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comment How does Mathematica understand branchcuts of the complex logarithm?
The paper Can your computer do complex analysis? is also relevant to this discussion.
comment Finding the intersection of a curve with an interpolation function
The RootSearch package <library.wolfram.com/infocenter/MathSource/4482/>; by Ted Ersek returns all the roots over a specified range as a list of replacement rules This package .A RootsInRange function was presented in "Finding Roots in an Interval" in The Mathematica Journal 7(2), 1998: should be sufficient: