So I'm doing work on the Fibonacci Numbers, and I came across this problem for the generating function enter image description here

for the recursive fibonacci numbers. This is the problem

I have two questions: 1. Why is it useful to use a complex variable $z^n$ as apposed to a real variable $x^n$? 2. What does it mean by derive an identity for $f_n$? Note that $f_0 = 0$, $f_1 = 1$, $f_n = f_{n-1} + f_{n-2}$ for $n\ge2$.

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    $\begingroup$ If you are going to refer to Theorem 9.9, don't you think it might be a good idea to state which book that comes from? - Theorem 9.9 in my book has nothing to do with complex variables and Fibonacci numbers ... $\endgroup$ – Old John Jan 16 '13 at 23:13
  • $\begingroup$ I apoligize. It can be found here: math.sfsu.edu/beck/papers/complex.pdf $\endgroup$ – Anthony Peter Jan 16 '13 at 23:14
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    $\begingroup$ @Old John, I take your point, but the Residue Theorem in your book is quite likely to be the same as the Residue Theorem in OP's book. $\endgroup$ – Gerry Myerson Jan 16 '13 at 23:15
  • $\begingroup$ Anthony, how would you propose to integrate a function of a real variable around a circle in the complex plane? $\endgroup$ – Gerry Myerson Jan 16 '13 at 23:19
  1. It's much easier to understand power series in the context of complex analysis than real analysis. (The sum of a power series is an analytic function, and those are exactly the ones you study in complex analysis. Also, the radius of convergence is directly related to the singularities of this analytic function.)

  2. You want to find a formula for $f_n$. Have you tried following the hint? (Recall Cauchy's integral formula to relate the integral to the value of $f_n$.)

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  • $\begingroup$ In what manner is the radius of convergence related to the singularities? I personally know that it is, but in what way? Also, about number two, I still don't understand what it's asking because obviously the formula for fn is fn= fn-1 + fn-2. $\endgroup$ – Anthony Peter Jan 16 '13 at 23:19
  • $\begingroup$ @AnthonyPeter The power series of $f(z)$ centered at $z=a$ converges on the largest disc $\{ |z-a| = R \}$ on which $f$ is analytic. I.e. (somewhat sloppily) the radius of convergence is the distance to the closest singularity. $\endgroup$ – mrf Jan 16 '13 at 23:22
  • $\begingroup$ Anthony, what's wanted is a formula for $f_n$ that depends only on $n$, and not on previous values of $f$. $\endgroup$ – Gerry Myerson Jan 16 '13 at 23:24
  • $\begingroup$ @GerryMyerson Oh, that would make much more sense. How would the provided hint help do this though? $\endgroup$ – Anthony Peter Jan 16 '13 at 23:29
  • $\begingroup$ Anthony, have you tried doing the integration? Or maybe what you are missing is the connection between $F(z)$ and the function you are asked to integrate --- you should think about that. $\endgroup$ – Gerry Myerson Jan 16 '13 at 23:36

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