Fibonacci Generating Function of a Complex Variable

Question

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 $n\ge2$.

Answer 1

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 $f_n$.)