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Math Help - How to use Cauchy Riemann

  1. #1
    Junior Member
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    How to use Cauchy Riemann

    I have reached a step in the question where I have to apply the Cauchy Riemann formulas to:

    (-6x-2By)^2+(-12x+4By)i

    How would I apply the Cauchy Riemann formula to answer the rest of the question?

    And how would I know if it is analytic?

    The cauchy-riemann formula:

    The function f(z)= g(x,y)+ h(x,y)i is differentiable with respect to z= x+ iy only if it satisfies the "Cauchy-Riemann equations":

    \frac{\partial g}{\partial x}=\frac{\partial h}{\partial y}

    and

    \frac{\partial g}{\partial y}=-\frac{\partial h}{\partial x}
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  2. #2
    Junior Member piglet's Avatar
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    Quote Originally Posted by zizou1089 View Post
    I have reached a step in the question where I have to apply the Cauchy Riemann formulas to:

    (-6x-2By)^2+(-12x+4By)i

    How would I apply the Cauchy Riemann formula to answer the rest of the question?

    And how would I know if it is analytic?

    The cauchy-riemann formula:

    The function f(z)= g(x,y)+ h(x,y)i is differentiable with respect to z= x+ iy only if it satisfies the "Cauchy-Riemann equations":

    \frac{\partial g}{\partial x}=\frac{\partial h}{\partial y}

    and

    \frac{\partial g}{\partial y}=-\frac{\partial h}{\partial x}
    You do precisely what you've wrote down. let  g = real part of your function i.e.  (-6x-2By)^2 and let  h = imaginary part of your function i.e.  (-12x+4By)

    Now it's just straight-forward partial differentiation

    Find  \frac{\partial g}{\partial x} . This means you need to differenciate g w.r.t x.

    so  \frac{\partial g}{\partial x} =  -12(-6x-2By)

    and as  \frac{\partial h}{\partial y} is the partial derivative of h w.r.t y

    we get  \frac{\partial h}{\partial y} = 4B

    Clearly  \frac{\partial g}{\partial x}\ne \frac{\partial h}{\partial y}

    A function is analytic in the complex plane if the BOTH caucy riemann equations are satisfied. So since the first C-R equation i checked is not satisfied you can say that the function is not analytic
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