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

Insights Things Which Can Go Wrong with Complex Numbers - Comments

  1. Jan 14, 2016 #1

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

  2. jcsd
  3. Jan 14, 2016 #2

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    I thought Log(z) was defined to return the principal value, i.e. In the range ##(-\pi,\pi]##, while log(z) is left as shorthand for the set of values which satisfy ##z=e^w##. Thus ##log(z)=Log(z)+2\pi n i##.

    Likewise, ##\sqrt .## is defined to return a complex number with argument in the range ##(-\pi,\pi]##. A difficulty here is that there is no corresponding shorthand (is there?) for the set of solutions to the square root operation.

    It might be interesting to develop some generic rules for multivalued functions. E.g. If f() is such an operation, we might write {f(x)} for the set of values and F(x) for the principal value. If f distributes across multiplication (e.g. raising to a power, ##(ab)^c=a^cb^c##) then we could write ##F(ab)\in \{f(ab)\}\subseteq f(a)f(b)##.
     
  4. Jan 16, 2016 #3

    klotza

    User Avatar
    Gold Member

    Last edited by a moderator: Jan 16, 2016
  5. Jan 18, 2016 #4

    jasonRF

    User Avatar
    Science Advisor
    Gold Member

    I really like this insight. I wish I had seen a writeup like this 25 years ago while first learning about complex numbers; it wasn't until taking an elective in complex analysis my senior year of college that I finally started to get a handle on this. A link to this insight should become the standard reply to these kinds of questions that show up in the forums.

    The use of Log versus log may be a little non-standard. I must admit that when I first skimmed the article I assumed the capital version was the principal branch, but I can never remember how people define the principle branch anyway (##-\pi \leq \theta < \pi##; ##-\pi < \theta \leq \pi##; ##0 \leq \theta < 2 \pi##, etc) so I always have to check how any given author defines it. When I fully read the insight the notation is clearly defined so I have no problem with it at all.

    Great work!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Things Which Can Go Wrong with Complex Numbers - Comments
Loading...