Find Cauchy principal values of $\int_{-\infty}^{\infty}\frac{x}{(x^2+4)(x^2-2x+5)}\,dx$

Question

I have been asked to find the Cauchy principal vlaues of the following problem using residues: $\int_{-\infty}^{\infty}\frac{x}{(x^2+4)(x^2-2x+5)}\,dx$

So far I have taken $\oint_C\frac{z}{(z^2+4)(z^2-2z+5)}\,dz$

which gives me:

$2\pi i\sum \operatorname{Res}\biggl[\frac{z}{(z^2+4)(z^2-2z+5)}\biggl]$

Where do I go from here?

Answer 1

factor the denominator.

$\frac {z}{(z+2i)(z-2i)(z-1+2i)(z-1-2i)}$

We only care about the poles in the upper half-plane.

You have poles at $z = 2i, z = 1+2i$

at $z=2i$

$Res f(z) = \lim_\limits{z\to 2i} (z-2i)f(z) = \frac {2i}{(4i)(-1+4i)(-1)} = \frac {1+4i}{2(17)} = \frac {1}{34} + \frac {2i}{17} $

and at $z = 1+2i$

$Res f(z) = \frac {1+2i}{(1+4i)(1)(4i)} = \frac {(1+2i)(1-4i)}{(17)(4i)} = \frac {9 - 2i}{(17)(4i)} = -\frac{9i}{68} - \frac {1}{34}$

add them together

$2\pi i \left(\frac {-i}{68}\right) = \frac {\pi}{34}$