# Principal value and integral of 1/z

by wasia
Tags: 1/z, principal, principal value
 P: 53 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?
 P: 2,159 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.

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