When I learn QFT, I am bothered by many problems in complex analysis.
1) 1x−x0+iϵ=P1x−x0−iπδ(x−x0)
I can't understand why 1/x can have a principal value because it's not a multivalued function. I'm very confused. And when I learned the complex analysis, I've not watched this formula, can anybody tell me where I can find this formula's proof.
2) ddxln(x+iϵ)=P1x−iπδ(x)
3) And I also find this formula. Seemingly f(x) has a branch cut, then f(z)=1π∫∞Zdz′Imf(z′)z′−z
Now I am very confused by these formula, because I haven't read it in any complex analysis book and never been taught how to handle an integral with branch cut. Can anyone give me the whole proof and where I can consult.
Answer
The first equation, 1x−x0+iϵ=P1x−x0−iπδ(x−x0)
This can be proved by constructing a semicircular contour in the upper half-plane of radius ρ→∞, with an indent placed at x0, making use of the residue theorem adapted to semi-circular arcs. See Saff, Snider Fundamentals of Complex Analysis, Section 8.5 Question 8.
The third one is the Kramers-Kronig relation, as Funzies mentioned.
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