When I learn QFT, I am bothered by many problems in complex analysis.

1) $$\frac{1}{x-x_0+i\epsilon}=P\frac{1}{x-x_0}-i\pi\delta(x-x_0)$$

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) $$\frac{d}{dx}\ln(x+i\epsilon)=P\frac{1}{x}-i\pi\delta(x)$$

3) And I also find this formula. Seemingly $f(x)$ has a branch cut, then $$f(z)=\frac{1}{\pi}\int_Z^{\infty}dz^{\prime}\frac{{\rm Im} f(z^{\prime})}{z^{\prime}-z}$$ Can anyone can tell the whole theorem and its proof, and what it wants to express. enter image description here

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.

  • 2
    Unfortunately, there are two completely unrelated meanings of the term "principal value". The kind referred to here is the Cauchy principal value, which assigns values to otherwise undefined improper integrals. This has nothing to do with the principal value you had in mind, which is for selecting single-valued branches of multi-valued functions. I know, it's stupid. You'd think someone would have fixed all these weird ambiguities by now, but alas math is not French. – David H Mar 30 '14 at 8:53
  • @DavidH Thanks a lot! Then the last question, can you give me some clues? – 346699 Mar 30 '14 at 11:40
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    @user34669 I think the last expression goes by the name of "Kramers-Kronig" relation, it is a way to express a complex function in its real or imaginary part. So with either the real or imaginary part, you can reconstruct the whole function. For a proof, see en.wikipedia.org/wiki/Kramers%E2%80%93Kronig_relations – Funzies Mar 30 '14 at 12:11
  • More answers are at physicsoverflow.org/10327 – Arnold Neumaier Nov 1 '15 at 16:36
up vote 10 down vote accepted

The first equation, $$\frac{1}{x-x_0+i\epsilon}=P\frac{1}{x-x_0}-i\pi\delta(x-x_0)$$ is actually a shorthand notation for its correct full form, which is $$\underset{\epsilon\rightarrow0^+}{lim}\int_{-\infty}^\infty\frac{f(x)}{x-x_0+i\epsilon}\,dx=P\int_{-\infty}^\infty\frac{f(x)}{x-x_0}\,dx-i\pi f(x_0)$$ and is valid for functions which are analytic in the upper half-plane and vanish fast enough that the integral can be constructed by an infinite semicircular contour.

This can be proved by constructing a semicircular contour in the upper half-plane of radius $\rho\rightarrow\infty$, with an indent placed at $x_0$, 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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