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Math Help - Approaching complex problem.....

  1. #1
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    Approaching complex problem.....

    Lets say we have a function:  f(z) = x+i|y| .

    We want to find all point that is it differentiable. Right of the bat we know that it is differentiable for  y >0 . For  f(z) = x+iy is a "direct" function of  x+iy . E.g. we do not have to appeal to CR equations.

    Also it is not differentiable for  y<0 since  f(z) = x-iy is not a "direct" function of  x+iy .


    Now for  y = 0 can we use the same argument to show that  f(z) = x+i|y| is not differentiable there?

    Because then we have  f(z) = x . And this is not a "direct" function of  x+iy .
    Last edited by heathrowjohnny; February 8th 2009 at 01:41 PM.
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  2. #2
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    I mean analytic takes care of everything. Its stronger then differentiability. Thats what "function of z" is.
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  3. #3
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    Quote Originally Posted by heathrowjohnny View Post
    Lets say we have a function:  f(z) = x+i|y| .

    We want to find all point that is it differentiable. Right of the bat we know that it is differentiable for  y >0 . For  f(z) = x+iy is a "direct" function of  x+iy . E.g. we do not have to appeal to CR equations.

    Also it is not differentiable for  y<0 since  f(z) = x-iy is not a "direct" function of  x+iy .


    Now for  y = 0 can we use the same argument to show that  f(z) = x+i|y| is not differentiable there?

    Because then we have  f(z) = x . And this is not a "direct" function of  x+iy .
    For y=0 you can use the definition of differenciable, \lim_{z\to z_0} \frac{f(z)-f(z_0)}{z-z_0}. Now z=x+iy and z_0 = x. Therefore, \lim_{(x,y)\to (x_0,0)}\frac{x+i|y| - x}{iy} = \lim_{y\to 0}\frac{y}{|y|}.
    And this limit does not exist.
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  4. #4
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    so you can't just say that  f(z) = x is not an "analytic function" of  x+iy ?
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