Why are non-intersecting branch cuts necessary for multiple-valued functions?

[modnarandom]

Why is it necessary that branch cuts for multiple-valued functions are non-intersecting? Does this have to do with needing each sheet for one value (ex. for positive/negative square roots)?

 

[chiro]

Hey modnarandom.

The main reason of constructing branch cuts is find the regions where an inverse exists. In a multi-valued function (a complete misnomer because it acts like a function that produces a unique output and not two outputs), you have the problem where you don't have a bijectivity (i.e. 1-1 which means inverse defined over the whole domain/range pair).

So you create a branch cut that deals with defining separate branches that allow you to restrict the domain for that branch so that an inverse exists. So your branches will be mutually exclusive (i.e. given branch cuts corresponding to a collection of sets Cⁱ then C^x Intersection C^y = empty set).

If your example of the square root, you will have two disjoint branches corresponding to positive and negative values.

This graph is a good way of showing this:

http://en.wikipedia.org/wiki/Branch_point#Branch_cuts

 

[modnarandom]

Thanks! I think that makes a lot more sense now. ^_^