In Brown and Churchill's book, the concept of multivalued functions is not discussed in a very rigorous way (if at all). But I can see that branch cuts have importance in complex analysis, so I want to clarify my understanding of multivalued functions.
Is there a rigorous development of the definition of a multivalued function somewhere, along with branch cuts? Or is the whole idea of a multivalued function just a way of saying, "Hey, there's no unique way of defining the logarithm function here, so we're going to use whatever is convenient at the time"? And if the latter, where does a rigorous understanding of branch cuts fit in? Or are they also more of an intuitive term rather than a real defined mathematical object?
If they are rigorous, would a multivalued function be something like $f: \mathbb{C} \to \mathbb{C}^\infty$? I've never seen an infinite dimensional space before so I don't really know how that is developed.