We're currently dealing with Cauchys principal value and I'm going through some textbook examples step by step to try and understand how they work.
Sadly there is step which I can't seem to understand, it's the following: $$\lim\limits_{\epsilon\to0^+}\left[\int\limits_{\epsilon}^{\infty} \frac{1-e^{ix}}{x^2}dx+\int\limits_{-\epsilon}^{-\infty} \frac{1-e^{ix}}{x^2}dx\right] =$$ $$\int\limits_{C}\frac{1-e^{ix}}{x^2}dx-\lim\limits_{R\to\infty}\int\limits_{C_1}\frac{1-e^{ix}}{x^2}dx -\lim\limits_{r\to 0}\int\limits_{C_2}\frac{1-e^{ix}}{x^2}dx$$
With $C_1$ being the top part of a half circle with Radius $R$ (in the limes $R\to\infty)$ in the complex plane ($x=Re^{i\theta},0<\theta<\pi$).
$C_2$ being the top part of a half circle with Radius $r$ (in the limes $r\to0)$ in the complex plane ($x=re^{i\theta},0<\theta<\pi$)
The only other example of the Cauchy principal value I have seen was the one with $\int\limits_{-\infty}^{\infty}\frac{1}{x} \ dx $, but I don't know how the concepts used there apply to this example problem.
Why do the two integrals get splitted into three and where does the curve $C$ come from?
I would really appreciate any kind of help. Thanks.