Evaluate this complex Integral

[bugatti79]

Homework Statement

Use Cauchy's Integral Theorem to evaluate the following integral

$\int_0^{\infty} \frac{x^2+1}{(x^2+9)^2} dx$

Homework Equations

Res $f(z)_{z=z_0} = Res_{z=z_0} \frac{p(z)}{q(z)}=\frac{p(z_0)}{q'(z_0)}$

The Attempt at a Solution

I determine the roots of the denominator to be $x=\pm 3i$.
How do I convert these into polar form. I know $z=re^{i\theta}$

Do I need to convert these into $z=e^{f(i\theta)}$?

 

[Robert1986]

I wouldn't convert this to polar form. You need to figure out how to write $f$ as a rational function like $p/q$ that you have above. Where $p$ is analytic and non-zero at $z_0$.

 

[bugatti79]

Homework Statement

Use Cauchy's Integral Theorem to evaluate the following integral

$\int_0^{\infty} \frac{x^2+1}{(x^2+9)^2} dx$

Homework Equations

Res $f(z)_{z=z_0} = Res_{z=z_0} \frac{p(z)}{q(z)}=\frac{p(z_0)}{q'(z_0)}$

The Attempt at a Solution

I determine the roots of the denominator to be $x=\pm 3i$.
How do I convert these into polar form. I know $z=re^{i\theta}$

Do I need to convert these into $z=e^{f(i\theta)}$?

I wouldn't convert this to polar form. You need to figure out how to write $f$ as a rational function like $p/q$ that you have above. Where $p$ is analytic and non-zero at $z_0$.

I realize the pole is second order therefore using the residue formula for second order pole $z=3i$ we get

$=lim_{z \to 3i} \frac{d}{dz} \frac{ z^2+1}{(z^2+9)^2} = \frac{-2z(z^2-7)}{(z^2+9)^3} |_{z=3i}$

but we are going to get 0 in the denominator here..?