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Math Help - In complex analysis...

  1. #1
    MHF Contributor kalagota's Avatar
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    SINCE THERE ARE REPLIES: In complex analysis...

    are these two terms equivalent: analytic and differentiable?
    Last edited by kalagota; July 5th 2008 at 09:15 PM.
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  2. #2
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    Quote Originally Posted by kalagota View Post
    are these two terms equivalent: analytic and differentiable?
    No.

    Counter example: f(z) = \left | z \right|^2.

    A function f of the complex variable z is analytic at a point z_{0} if its derivative exists not only at z_{0} but at each point z in some neighbourhood of z_{0}.

    The function f(z) = \left | z \right|^2 is not analytic at any point since its derivative exists only at z = 0 and not throughout any neighbourhood (proof available upon request). So it's differentiable at z = 0 but not analytic at z = 0.
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  3. #3
    MHF Contributor kalagota's Avatar
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    now, after reading this, i got confused..

    Complex Differentiable -- from Wolfram MathWorld

    Analytic Function -- from Wolfram MathWorld

    Quote Originally Posted by mr fantastic View Post
    No.

    Counter example: f(z) = \left | z \right|^2.

    A function f of the complex variable z is analytic at a point z_{0} if its derivative exists not only at z_{0} but at each point z in some neighbourhood of z_{0}.

    The function f(z) = \left | z \right|^2 is not analytic at any point since its derivative exists only at z = 0 and not throughout any neighbourhood (proof available upon request). So it's differentiable at z = 0 but not analytic at z = 0.
    you don't have to prove.. i know how to do it.. thanks anyways..
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    Super Member Aryth's Avatar
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    A function that is analytic is also complex differentiable. But a function that is complex differentiable may not always be analytic.

    In order for a complex differentiable function to be analytic it must be complex differentiable at every point of the region in question.
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  5. #5
    MHF Contributor kalagota's Avatar
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    thanks for that but it doesn't help much since you just stated the definition..

    to give you idea of what i am talking about, for a given function, you are to show that the Cauchy-Riemann equations is satisfied at a point, say z0 but also show that it is not differentiable at that point.

    consider this thread.. http://www.mathhelpforum.com/math-he...-analysis.html

    basically, the question is similar except at the last. instead of showing that it is not holomorphic, you have to show that it is not differentiable..
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    A function is differenciable at a point z_0 iff it is defined in the neighborhood of this point and \lim_{z\to z_0} \tfrac{f(z)-f(z_0)}{z-z_0} exists. A function is analytic at a point z_0 iff there is a neighborhood around z_0 such that the function is differenciable at all those point in the neighborhood. The notion of a function being differenciable at only one point and nowhere else around the neighborhood really does not appear in complex analysis. Most functions that are considered are also differenciable in the neighorhood as well. It turns out that if a function is analytic then it must be infinitely differenciable and the Taylor series must converge to the function. One of the many shocking results in complex analysis. Therefore, the term "analytic" is consistent with the other meaning of the word "analytic" i.e. having a power series expansion.
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    Quote Originally Posted by ThePerfectHacker View Post
    [snip]
    The notion of a function being differenciable at only one point and nowhere else around the neighborhood really does not appear in complex analysis. Most functions that are considered are also differenciable in the neighorhood as well.
    [snip]
    The function f(z) = \left | z \right|^2 appears in many tetxbooks. It provides the necessary counter-example to the statement that analytic and differentiable are equivalent.
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    Quote Originally Posted by mr fantastic View Post
    The function f(z) = \left | z \right|^2 appears in many tetxbooks. It provides the necessary counter-example to the statement that analytic and differentiable are equivalent.
    I meant by that that the functions that are usually studied on complex analysis do not have this behavior. Like all the theorems that are encountered avoid this.
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  9. #9
    MHF Contributor kalagota's Avatar
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    now, i have a question:
    what if i found that it is not analytic at a point, can i conclude that it is not differentiable at that point?
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    Quote Originally Posted by kalagota View Post
    now, i have a question:
    what if i found that it is not analytic at a point, can i conclude that it is not differentiable at that point?
    Look at what Mr.Fantastic did.
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  11. #11
    MHF Contributor kalagota's Avatar
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    where? this one?

    No.

    Counter example: .

    A function f of the complex variable z is analytic at a point if its derivative exists not only at but at each point z in some neighbourhood of .

    The function is not analytic at any point since its derivative exists only at and not throughout any neighbourhood (proof available upon request). So it's differentiable at but not analytic at .
    he showed that the function is not differentiable at a neighborhood and then concluded not analytic..

    what if, i found it not analytic at the point? i can conclude that there exists a neighborhood of the point that is not differentiable, right? but, can i conclude that it is not differentiable at the point?

    (waah, i am tempted to put the original question i have, but due to intellectual honesty, i can't! )

    EDIT: this is the original question but i did not include the function and the point..

    Given this function f(z) = ????, show that it is continuous and satisfies Cauchy-Riemann equations at z=???? but is not differentiable there.

    i am having trouble with the last part, that is to show that it is not differentiable there..

    what i did, i showed that it is not analytic at that point. can i conclude that it is not differentiable there? if not, what can i do?

    thanks a lot!!
    Last edited by kalagota; July 6th 2008 at 09:16 PM. Reason: added some things...
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