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Principal value and integral of 1/z 
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#1
Mar610, 07:14 AM

P: 53

I am slightly confused about the definition of principle value. If we have an integral
[tex]\int 1/z,[/tex] where the integration from [tex]\infty[/tex] to [tex]\infty[/tex] is implied, then by Cauchy integral theorem we know that the principle value [tex]P \int 1/z=i\pi.[/tex] However, I would like to write down this principle value explicitly. My best shot is [tex] \lim_{\epsilon\rightarrow0}\lim_{R\rightarrow\infty}\int_{R}^{\epsilon}1/z+\int_{\epsilon}^{R}1/z.[/tex] Assuming that this is correct (is it?) I can (can I?) calculate the integrals first and take limits afterwards. I get [tex]\lim_{\epsilon\rightarrow0}\lim_{R\rightarrow\infty} \ln\left(\frac{\epsilon}{\epsilon}\right) + \ln\left(\frac{R}{R}\right)=2\ln(1)=0.[/tex] Can you tell me what am I doing wrong? 


#2
Mar610, 08:08 AM

P: 2,157

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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