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Math Help - Cauchy-Riemann equations and holomorphic functions

  1. #1
    Junior Member CuriosityCabinet's Avatar
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    Cauchy-Riemann equations and holomorphic functions

    Consider the function defined by f(x+iy) = \sqrt(\left |x  \right |\left |y  \right |). Show that f satisfies the Cauchy-Riemann equations at the origin yet it is not holomorphic at zero.

    I can show that it satisfies the C-R equations but unsure about showing it is not holomorphic at zero. Please help?
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    Re: Cauchy-Riemann equations and holomorphic functions

    f(z)-f(0)/(z-0) = f(z)/z
    let z goes to 0 from the positive x axis, we have f(z)/z = 0/z = 0
    let z goes to 0 from the positive part of the line y=x, we have f(z)/z = x/(x+ix)=(1-i)/2
    Thanks from CuriosityCabinet
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    Junior Member CuriosityCabinet's Avatar
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    Re: Cauchy-Riemann equations and holomorphic functions

    Quote Originally Posted by xxp9 View Post
    f(z)-f(0)/(z-0) = f(z)/z
    let z goes to 0 from the positive x axis, we have f(z)/z = 0/z = 0
    let z goes to 0 from the positive part of the line y=x, we have f(z)/z = x/(x+ix)=(1-i)/2
    Thanks, so because the left and right-hand side limits are different, f is not holomorphic? Also, what is the first line representing?
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    Re: Cauchy-Riemann equations and holomorphic functions

    A function, of a complex variable, is "holomorphic" at a point if and only if it is differentable there. What xxp9 did is show that the limit defining the derivative does not exist by showing that the limits, as z approaches 0 from two different directions, give different resuts
    Thanks from xxp9 and CuriosityCabinet
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