- #1

parton

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## Homework Statement

Hi!

I have a little problem understanding a proof ('Wick rotation') which I found in a textbook on QFT.

Assume: f and g are polynomials with degree(g)-degree(f) >= 2

and f/g has no poles on the closed 1. and 3. quadrant.

proposition:

[tex] \int_{-\infty}^{+\infty} \mathrm{d}x \dfrac{f(x)}{g(x)} = i \int_{-\infty}^{+\infty} \mathrm{d}x \, \dfrac{f(ix)}{g(ix)} [/tex]

proof (just until the part I don't undestand):

We rotate the real axis into the imaginary axis (counter-clockwise). Because the rotating real axis does not hit poles of f/g we can write

[tex] \int_{-\infty}^{+\infty} \mathrm{d}x \dfrac{f(x)}{g(x)} = \int_{-i\infty}^{+i\infty} \mathrm{d}x \dfrac{f(x)}{g(x)} [/tex]

by the Cauchy theorem on the path-independence of integrals over holomorphic functions.

Could anyone explain this last step to me?

## Homework Equations

## The Attempt at a Solution

I heard about path-independence, but I thought we need two paths with identical initial and end point, but here it is obviously not the case... maybe I am missing something