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Thread: Complex Analysis - Partial Fraction Decomposition

  1. #1
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    Complex Analysis - Partial Fraction Decomposition

    Hey guys. This might not be quite the right place on the forum, but here goes. I have this assignment that I have almost completed, but I can't quite figure out the last bit. Could anyone give me a helping hand?

    Prove that$\displaystyle F_0 = 0$, $\displaystyle F_1 = 1$ and $\displaystyle F_n =F_{n-1} + F_{n-2}$ (This is the sequence of fibonacci numbers: 0, 1, 1, 2, 3, 5, 8,...).


    Find the partial fraction decomposition of the rational function and use it to prove Binet's formula:

    (1)
    $\displaystyle F_n = \frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^n - \left(\frac{1-\sqrt{5}}{2}\right)^n\right]$

    Alright, I have shown that $\displaystyle \alpha_1 = \frac{-1 + \sqrt{5}}{2}$ and $\displaystyle \alpha_2 = \frac{-1 - \sqrt{5}}{2}$ are simple poles of $\displaystyle f(x)$ and I've found out that the partial fraction decomposition of $\displaystyle f(z)$ is:

    (2)
    $\displaystyle f(z) = \frac{\frac{1+\sqrt{5}}{2\sqrt{5}}}{z-\alpha_1} - \frac{\frac{1-\sqrt{5}}{2\sqrt{5}}}{z-\alpha_2} $

    I just have no idea how to use that, to show (2) from (1). Can someone give me a clue?

    /Morten
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  2. #2
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    Re: Complex Analysis - Partial Fraction Decomposition

    Find the partial fractions decomposition of what rational function? I don't see anywhere that you say what rational function you are talking about.
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  3. #3
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    Re: Complex Analysis - Partial Fraction Decomposition

    Arh sorry.. Apparently it only pasted half of the assignment into the post. Here is the first half of the assignment:

    Prove that the function $\displaystyle z/(1-z-z^2)$ has simple poles at $\displaystyle z = (-1 \pm \sqrt{5})/2$ and determine the radius of convergence of the power series:

    $\displaystyle \frac{z}{1-z-z^2} = \sum_{n = 0}^{\infty}F_nz^n$

    So the rational function is $\displaystyle f(z) = \frac{z}{1-z-z^2}$
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  4. #4
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    Re: Complex Analysis - Partial Fraction Decomposition

    Surely you understand that the poles are where the denominator is 0...
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