Using the Residue Theorem for Real Integrals

[liorda]

Homework Statement

$$I=\int_{-\infty}^{\infty} { dx \over {5x^2+6x+5}}$$

Homework Equations

The residue theorem.

The Attempt at a Solution

I can't use the residue theorem since the denominator has real zeros. How should I solve this?

 

[Hootenanny]

Homework Statement

$$I=\int_{-\infty}^{\infty} { dx \over {5x^2+6x+5}}$$

Homework Equations

The residue theorem.

The Attempt at a Solution

I can't use the residue theorem since the denominator has real zeros. How should I solve this?

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" and in particular at second 7.6.

 

[awkward]

Maybe you should double-check that business about the real zeros.

 

[Hootenanny]

Maybe you should double-check that business about the real zeros.

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

 

[liorda]

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

Thanks guys.

 

[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...