Are there any good books/online resources for learning about branch cuts at the level of introductory undergraduate complex analysis.
Thanks in advance.
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When I was an undergrad I really enjoyed Palka's treatment of this; it is definitely accessible and rigorous.
See http://www.cds.caltech.edu/~nair/abel.pdf It actually doesn't require any complex analysis really, but it explains branch cuts and points for Riemann surfaces of polynomials, so it's a great introduction. It also uses these to prove an important theorem.
I recommend to go through chapter VI: Multifunctions and chapter VII: The Logarithmic Function of Visual Complex Analysis by T. Needham.
VI.1 Example: Fractional powers, VI.2 Single-Valued Branches of a Multifunction, VI.3 Relevance to Power Series, VI.4 An Example with Two Branch Points
VII.1 Inverse of the Exponential Function, VII.2 The Logarithmic Power Series, VII.3 General Powers
provide a guided tour regarding branch cuts and related aspects with plenty of illuminating pictures.