\]
Solution
Applying the rational function formula: \[ \Res_{z=z_0}{1\over 1+z^2} &= {1\over 2z}\evalfrom_{z= z_0} \implies\\ \Res_{z=i}f(z) &= {1\over 2i} = -{i\over 2} \\ \Res_{z=-i}f(z) &= -{1\over 2i} = {i\over 2} .\]