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Math Help - Differentiation using Cauchy Riemann Equations

  1. #1
    Senior Member bugatti79's Avatar
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    Differentiation using Cauchy Riemann Equations

    FOlks,

    Use the C-R equations to determine whether f(z)=u(x,y)+iv(x,y) is complex analytic for the following, if so calculate f'(z).

    a) \frac{x-iy}{x^2+y^2}

    1) Is complex analytic the same as complex differentiable?

    2) I determine f(z) = \frac{x}{x^2+y^2} +i \frac{-y}{x^2+y^2} and f'(z) =u_x+i v_x

    Using the quotient rule I determine u_x=v_y= \frac{-x^2+y^2}{(x^2+y^2)^2} and u_y=-v_x= \frac{-2xy}{(x^2+y^2)^2}

    Therefore f'(z)= \frac{-x^2+y^2}{(x^2+y^2)^2}+ i \frac{2xy}{(x^2+y^2)^2}...?
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  2. #2
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    Re: Differentiation using Cauchy Riemann Equations

    1) On Wikipedia: "a function that is complex-differentiable in a whole domain (holomorphic) is the same as an analytic function."
    2)
    du/dx = dv/dy => True
    -dv/dx => True

    Thus, f'(z) = (y^2-x^2)/(x^2+y^2)^2 + (2 i x y)/(x^2+y^2)^2

    Theorem 3.3.4 If verifies the Cauchy-Riemann Formulas at and if the partial derivatives of and are continuous at , then is derivable at and . [source]
    Last edited by MrCryptoPrime; April 18th 2012 at 02:24 PM.
    Thanks from bugatti79
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