2nd order pole while computing residue in a complex integral

[Karthiksrao]

Hello,

I am trying to understand how to get the residue as given by wolfram :

http://www.wolframalpha.com/input/?i=residue+of+e^{Sqrt[x^2+++1]}/(x^2+++1)^2

The issue I am facing is - since it is a second order pole, when I try to different e^{\sqrt{x^+1}} I get a \sqrt{x^+1} in the denominator. And hence I cannot substitute x = i to get the residue. How do I deal with a pole which comes about on differentiating?

Thanks!

 

[Svein]

The standard way to deal with a second-order pole is to expand the nominator using Taylor's theorem around the pole. Dividing by the denominator will give you the value of the residue. Take the pole at z=i: e^{\sqrt{z^{2}+1}}=e^{0}+f_{1}(z)(z-i)+f_{2}(z)(z-i)^{2}. Dividing by (z-i)^{2}, we get \frac{e^{\sqrt{z^{2}+1}}}{(z-i)^{2}}=\frac{e^{0}}{(z-i)^{2}}+\frac{f_{1}(z)}{(z-i)}+f_{2}(z). So, f₁(i) is the residue at z=i.