Thread: area defined as variable resedue question..

1. area defined as variable resedue question..

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

in this area

$
\gamma _r=\left \{ |z|=r \right \},n$

i need to find the points which turn to zero in the denominator
and non zero in the numerator.
i got two such points
$z=\pm \sqrt{n}$
by using this formula
$res(\sqrt{a})=\frac{p(a)}{q(a)'}$
$res(\sqrt{n})=\frac{1}{4\pi i}$
$res(-\sqrt{n})=\frac{1}{4\pi i}$

the third point is z=0 but for it we have both numerator and denominator 0
i calculated the residium for it by $res(f(x),a)=\lim_{x->a}(f(x)(x-a))$ 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

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

in this area

$
\gamma _r=\left \{ |z|=r \right \},n$
This simply makes no sense. You say you want to integrate "from plus to minus infinity", but then you say that |z|= r. They can't both be true.

Perhaps you mean that you want to integrate around the circle z= |r|
And saying that " $n< r^2< n+1$" simply says that $\sqrt{n}< r< \sqrt{n+1}$

i need to find the points which turn to zero in the denominator
and non zero in the numerator.
i got two such points
$z=\pm \sqrt{n}$
by using this formula
$res(\sqrt{a})=\frac{p(a)}{q(a)'}$
$res(\sqrt{n})=\frac{1}{4\pi i}$
$res(-\sqrt{n})=\frac{1}{4\pi i}$
$e^{2\pi i z^2}- 1$ is 0 only when $e^{2\pi iz^2}= 1$ which happens only when $2\pi i z^2$ is a multiple of $2\pi i$: $z^2= m$ for some integer m so the singularities are at $-\sqrt{m}$ and $\sqrt{m}$. If you are integrating around the path |z|= r, with $
\sqrt{n}< r< \sqrt{n+1}$
, then there are two poles for every positive integer $m\le n$

the third point is z=0 but for it we have both numerator and denominator 0
i calculated the residium for it by $res(f(x),a)=\lim_{x->a}(f(x)(x-a))$ 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