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Math Help - Complex Function Differentiability

  1. #1
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    Complex Function Differentiability

    First off, sorry if this is in the wrong section it could be in calculus I guess, but my unit is titled complex analysis so I thought it might go in here.

    So I have just started Complex analysis for the first time and just needed some help.

    When I am required to find all the points in the complex plane at which a complex function of several variables (i.e f(x+iy)=u(x,y)+iv(x,y)) is differentiable, what steps do i take in determining the answer. When dealing with real numbers i would usually find the answer via inspection or asymptotes ect. But i am having trouble picturing it on the complex plane. Do i use the Cauchy-Riemann Equations to test for analyticity?

    For example f(x+iy)= 2(x^2-y^2) -i4xy , has the answer differentiable at z=0. but how do i determine this.

    Any help would be greatfully appreciated as my lecturer has not covered this yet but trying to wrap my head around it
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: Complex Function Differentiability

    Quote Originally Posted by olski1 View Post
    First off, sorry if this is in the wrong section it could be in calculus I guess, but my unit is titled complex analysis so I thought it might go in here.

    So I have just started Complex analysis for the first time and just needed some help.

    When I am required to find all the points in the complex plane at which a complex function of several variables (i.e f(x+iy)=u(x,y)+iv(x,y)) is differentiable, what steps do i take in determining the answer. When dealing with real numbers i would usually find the answer via inspection or asymptotes ect. But i am having trouble picturing it on the complex plane. Do i use the Cauchy-Riemann Equations to test for analyticity?

    For example f(x+iy)= 2(x^2-y^2) -i4xy , has the answer differentiable at z=0. but how do i determine this.

    Any help would be greatfully appreciated as my lecturer has not covered this yet but trying to wrap my head around it
    I think you should read here:

    Cauchy
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  3. #3
    MHF Contributor chisigma's Avatar
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    Re: Complex Function Differentiability

    Quote Originally Posted by olski1 View Post
    First off, sorry if this is in the wrong section it could be in calculus I guess, but my unit is titled complex analysis so I thought it might go in here.

    So I have just started Complex analysis for the first time and just needed some help.

    When I am required to find all the points in the complex plane at which a complex function of several variables (i.e f(x+iy)=u(x,y)+iv(x,y)) is differentiable, what steps do i take in determining the answer. When dealing with real numbers i would usually find the answer via inspection or asymptotes ect. But i am having trouble picturing it on the complex plane. Do i use the Cauchy-Riemann Equations to test for analyticity?

    For example f(x+iy)= 2(x^2-y^2) -i4xy , has the answer differentiable at z=0. but how do i determine this.

    Any help would be greatfully appreciated as my lecturer has not covered this yet but trying to wrap my head around it
    A complex variable function f(z)= u(x,y) + i\ v(x,y) is differenziable in a point z=z_{0}= x_{0}+i\ y_{0} only if there the 'Cauchy-Riemann equations' are satisfied...

    \frac{\partial u}{\partial x}= \frac{\partial v}{\partial y}\ ,\ \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x} (1)

    In particular for u(x,y)= 2\ (x^{2}-y^{2}) and v(x,y)= 4\ x\ y the (1) are satisfied in the whole complex plane... in Your example however is u(x,y)= 2\ (x^{2}-y^{2}) and v(x,y)= - 4\ x\ y and the (1) are satisfied only for x=y=0...

    Kind regards

    \chi \sigma
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  4. #4
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    Re: Complex Function Differentiability

    Thanks Guys, thats what i was sort of thinking. Got it now! Man i love this forum
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