Results 1 to 5 of 5

Math Help - Conformal map of this region

  1. #1
    Junior Member
    Joined
    May 2010
    Posts
    63

    Conformal map of this region

    Hi everyone,

    I have the following region S = { z : 0 < Im(z) < PI } in C and I'm looking for its image under z --> w = (1+ie^z)/(1-ie^z).
    I'm aware of a technique of finding images by looking at what happens to the boundaries of the region being transformed.

    In this case, it looks like I have two lines bounding S, L1: Im(z) = 0 and L2: Im(z) = PI

    The transformation itself doesn't look too simple. I started wondering whether I could break up the transformation into smaller, logical steps for visualization. Such as:

    z --> k = e^z

    k --> m = ik

    m --> n = (1+m)/(1-m)


    If I were take the boundaries of S through these simpler transformations in that order, would I get the correct boundaries of the image of S under z --> w?

    On a side note, I can visualize the geometric effect of the first two of my transformations but not the third one. What does it do?

    As always - thank you!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Mar 2010
    From
    Beijing, China
    Posts
    293
    Thanks
    23

    Re: Conformal map of this region

    let f(z)=(1+iz)/(1-iz)=-(z-i)/(z+i), then w = f o exp(z).
    exp maps the strip S to the upper half plane, excluding {0}.
    f maps the upper half plane to the unit disk, excluding {1, -1}

    To visualize f, try to transform it on to the Riemann sphere using Stereographic projection
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    May 2010
    Posts
    63

    Re: Conformal map of this region

    I figured out how you got that answer with a little help from the textbook. Thanks a lot!
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    May 2010
    Posts
    63

    Re: Conformal map of this region

    Instead of starting a new thread, I'll ask another conformal map question here.

    I want to find a map from S1 = { Im(z) < a } to S2 = D(b;1)

    Could I map S1 onto the lower open half-plane via translation by 'a', then map the lower open half-plane onto the unit disk and then finally map the unit disk onto D(b;1) by translating it by 'b' units? Is my strategy valid? (it doesn't matter if it's efficient)

    Thanks again.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Senior Member
    Joined
    Mar 2010
    From
    Beijing, China
    Posts
    293
    Thanks
    23

    Re: Conformal map of this region

    that should work
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. conformal map, sector
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: March 7th 2010, 12:03 PM
  2. conformal self-map
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: February 10th 2010, 11:58 AM
  3. how to show this is a conformal map
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 11th 2009, 02:48 PM
  4. Conformal mapping
    Posted in the Calculus Forum
    Replies: 0
    Last Post: December 9th 2008, 06:07 PM
  5. Conformal Mapping
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 6th 2008, 11:20 PM

Search Tags


/mathhelpforum @mathhelpforum