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?

Answers and Replies

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.