Problem with contour integration

[Toyona10]

Homework Statement

Evaluate \intdx/(x²+2x+2) (limits from -∞ to ∞) by contour integration

Ok so, this is what i did...:

we have ---> \Gamma : |z|=R

\ointdz/(z²+2z+2) =
\intdx/x²+2x+2 (limits -R to R) +\intdz/(z²+2z+2) (lower limit \Gamma)--> (1)

[where f(z)=1/(z²+2z+2)]

we have, lim_z→∞ zf(z) = lim_z→∞ z/(z²+2z+2)
= 0
therefore,
lim_R→∞ \intdz/(z²+2z+2) [lower limit\Gamma
= 0

Taking the limit R→∞ on both sides of (1)

\ointdz/(z²+2z+2)= \intdx/x²+2x+2 [limits from -∞ to ∞) + 0


where f(z)= 1/z²+2z+2 so for finding out the poles of f(z)...
yeah this is where I am stuck, because in the previous one we did f(z) was→1/z⁴+1

so there we solved for z the way we solve for the roots of a complex number...
all that k= n-1 and nth root of z = re^{i(θ+2k∏)1/n}...

so how do we solve in this case?

Thanks a lot in advance...

 

[Dick]

To find the poles of 1/(z^2+2z+2) you need to find the roots of z^2+2z+2=0. That shouldn't be hard for you. It's a quadratic equation.

 

[Toyona10]

To find the poles of 1/(z^2+2z+2) you need to find the roots of z^2+2z+2=0. That shouldn't be hard for you. It's a quadratic equation.

okay...after solving quadratically, then we go for the residues, right? But i don't get the answer I am supposed to...its supposed to be π...

 

[micromass]

okay...after solving quadratically, then we go for the residues, right? But i don't get the answer I am supposed to...its supposed to be π...

Tell us what you did!

 

[Dick]

okay...after solving quadratically, then we go for the residues, right? But i don't get the answer I am supposed to...its supposed to be π...

Sure, it's pi. Show why you didn't get pi. What are the residues and what kind of contour are you picking? Can you show your work?

 

[Toyona10]

Tell us what you did!

I completed the square for that qudratic equation and its like this...→(z+1)²+1
then what? we find the residue for z→-1 to the order of 2?...

 

[Dick]

I completed the square for that qudratic equation and its like this...→(z+1)²+1
then what? we find the residue for z→-1 to the order of 2?...

You find the roots where the poles are and pick a contour first. Then decide what residues you need to evaluate.