Using the Residue Theorem for Real Integrals

1. Sep 14, 2011

liorda

1. The problem statement, all variables and given/known data
$$I=\int_{-\infty}^{\infty} { dx \over {5x^2+6x+5}}$$

2. Relevant equations
The residue theorem.

3. The attempt at a solution
I can't use the residue theorem since the denominator has real zeros. How should I solve this?

2. Sep 14, 2011

Hootenanny

Staff Emeritus
Yes you can use the residue theorem, you just need to "go around" the poles, i.e. deform your usual semi-circular contour around the poles. Have a look http://www.nhn.ou.edu/~milton/p5013/chap7.pdf" [Broken] and in particular at second 7.6.

Last edited by a moderator: May 5, 2017
3. Sep 14, 2011

awkward

4. Sep 14, 2011

Hootenanny

Staff Emeritus
Good catch, I didn't even think to look!

5. Sep 15, 2011

liorda

oops. From now on, I'll never try to guess root. Quadratic equation, we meet again.

Thanks guys.

6. Sep 15, 2011

dynamicsolo

There is also the quaint old "complete the squares" technique:

$$\int_{-\infty}^{\infty} \frac{dx}{5x^2+6x+5} = \frac{1}{5} \int_{-\infty}^{\infty} \frac{dx}{ (x^2+\frac{6}{5}x+\frac{9}{25}) + (1 - \frac{9}{25} ) }$$

$$= \frac{1}{5} \int_{-\infty}^{\infty} \frac{dx}{ (x+\frac{3}{5})^{2} + (\frac{4}{5})^{2} } .$$

I believe quadratic polynomials in the denominator never require complex-analytic techniques, though you certainly aren't forbidden to use them...