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Math Help - Show that f(Z) = Z(hat) is not differentiable.

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    Show that f(Z) = Z(hat) is not differentiable.

    Hi ppl, would love the solutuin to "show that f(Z) = Z(hat) is not differentiable"
    Last edited by mr fantastic; March 25th 2011 at 03:15 AM. Reason: Title.
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    Quote Originally Posted by davy337 View Post
    Hi ppl, would love the solutuin to "show that f(Z) = Z(hat) is not differentiable"

    I think you mean f(z)=\bar{z}=x-iy . Use the Cauchy Riemann equations for u=x,v=-y .
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    Quote Originally Posted by davy337 View Post
    Hi ppl, would love the solutuin to "show that f(Z) = Z(hat) is not differentiable"
    The previous poster has given help. If you need more help, please show all your work and say where exactly you are stuck.
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    Or you could use the reasoning that leads to the Cauchy-Riemann equations. Let z= x+ iy where x and y are real numbers. Then \overline{z}= x- iy. To find the derivate at, say z_0= x_0+ iy_0 we form the "difference quotient" \frac{f(z)- f(z_0)}{z- z_0}=\frac{x- iy- (x_0- iy_0)}{x+ iy- (x_0+ iy_0)}= \frac{x- x_0+ i(-y- y_0)}{x-x)+ i(y- y_0)} and take the limit as z goes z_0= x_0+ iy_0. Since that is a two dimensional limit, in order that the derivative exist, the limit must be the same as we approach from any direction.

    In particular, what do we get if we approach (x_0, y_0) along the line (x, y_0) ?

    What do we get if we approach (x_0, y_0) along the line (x_0, y)? Are those the same?
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    Quote Originally Posted by FernandoRevilla View Post
    I think you mean f(z)=\bar{z}=x-iy . Use the Cauchy Riemann equations for u=x,v=-y .
    There is also a useful (but perhaps not commonly known) Cauchy-Riemann relation for when you have a function of z and \overline{z}:

    \displaystyle \frac{\partial f}{\partial \overline{x}} = 0.
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