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Math Help - Show function satisties C-R relations, but is not analytic

  1. #1
    Junior Member platinumpimp68plus1's Avatar
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    Show function satisties C-R relations, but is not analytic

    Regarding the complex function f(z)=sqrt(|xy|), where z=x+iy, and its behavior at the origin.

    I'm really confused - how can this possibly satisfy the Cauchy-Riemann relations? There is no imaginary part, so u(x,y)=sqrt(|xy|) and v(x,y)=0... all of v's partials will be 0, but u's are not, so they can't be equal.

    Does the question mean to find the partials of u(0,0) and v(0,0) and show they are equal? ... This seems silly.

    I know it can't be analytic at the origin, because the derivative is undefined there, but I don't understand how it satisfies C-R! Please help shed some light.
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  2. #2
    Senior Member
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    Quote Originally Posted by platinumpimp68plus1 View Post
    Regarding the complex function f(z)=sqrt(|xy|), where z=x+iy, and its behavior at the origin.

    I'm really confused - how can this possibly satisfy the Cauchy-Riemann relations? There is no imaginary part, so u(x,y)=sqrt(|xy|) and v(x,y)=0... all of v's partials will be 0, but u's are not, so they can't be equal.

    Does the question mean to find the partials of u(0,0) and v(0,0) and show they are equal? ... This seems silly.

    I know it can't be analytic at the origin, because the derivative is undefined there, but I don't understand how it satisfies C-R! Please help shed some light.
    u(x,y)=\sqrt{(|xy|)}, then

    u_{x} |_{(0,0)}= \lim_{\triangle x \to 0} \frac{u(\triangle x + x,y) - u(x,y)}{\triangle x}

    = \lim_{\triangle x \to 0} \frac{\sqrt{(|(x + \triangle x)y|)} - \sqrt{(|xy|)}}{\triangle x}

    =\sqrt{|y|} \left(\lim_{\triangle x \to 0} \frac{\sqrt{(|(x + \triangle x)|)} - \sqrt{(|x|)}}{\triangle x}\right)

    substitute y = 0 and we have

    u_{x} |_{(0,0)}=0 \cdot \left(\lim_{\triangle x \to 0} \frac{\sqrt{(|(x + \triangle x)|)} - \sqrt{(|x|)}}{\triangle x}\right) = 0 = v_y
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