Residue for $\int_{-\infty}^{\infty} \frac{1}{\left(x^{2}+1\right)^{3}} d x$ problem with second derivative.

Question

Compute residue for $\int_{-\infty}^{\infty} \frac{1}{\left(x^{2}+1\right)^{3}} d x$

$$ f(z) = \frac{1}{\left(z^{2}+1\right)^{3}} $$ I have problem with second derivative: Where I compute $\operatorname{Res}[f(z), i] = \frac{1}{2!}\lim_{z\mapsto i } (z-i)^{3} \frac{d^2}{dz^{2}}f(z) $

but second derivative is: $$ \frac{d^2}{dz^{2}}f(z) =\frac{6 \left(7 z^2-1\right)}{\left(z^2+1\right)^5}= \frac{6 \left(7 z^2-1\right)}{\left(z+i)^5(z-i\right)^5} $$

Where is a mistake?

Answer 1

The mistake is that you computed

$$(z-i)^{3} \frac{d^2}{dz^{2}}f(z)$$

instead of

$$\frac{d^2}{dz^{2}}\Big[(z-i)^{3}f(z)\Big].$$