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(Complex Variables) Differentiability of Arg z

  1. Feb 20, 2006 #1
    I am proving that the function f(z) = Arg z is nowhere differentiable by using the definiton of a derivative. I let z = x + yi. Then, if the limit exists, we have

    f'(z) = lim (/\z -> 0) ( f(z + /\z) - f(z) ) / /\z.

    (Note that /\ is the triangle symbol)
    Also, let /\z = p + iq, where p and q are real values.

    Arg z = Tan^-1 (y/x)...how will I continue from here?
     
  2. jcsd
  3. Feb 23, 2006 #2
    so you want to get

    [tex]
    f'(z) = \lim_{\Delta z \rightarrow 0} \frac{f(z+\Delta z) - f(z)}{\Delta z}
    [/tex]

    Express the limit in terms of [tex]u(x_0,y_0) and v(x_0,y_0) [/tex], that is,[tex]x_0, y_0, \Delta x, \Delta y [/tex]. then evaluate the limit using 2 approaches: when [tex]\Delta x = 0[/tex] and [tex]\Delta y = 0[/tex]. If f(z) = Arg z is differentiable, the derivative should be equal in both cases.
     
    Last edited: Feb 23, 2006
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