i need to calculate this integral from plus to minus infinity$\displaystyle f(z)=\frac{z}{e^{2\pi iz^2}-1}\\$

in this area

$\displaystyle

\gamma _r=\left \{ |z|=r \right \},n<r^2<n+1

$

i need to find the points which turn to zero in the denominator

and non zero in the numerator.

i got two such points

$\displaystyle z=\pm \sqrt{n}$

by using this formula

$\displaystyle res(\sqrt{a})=\frac{p(a)}{q(a)'}$

$\displaystyle res(\sqrt{n})=\frac{1}{4\pi i}$

$\displaystyle res(-\sqrt{n})=\frac{1}{4\pi i}$

the third point is z=0 but for it we have both numerator and denominator 0

i calculated the residium for it by $\displaystyle res(f(x),a)=\lim_{x->a}(f(x)(x-a))$ formula

but then

my prof says some stuff that involves the area

he says that my points are 0 +1 -1 +2^(0.5) -2^(0.5) etc.. because the denominator goes to zero

for each point have a residiu and i need to sum the residiums inside.

but here the area is not defined

its not like (by radius 3)

i dont know what point are inside the area