# Problem with contour integration

1. Apr 17, 2012

### Toyona10

1. The problem statement, all variables and given/known data
Evaluate $\int$dx/(x2+2x+2) (limits from -∞ to ∞) by contour integration

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

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

$\oint$dz/(z2+2z+2) =
$\int$dx/x2+2x+2 (limits -R to R) +$\int$dz/(z2+2z+2) (lower limit $\Gamma$)--> (1)

[where f(z)=1/(z2+2z+2)]

we have, limz→∞ zf(z) = limz→∞ z/(z2+2z+2)
= 0
therefore,
limR→∞ $\int$dz/(z2+2z+2) [lower limit$\Gamma$
= 0

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

$\oint$dz/(z2+2z+2)= $\int$dx/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 im stuck, cuz 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 = rei(θ+2k∏)1/n......

so how do we solve in this case?

2. Apr 17, 2012

### 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.

3. Apr 17, 2012

### Toyona10

okay...after solving quadratically, then we go for the residues, right? But i dont get the answer im supposed to...its supposed to be π...

4. Apr 17, 2012

### micromass

Staff Emeritus
Tell us what you did!!

5. Apr 17, 2012

### Dick

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?

6. Apr 17, 2012

### Toyona10

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?....

7. Apr 17, 2012

### Dick

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