Results 1 to 3 of 3

Math Help - complex numbers - winding number

  1. #1
    Newbie
    Joined
    Nov 2010
    Posts
    19

    complex numbers - winding number

    Hi all,
    I need to prove the following lemma in attachment . Our instructor wants us to do it in the following way:
    Let A be the set of all  x  \in [0,1] with the property that there exists a continuous function
     \theta : [0,x] -> R such that   \theta (0) = \theta_{0}
    and

     \gamma (t) =  |  \gamma (t) | e^{i \theta (t) }    t \in [0,x]

    Prove that A is nonempty , closed and relatively open in [0,1] and conclude that A = [0,1] since [0,1] is connected.

    Can anyone help?? I don't even have a slight idea how to proceed.
    Attached Thumbnails Attached Thumbnails complex numbers - winding number-screenshot-complex-made-simple-google-books-mozilla-firefox-1.png  
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Apr 2009
    From
    México
    Posts
    721
    Notice you can bound \gamma away from zero and this together with the unifrom continuity of said function allows you to pick a neighbourhood (in [0,1] ) of any point in A on which there is a branch of logarithm defined. There may be some ambiguity over multiples of 2\pi but the exponential doesn't see them, so you can ignore them and take the "true" extension. This proves A is both open and closed, and A is never empty because 0 is always in your set. I leave the formal proof to you.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Nov 2010
    Posts
    19
    OK, I saw that the set A is open. How do we see it's closed? I can't see why the complement should be open.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: September 27th 2010, 04:14 PM
  2. Complex numbers-finding real number pairs
    Posted in the Calculus Forum
    Replies: 9
    Last Post: March 20th 2010, 11:36 PM
  3. winding number and jacobian
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: March 25th 2009, 11:56 AM
  4. Urgent, residue and winding numbers
    Posted in the Calculus Forum
    Replies: 8
    Last Post: November 12th 2008, 01:04 PM
  5. Replies: 1
    Last Post: May 24th 2007, 04:49 AM

Search Tags


/mathhelpforum @mathhelpforum