Principal value and integral of 1/z

[wasia]

I am slightly confused about the definition of principle value. If we have an integral
$$\int 1/z,$$
where the integration from $$-\infty$$ to $$\infty$$ is implied, then by Cauchy integral theorem we know that the principle value
$$P \int 1/z=i\pi.$$

However, I would like to write down this principle value explicitly. My best shot is
$$\lim_{\epsilon\rightarrow0}\lim_{R\rightarrow\infty}\int_{-R}^{-\epsilon}1/z+\int_{\epsilon}^{R}1/z.$$

Assuming that this is correct (is it?) I can (can I?) calculate the integrals first and take limits afterwards. I get

$$\lim_{\epsilon\rightarrow0}\lim_{R\rightarrow\infty} \ln\left(-\frac{\epsilon}{\epsilon}\right) + \ln\left(-\frac{R}{R}\right)=2\ln(-1)=0.$$

Can you tell me what am I doing wrong?

 

[Count Iblis]

You need to think about how to define the function ln(z). There is no unique definition, precisely because of the path dependence of the integral of 1/z.