I am new to complex analysis and just about starting to understand contour integrals.
There is a branch cut on negative real axis. The contour of integration I am interested in is made up of multiple segments, of which my question is regarding the following 2 segments (the ones one either side of the negative real axis where the branch cut lies).
- Segment A - Starts from $-\infty + i \epsilon$ and ending in $-\epsilon + i \epsilon$
- Segment B - Starts from $-\epsilon - i \epsilon$ and ending in $-\infty - i \epsilon$
Is the following right?
$$\lim_{\epsilon \rightarrow 0}\int_{A} f(z) dz + \int_{B} f(z) dz = \int_{-\infty}^{0} f(x e^{i\pi}) dx + \int_{0}^{-\infty} f(x e^{-i\pi}) dx$$
I am confused if there is another $i2\pi$ that I need to use to account for the branch cut jump in phase.