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Math Help - Complex Harmonies

  1. #1
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    Complex Harmonies

    I've been in trouble enough lately, I figured I would get up to speed on a few things. Pondering elementary Complex Analysis, I made it only to page 53 before I managed to get stuck.

    I want a non-zero Harmonic Function that disappears on the hyperbola y = 1/x.

    I'm staring at f(z) = z^2.

    I created a lovely page (or 3) of algebraic manipulations that lead to interesting results and relationships, but don't solve the problem at hand. I figure I'm just missing something.

    Do I get a hint? What about f(z) = z^2 is supposed to impress me?
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  2. #2
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    I do not understand. You want an analytic function f:\mathbb{C}\mapsto\mathbb{C} so that f\not = 0 and f\left( x + i\frac{1}{x} \right)=0 for all real x\not = 0. Is that what you are asking? Because the word "harmonic" is strange here.

    Try f(z) = z^2 - \bar z^2 - 4i. This is not everywhere analytic but it almost works.
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  3. #3
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    What does "disappears on the hyperbola y = 1/x" mean here?
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  4. #4
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    Right. I seem already to have picked up the somewhat cheesy vocabulary of the author. Maybe I need a new book. Also, it seems I am assuming you can see what I'm seeing. How many students have I told not to do that?!

    In any case:

    1) It's u(x,y) on \Re^{2}\mapsto\Re^{2}
    2) "disappears" or "vanishes" seems to mean "u(x,y)= 0"

    So u(x,1/x) = 0, u(x,y) is not constant, and u(x,y) satisfies Laplace's Equation, second partial derivatives sum to zero.

    I have a hint, "Look at f(z) = z^2". Sadly, this "look" doesn't seem to be clarifying anything.
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  5. #5
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    Can you try u(x,y) = 2xy - 2? It is harmonic.
    The conjugate is v(x,y) = y^2  - x^2 .
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  6. #6
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    Hmpf. I still don't see why the author wanted me to look at z^2 for a clue. I wasn't expecting a unique result, but it seems I haven't found the one the author was hinting at.

    Time to move on.
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  7. #7
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    Quote Originally Posted by TKHunny View Post
    Hmpf. I still don't see why the author wanted me to look at z^2 for a clue.
    Actually I did use the authorís hint.
    If v is the harmonic conjugate of u then -u is the harmonic conjugate of v.
    z^2  = (x^2  - y^2 ) + i(2xy)
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  8. #8
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    Good call. I begin to see. I'm sure I'll be back in another 50 pages.
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