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Math Help - non-existence of an antiderivative of a complex function

  1. #1
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    non-existence of an antiderivative of a complex function

    Explain in detail why the fact that the integral S(1/z)dz=1/(2(pi)i) taken on the unit circle with center zero i.e lzl=1 ensures that f(z)=1/z does not have an antiderivative on the region {z: z does not equal 0}.

    my thoughts
    f(z) is continuous on the region as the region does not include z=0 and the contour lzl=1 lies entirely in the region so the integral should be path independent and so the integral should have an antiderivative (i.e logz) and the integral around closed surves lying entirely in the region should all have values of 0 but 1/(2(pi)i) does not equal 0 so the antiderivative does not exist (also at (1,0) on the circle 1/(2(pi)i) is undefined).

    Is this reasoning correct and complete? If not could someone please help me.

    Thanks in advance.
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  2. #2
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    Quote Originally Posted by chocaholic View Post
    Explain in detail why the fact that the integral S(1/z)dz=1/(2(pi)i) taken on the unit circle with center zero i.e lzl=1 ensures that f(z)=1/z does not have an antiderivative on the region {z: z does not equal 0}.

    my thoughts
    f(z) is continuous on the region as the region does not include z=0 and the contour lzl=1 lies entirely in the region so the integral should be path independent and so the integral should have an antiderivative (i.e logz) and the integral around closed surves lying entirely in the region should all have values of 0 but 1/(2(pi)i) does not equal 0 so the antiderivative does not exist (also at (1,0) on the circle 1/(2(pi)i) is undefined).

    Is this reasoning correct and complete? If not could someone please help me.

    Thanks in advance.

    It looks fine but slightly too wordy: enough to say that the function's analytic on the given

    region and if it had an antiderivative then the given (closed-path) integral shoud be zero.

    Tonio
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  3. #3
    MHF Contributor FernandoRevilla's Avatar
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    Only as a complement: the differential form w=dz/z is closed in \mathbb{C}-\{0\} i.e. for every z_0\neq 0 there exists a determination of \log z in a neighbourhood of z_0 and that determination is a a primitive of w , but that primitive does not exist on \mathbb{C}-\{0\} as already has been pointed out.


    Fernando Revilla
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