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Complex Taylor series

  1. Mar 28, 2007 #1
    TASK:
    Assuming a complex function f(z) can be expanded as a Taylor series around z=0, i.e.:

    [tex]f(z)=\sum_{n=0}^{\infty}a_{n}z^n[/tex]

    Setting z=r*exp(i*theta), assuming a_n is real, find real part u(r, theta), imaginary part v(r,theta).

    Comment the result, especially for r=1.


    MY SOLUTION:
    [tex]f(z)=\sum_{n=0}^{\infty}a_{n}r^n(e^{i\theta})^n = \sum_{n=0}^{\infty}a_{n}r^n(cosn\theta+isinn\theta)[/tex]

    [tex]u(r,\theta)=\sum_{n=0}^{\infty}a_{n}r^n(cosn\theta)[/tex]
    [tex]v(r,\theta)=\sum_{n=0}^{\infty}a_{n}r^n(sinn\theta)[/tex]


    So I am to comment on the result.. What is there to say? I have "decomposed" the complex series into two real series, but this is kindo given; i'm sure they're asking for another comment.. Ideas?

    Is this an "optimal" solution BTW, or are there any other possibilities? Later on I'm asked to find real part of a expansion of ln.
     
  2. jcsd
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