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I Is long division for a non-polynomic function possible?

  1. Mar 4, 2016 #1
    My question is: suppose you have a function ##F(x)## which has an asymptote at ##x=x*##. Can you decompose ##F(x)## so that
    $$F(x) = G(x) + H(x)$$
    where ##G(x)## is defined at ##x=x*## and ##H(x)## contains the asymptotic behaviour at ##x=x*## and goes to ##0## at plus or minus ##\inf##? This is commonly done when ##F(x)## is a polynomial in order to approximate ##F(x)##'s behaviour with the function ##G(x)## when ##x## is sufficiently far away from ##x^{*}##.

    I have a very complicated function ##F(x)## that I would like to extract ##G(x)## from. If you're curious here it is:

    $$F(x) = \frac{(2 a_1 z^{-a_2} -2 a_1 z^{-a_1} + z^{-a_1} - z^{-a_2}+ a_1^2 z^{-a_1} - a_2^2 z^{-a_2}) ln(z)^2 + (2 a_2 z^{-a_2} -2 a_1 z^{-a_1}+2 z^{-a_2} - 2 z^{-a_1}) ln(z) + 2 z^{-a_1} - 2 z^{-a_2}}
    {(a_1 z^{-a_1} - a_2 z^{-a_2} + z^{-a_2} - z^{-a_1}) ln(z)^2 + (z^{-a_1}-z^{-a_2}) ln(z)}$$

    where
    $$z = 10^{x-x^{*}}$$
    and ##a_2>a_1>0## and ##x^*>0##. I've attached a graph for an instance of the function where ##x^{*}=2##. As you can see there is an asymptote at ##x=2##. I want either ##G(x)## or the slope of ##G(x)## at ##x=x^{*}##. Has anyone encountered a similar problem?

    Screen%20Shot%202016-03-04%20at%207.08.13%20PM.png
     
    Last edited: Mar 5, 2016
  2. jcsd
  3. Mar 5, 2016 #2

    mfb

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    2016 Award

    Staff: Mentor

    Such a decomposition is always possible and never unique (e.g. for every solution you can add a Gaussian to G and subtract it from H). That also means that the slope of G(x) is not fixed at x=x*, it can have any value you like.

    To find a possible decomposition, it is sufficient to describe the behavior either for x around x* or for x to +- infinity. The latter is probably easier here as it does not involve finding x*.
     
  4. Apr 15, 2016 #3
    Thank you very much for your insight. You made the arbitrariness of G(x) very clear to me. Sorry it took so long to get back to you.
     
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