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In the book Many-Body Physics by Coleman, on page 110 there is the following statement:

Using Cauchy's principal part equation, $1/(x-i \delta) = P(1/x) + i \pi \delta(x)$, where $P$ is the principal part.

Here $\delta$ is a number and $\delta(x)$ I presume to be the Dirac delta. I am not sure what this means. I assume it is related to the principal part of a function but, otherwise, I don't know how to obtain this. Help would be appreciated.

Daniele Tampieri
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user110503
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1 Answers1

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This is what’s called the Sokhotski-Plemelj theorem; you can check Wikipedia's article about it.

Albert
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QuantumEyedea
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  • Thanks. The only issue I have is that here it is used without the integration. How do we know that if the integral is the same the integrands will be too? – user110503 Jun 30 '18 at 16:38
  • Note that the author, as well as Srednicki in Quantum Field Theory, calls "principal part" to what often in literature---including Wikipedia---is called Cauchy principal value. Shokhotski-Plemelj theorem specifically implies the expression cited by the OP, which indeed shows no integral. – Albert Apr 26 '23 at 11:42