Why Can't I Use Jordan's Lemma to Compute This Improper Integral?

[cyril14]

Why I cannot use Jordan lemma to compute improper integral

$$\int_{-\infty}^{\infty} f(z) \hbox{\ d}(z)$$

of a function like

$$f(z)=\frac{\exp(-|z|)}{(a^2+z^2)} \mbox{\ for } a>0$$

Such a function is finite and continuous for $$|z|>a$$ and $$z f(z)$$ vanishes for $$z \to \infty$$.

I know, that this function is not differentiable in z=0, but it seems to me that this is not a cause of problems, as there exists contour integral along the upper half-circle around the origin, and its limit for vanishing diameter is 0.

Can somebody explain me why I cannot use Jordan lemma in this case?

Thanks in advance,
Cyril Fischer

 

[fresh_42]

Have you checked whether the convergence of $zg(z)$ is uniformly?