Calculating the residue of a complex function

[Robin04]

Homework Statement

Calculate the residue of the following function at its singularities: $f(z)=\frac{e^{i\omega z}\lambda z}{(z^2+\lambda^2)^2}$

Relevant Equations

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The singularities occur at $z = \pm i\lambda$. As $\frac{d}{dz}(z^2+\lambda^2)^2|_{z=\pm i\lambda}=0$, these singularities aren't first order and the residues cannot be calculated with differentiating the denominator and evaluating it at the singularities. What is the general method to determine the Laurent series?

 

[fresh_42]

Wikipedia has your example, https://de.wikipedia.org/wiki/Residuensatz, but I'm too lazy to translate and adapt it to your exact situation. I suggest to use the translation functionality in chrome. It worked quite well here: