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Laurent's Theorem watch

    • Thread Starter

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

    Part (A)
    Suppose f(z) = u(x,y) + iv(x,y)\;and\;g(z) = v(x,y) + iu(x,y) 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 D'(0,R) = \left\{ {z \in C:0 < |z| < R} \right\} where R>0 is fixed. What is the formulae for c_n in the Laurent expansion:

f(z) = \sum\limits_{n = - \infty }^\infty {c_n z_n }.

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

    - Well I know that:
    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 )}}

    Any suggestions from here?

    PART (C)
    Find the maximal radius R>0 for which the function 

f(z) = (\sin z)^{ - 1} 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

    (A) Look to show u_x = u_y = 0
    (B) Apply an estimation theorem to that expression for c_n
    (C) Do you know the sinz series?
    • Thread Starter

    hmm so for part (A)
    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?
    • Thread Starter

    for part (C):
    yes the sin z series is:

       \sin z = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\mbox{ for all } x\!

    what would be the next step please?

    Personally, I'd look at sin z / z.

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