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

E^(-i * x) not well-defined. Why?

  1. Nov 4, 2012 #1
    e^(-i * x) not well-defined. Why??

    Hi, Just saw this as a step in an example that demonstrates the differentiability of holomorphic function. But I can't for the life of me figure out why e^(-2iθ) is ill-defined.
     
  2. jcsd
  3. Nov 4, 2012 #2
    Re: e^(-i * x) not well-defined. Why??

    What do you mean it is ill-defined. Why do you say that? It's well defined in the opinions of many smart people. I can't go beyond that until I know what objection there is to the conventional definition there is.
     
  4. Nov 6, 2012 #3
    Re: e^(-i * x) not well-defined. Why??

    It is well defined as a function of a real variable theta. But as a function on the plane, considering theta=theta(z), it has a problem at z=0. Does this answer your question? It is hard to tell without more context.

    As a function of z, this function is the complex conjugate of (z/|z|)^2
     
  5. Nov 8, 2012 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Re: e^(-i * x) not well-defined. Why??

    The standard definition of "function" for real numbers requires that "if x= y, then f(x)= f(y)"- i.e. that f is "well-defined". For functions of complex variables, that is simply too restrictive. "Functions" that we would like to be able to use, such as [itex]e^x[/itex] would no longer be "functions". So we drop that requirement.
     
  6. Nov 8, 2012 #5
    Re: e^(-i * x) not well-defined. Why??

    Some functions are given the requirement of Principal Value to make some functions of a complex variables become actual functions.

    I.e , let z=Re^(ix), and restrict x to be in
    (-pi,pi].

    This restriction works since e^(ix)=e^(ix+i*2n*pi) for all integers n.

    On the other hand, it makes functions behave less like we would wabt them to.

    That is, Log(xy)=/=Log(x)+Log(y) generally for principal value logarithm. On the other hand, log(xy)=log(x)+log(y).

    (correct me if I'm wrong as I am just typing from memory)

    Edit: x,y is a complex number for the logarithm examples.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: E^(-i * x) not well-defined. Why?
  1. Integral of x²e^-x² (Replies: 19)

  2. Integral of e^x/x (Replies: 15)

Loading...