# Fibonacci Generating Function of a Complex Variable

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

for the recursive fibonacci numbers.

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 $n\ge2$.

• 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 ... – Old John Jan 16 '13 at 23:13
• I apoligize. It can be found here: math.sfsu.edu/beck/papers/complex.pdf – Anthony Peter Jan 16 '13 at 23:14
• @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. – Gerry Myerson Jan 16 '13 at 23:15
• Anthony, how would you propose to integrate a function of a real variable around a circle in the complex plane? – Gerry Myerson Jan 16 '13 at 23:19

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 $f_n$.)
• @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. – mrf Jan 16 '13 at 23:22
• Anthony, what's wanted is a formula for $f_n$ that depends only on $n$, and not on previous values of $f$. – Gerry Myerson Jan 16 '13 at 23:24
• 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. – Gerry Myerson Jan 16 '13 at 23:36