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Math Help - Inverse of a Complex function

  1. #1
    Len
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    Inverse of a Complex function

    Let X belong to C denote the set of points z such that |z|< 2, and define

    f : X --> C by the formula

    f(z)= (3z+i)/(3-iz)

    Find a formula for the inverse map g=f^-1


    Could really use some help, complex numbers are the introduction to my course and I'm kind of stuck. Any and all help is appreciated!
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  2. #2
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    Re: Inverse of a Complex function

    Quote Originally Posted by Len View Post
    Let X belong to C denote the set of points z such that |z|< 2, and define

    f : X --> C by the formula

    f(z)= (3z+i)/(3-iz)

    Find a formula for the inverse map g=f^-1


    Could really use some help, complex numbers are the introduction to my course and I'm kind of stuck. Any and all help is appreciated!
    The inverse function will have the f and z values swap...
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  3. #3
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    Re: Inverse of a Complex function

    to amplify ProveIt's response, let f(z) = w, so we have

    w = (3z+i)/(3-iz).

    now solve for z in terms of w. the result will be z = g(w), for some function g, which is the function you are looking for.
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  4. #4
    MHF Contributor chisigma's Avatar
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    Re: Inverse of a Complex function

    Quote Originally Posted by Deveno View Post
    to amplify ProveIt's response, let f(z) = w, so we have

    w = (3z+i)/(3-iz).

    now solve for z in terms of w. the result will be z = g(w), for some function g, which is the function you are looking for.
    To amplify Deveno's response, let...

    w=f(s)= c_{1}\ s + c_{2}\ s^{2} + c_{3}\ s^{3} +...\ ;\ c_{1}\ne 0 (1)

    ... which is analytic for |s|<r , its inverse function is...

    s=f^{-1}(w)= d_{1}\ w + d_{2}\ w^{2}+ d_{3}\ w^{3}+... (2)

    ... where...

    d_{n}= \frac{1}{n!}\ \lim_{s \rightarrow 0} \frac{d^{n-1}}{d s^{n-1}}\ (\frac{s}{f(s)})^{n} (3)

    If w(z)= \frac{3 z+i}{3-i z} a function like (1) can be obtained setting s=z+\frac{i}{3} ...

    w(s)= \frac{s}{(1-\frac{i}{6})-i\ \frac{s}{3}} (4)

    Now You have to compute the coefficients d_{n} using (3)...

    \chi \sigma
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