Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Branch cuts for surfaces

  1. Aug 15, 2012 #1
    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)?
  2. jcsd
  3. Aug 15, 2012 #2


    User Avatar
    Science Advisor

    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 Cx Intersection Cy = 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:

  4. Aug 18, 2012 #3
    Thanks! I think that makes a lot more sense now. ^_^
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook