Jordan's Lemma - Example

Example

The function

satisfies the condition of Jordan's lemma with a = 1 for all R > 0 with R ≠ 1. Note that, for R > 1,

hence (*) holds. Since the only singularity of f in the upper half plane is at z = i, the above application yields

Since z = i is a simple pole of f and 1 + z2 = (z + i)(z - i), we obtain

$\operatorname{Res}(f,i)=\lim_{z\to i}(z-i)f(z) =\lim_{z\to i}\frac{e^{iz}}{z+i}=\frac{e^{-1}}{2i}$

so that

This result exemplifies how some integrals difficult to compute with classical tools are easily tackled with the help of complex analysis.

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