Binet's Formula

$$F_z=\frac{{\varphi }^z-{\psi }^z}{\sqrt{5}}$$

The Analytic Continuation of the Fibonacci Sequence.

Derivation

$$\text{Let }A=\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]$$ $$\left[\begin{array}{c}{F}_{z+1}\\ F_z\end{array}\right]=A^z\left[\begin{array}{c}{F}_1\\ F_0\end{array}\right]$$

See Diagonalization

$$\begin{array}{rl}A& =PD{P}^{-1}\\ & \\ P& =\left[\begin{array}{cc}\psi & \varphi \\ 1& 1\end{array}\right]\\ D& =\left[\begin{array}{cc}\psi & 0\\ 0& \varphi \end{array}\right]\\ P^{-1}& =\frac{1}{\sqrt{5}}\left[\begin{array}{cc}-1& \varphi \\ 1& \psi \end{array}\right]\\ \text{where }\varphi & =\frac{1+\sqrt{5}}{2}\\ \text{where }\psi & =\frac{1-\sqrt{5}}{2}\end{array}$$ $$\begin{array}{rl}{A}^z& =P{D}^z{P}^{-1}\\ & =\frac{1}{\sqrt{5}}\left[\begin{array}{cc}\psi & \varphi \\ 1& 1\end{array}\right]\left[\begin{array}{cc}\psi & 0\\ 0& \varphi \end{array}\right]\left[\begin{array}{cc}-1& \varphi \\ 1& \psi \end{array}\right]\\ & =\frac{1}{\sqrt{5}}\left[\begin{array}{cc}{\varphi }^{z+1}-{\psi }^{z+1}& {\psi }^z-{\varphi }^z\\ {\varphi }^z-{\psi }^z& 1\end{array}\right]\end{array}$$ $$\begin{array}{rl}\left[\begin{array}{c}{F}_{z+1}\\ F_z\end{array}\right]& =\frac{1}{\sqrt{5}}\left[\begin{array}{cc}{\varphi }^{z+1}-{\psi }^{z+1}& {\psi }^z-{\varphi }^z\\ {\varphi }^z-{\psi }^z& 1\end{array}\right]\left[\begin{array}{c}{F}_1\\ F_0\end{array}\right]\\ & =\frac{1}{\sqrt{5}}\left[\begin{array}{c}{\varphi }^{z+1}-{\psi }^{z+1}\\ {\varphi }^z-{\psi }^z\end{array}\right]\end{array}$$ $$\begin{array}{c}\therefore F_z=\frac{{\varphi }^z-{\psi }^z}{\sqrt{5}}\\ \text{QED}\end{array}$$