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Math Help - When exactly do we use the Cauchy-Riemann equations?

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    When exactly do we use the Cauchy-Riemann equations?

    I'm having a really hard time trying to figure out when to use the Cauchy-Riemann equations, u_x = v_y and u_y = -v_x. My textbook says that they are necessary, but not sufficient conditions for differentiability. My textbook also says that they can be used to show that a function is not differentiable at a point, by showing that the CR equations do not hold. So is that all they can be used for, to show non-differentiability?
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    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by Alexrey View Post
    So is that all they can be used for, to show non-differentiability?
    We can also use the definition of differentiability. For example f(z)=\bar{z}


    \dfrac{\overline{z+h}-\bar{z}}{h}=\dfrac{\bar{h}}{h}=e^{-2i\theta}

    so, the limit as h\to 0 depends on \theta as a consequence , f is not differentiable at z.
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    I know that we can use the above to prove that a function is not differentiable, but lets say that I get a question that says, "Prove that the following function is NOT differentiable", then instead of using the above method that you mentioned, could I straight away use the Cauchy-Riemann equations to prove the statement?

    Also my textbook says that if a function has continuous first order partial derivatives that satisfy the Cauchy-Riemann equations, then it is differentiable, so if I get a question that says, "The following function has continuous partial derivatives, prove that it is differentiable", then again can I use the CR equations instead of the usual method for proving differentiability that you used?
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    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by Alexrey View Post
    I know that we can use the above to prove that a function is not differentiable, but lets say that I get a question that says, "Prove that the following function is NOT differentiable", then instead of using the above method that you mentioned, could I straight away use the Cauchy-Riemann equations to prove the statement?

    Of course you can, f(z)=\bar{z}=x-iy\Rightarrow u_x=1\neq -1=v_y so, f is not differentiable at z=x+iy

    Also my textbook says that if a function has continuous first order partial derivatives that satisfy the Cauchy-Riemann equations, then it is differentiable, so if I get a question that says, "The following function has continuous partial derivatives, prove that it is differentiable", then again can I use the CR equations instead of the usual method for proving differentiability that you used?
    Yes, you can.
    Last edited by FernandoRevilla; May 18th 2011 at 02:34 AM.
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    Awesome, thanks very much!
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