I have been reading Chapter 7 of Peskin & Schroeder about full propagator, the Kallen Lehman spectral representation, and got stuck at the branch cut singularities and at the complex logarithm of negative numbers. I have posted in the Analysis forum (but have not recieved any answer) the following question: Does anyone know what a branch-cut singularity is? I have been trying to understand its importance in physics, but I got lost. I would guess that a singularity in physical context should mean that the value of a function should become very big near that singularity. But if we take complex logarithm, we can become big only in two cases, when the argument is either 0 or infinity. However, people choose the negative part of a real line in a complex plane as a branch cut for a complex logarithm, and say that this branch cut is a weak singulartiy compared to a simple pole. What do they mean by that? Can anyone comment something about the "weakness of branch-cut singularities" in QFT or overall in physics.