Dividing by variable when solving algebraic equation

[Alex K]

This kind of game of wandering around the complex plane to pick up to correct solution to a complex square root or a logarithm is super important in physics, by the way. For example, there's a theorem central to scattering theory and quantum theory called the "optical theorem" which tells you that the imaginary part gives you the total scattering cross section (basically the probability of two particles doing something when they meet), and the sign of that imaginary part tells you whether causality is preserved or not, because if you got it wrong, you'd have negative scattering cross section, which would mean the probability for something scattering is negative. The original prescription how to properly deal with paths through the complex plane in particle scattering is by Feynman himself, and is called the Feynman prescription accordingly.