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 " " simply says that

is 0 only when which happens only when is a multiple of : for some integer m so the singularities are at and . If you are integrating around the path |z|= r, with , then there are two poles for every positive integeri need to find the points which turn to zero in the denominator

and non zero in the numerator.

i got two such points

by using this formula

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

i calculated the residium for it by 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