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Homework Help: Complex Deriviative

  1. Aug 14, 2010 #1
    Find all points where the function has a derivative. At which of these points the function is analytical.


    [tex]

    f(z) = \left\{
    \begin{array}{ll}
    {z^2}sin(\frac{1}{|z^2|}) & z \neq 0 \\
    0 & z = 0}
    \end{array}
    \right.

    [/tex]

    I have tried deriving directly using the limit and also tried using Cauchy-Riemann, both tries led to complicated formulas.

    For example Cauchy-Riemann approach:
    [tex]
    f(a+ib) = \underbrace{ (a^2-b^2)sin(1/\sqrt{a^2+b^2})}_{u} +
    i\underbrace{{2ab}\cdot{sin(1/\sqrt{a^2+b^2})}}_{v}
    [/tex]
    Now I need to calculate dv/da ,du/da ,dv/db,du/db, but this seems like a headache.
     
    Last edited: Aug 14, 2010
  2. jcsd
  3. Aug 14, 2010 #2
    Why do you have square roots over the a^2 + b^2 expressions? Isn't there a |z^2| = |z|^2 = a^2 + b^2 term in the original function?

    Also taking partial derivatives is easy. Just check the Cauchy-Riemann equations for z = a + ib =/= 0. At the origin, you'll probably have to resort to the definition anyways. You might already know the answer at least for the origin since there is a real-valued function that is similarly defined and often used as a counterexample.
     
  4. Aug 14, 2010 #3
    Yes, my mistake, it should be a^2+b^2 without the root.
    But, still it does not make finding partial derivatives simple. For example, here is du/da calculated by WolframAlpha:
    attachment.php?attachmentid=27559&stc=1&d=1281804936.png

    Can you explain the real-valued function method? How do you use it as a counter example?
     

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