Laurent's Theorem

  1. Hi just a bit of help needed here as I don;t know where to start:

    Part (A)
    ----------------------------
    Suppose [itex]f(z) = u(x,y) + iv(x,y)\;and\;g(z) = v(x,y) + iu(x,y)[/itex] are analytic in some domain D. Show that both u and v are constant functions..?

    I guess we have to use the CRE here but not really sure how to approach this..?

    Part (B)
    ----------------------------
    Let f be a holomorphic function on the punctured disk [itex]D'(0,R) = \left\{ {z \in C:0 < |z| < R} \right\}[/itex] where R>0 is fixed. What is the formulae for c_n in the Laurent expansion:
    [itex]
    f(z) = \sum\limits_{n = - \infty }^\infty {c_n z_n }[/itex].

    Using these formulae, prove that if f is bounded on D'(0,R), it has a removable singularity at 0.

    - Well I know that:
    [itex]c_n = \frac{1}
    {{2\pi i}}\int\limits_{\gamma _r }^{} {\frac{{f(s)}}
    {{(s - z_0 )^{n + 1} }}} ds = \frac{{f^{(n)} (z_0 )}}
    {{n!}}[/itex].
    Any suggestions from here?


    PART (C)
    -------------------
    Find the maximal radius R>0 for which the function [itex]
    f(z) = (\sin z)^{ - 1}[/itex] is holomorphic in D'(0,R) and find the principal part of its Laurent expansion about z_0=0

    ??

    Any help would be greatly appreciated.

    Thanks a lot
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Dick

    Dick 25,887
    Science Advisor
    Homework Helper

    I'll start you out with the first one. CRE's for the f(z) tell you u_x=v_y and u_y=-v_x. CRE's for g(z) tell you v_x=u_y and v_y=-u_x. What happens when you put both of these together?
     
  4. Dick

    Dick 25,887
    Science Advisor
    Homework Helper

    For the second one, you might want to focus your efforts on proving that c_n=0 for n<0.
     
  5. hmm so for part (1)
    u_x = v_y = -u_x AND
    u_y = -v_x = v_x

    so u and v are constant because u_x = -u_x and -v_x = v_x

    is that correct?
     
  6. Dick

    Dick 25,887
    Science Advisor
    Homework Helper

    Yes. u_x=-u_x means u_x=0. The same for all of the other stuff. All of the partial derivatives are zero. Hence?
     
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