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Math Help - Complex Mapping

  1. #1
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    Complex Mapping

    Find the image of the following region




    under the action of f(z)=z^5.

    I am horrible with mappings. I wrote z^5=r^5 \cdot e^{5i\theta_0}, and tried points in both regions, and then I ended up with the whole complex plane as the output and I don't think this is right. How do I find the image of this region? Thanks for the help.
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  2. #2
    MHF Contributor

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    The line y= 3x corresponds to the set of complex number a+ 3ai= a(1+ 3i) for a any real number (do you see why?). z^5= (a+ 3ai)^5= a^5(1+ 3i)^5. Now you could, if you wanted, multiply that out directly or use the "polar form". It will, in any case, be a constant, say u+ vi, so z^5= a^5(u+ vi) which gives "parametric" equations [/tex]y= va^5[/tex] and x= ua^5. a^5= \frac{x}{u} (as long as u is not 0) so y= v\frac{x}{u}= \frac{v}{u} x another straight line. Similarly for y= x/2 which corresponds to the set of complex numbers a+ \frac{1}{2}ai.

    That is, z^5 maps that region, bounded by straight lines, into another region bounded by straight lines. You will need to compute (1+ 3i)^5 and (1+ \frac{1}{2})^5 to determine exactly which straight lines.
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