Principal value and integral of 1/z

In summary, the conversation is discussing the definition and calculation of the principle value for an integral of 1/z, using the Cauchy integral theorem. The speaker is attempting to explicitly write down the principle value and asks for clarification on their method. They are advised to consider how the function ln(z) is defined due to the path dependence of the integral.
  • #1
wasia
52
0
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?
 
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  • #2
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.
 

1. What is the principal value of 1/z?

The principal value of 1/z is a mathematical concept used to define the value of a complex function at a point where the function is not defined. In the case of 1/z, the principal value is defined as the limit of the function as it approaches the point in question from all directions.

2. How is the principal value of 1/z calculated?

The principal value of 1/z can be calculated using the Cauchy principal value formula, which involves taking the limit of the function as it approaches the point in question from the real axis and imaginary axis separately and then adding the two results together.

3. What is the relationship between the principal value and the Cauchy principal value of 1/z?

The principal value and the Cauchy principal value of 1/z are equivalent in most cases, but they differ at points where the function has a singularity or pole. In such cases, the Cauchy principal value is often used to define the integral of 1/z, while the principal value may not exist.

4. Why is the principal value of 1/z important in complex analysis?

The principal value of 1/z is important in complex analysis because it allows us to extend the definition of a complex function to points where it is not defined. This is especially useful in evaluating integrals involving complex functions, where the principal value can be used to handle singularities or poles.

5. Can the principal value of 1/z be used to evaluate all integrals involving complex functions?

No, the principal value of 1/z can only be used to evaluate integrals where the function has a simple pole, which means that the function approaches infinity at a single point. For integrals involving functions with more complex poles, other techniques such as the residue theorem may be used.

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