Results 1 to 2 of 2

Math Help - Complex Mappings

  1. #1
    Newbie
    Joined
    Feb 2010
    Posts
    6

    Complex Mappings

    Can anyone help me with these problems I am stuck on?

    1. Consider the complex mapping
    z -> f(z) = (1+z)/(1-z)
    Find the images of i and 1 − i. What are the images of the real and the
    imaginary axes?

    2. Find Mobius transformations f such that
    (a) f(0) = i, f(1) = 1, f(−1) = −1;
    (b) f(0) = infinity, f(1) = −1, f(infinity) = 1.

    3. Find a Mobius transformation mapping {z : Imz > 2} onto the disc
    {w : |w − 2| < 3}.

    Thanks for any help you can give.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by alawrie View Post
    Can anyone help me with these problems I am stuck on?

    1. Consider the complex mapping
    z -> f(z) = (1+z)/(1-z)


    \frac{z+1}{-z+1}=\frac{-|z|^2+1+2Im(z)\,i}{|-z+1|^2} , and now just substitute

    Find the images of i and 1 − i. What are the images of the real and the
    imaginary axes?

    2. Find Mobius transformations f such that
    (a) f(0) = i, f(1) = 1, f(−1) = −1;
    (b) f(0) = infinity, f(1) = −1, f(infinity) = 1.


    A Moebius transformtion is of the form z\mapsto \frac{az+b}{cz+d}\,,\,a,b,c,d\in\mathbb{C}\,,\,\,a  d-bc\neq 0 , so choose the coefficients accordingly. For example , for (a):

    \frac{b}{d}=i\,,\,\,\frac{a+b}{c+d}=1\,,\,\,\frac{-a+b}{-c+d}=-1 \Longrightarrow b=di\,,\,a+di=c+d\Longrightarrow a-c=d(1-i)\,,\,\,-a+di=c-d\Longrightarrow a+c=d(1+i)\longrightarrow \frac{a-c}{1-i}=\frac{a+c}{1+i}\Longrightarrow c=ai,

    and together with a+di=c+d we get a+di=ai+d\Longrightarrow (d-a)i=d-a\Longrightarrow a=d=-bi=-ci , and the transf. is z\mapsto\frac{z+i}{iz+1}

    Tonio

    3. Find a Mobius transformation mapping {z : Imz > 2} onto the disc
    {w : |w − 2| < 3}.

    Thanks for any help you can give.
    .
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Complex Mappings
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: November 18th 2010, 09:52 PM
  2. Complex analysis - mappings
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: January 24th 2010, 03:27 PM
  3. Mappings by 1/z
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 16th 2008, 11:22 AM
  4. mappings
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: August 2nd 2008, 10:20 AM
  5. Complex Analysis Question regarding mappings.
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 6th 2008, 12:46 AM

Search Tags


/mathhelpforum @mathhelpforum