bibkey: ozvatanpashaev2017binetcurves authors: Merve Özvatan and Oktay K. Pashaev year: 2017 title: Generalized Fibonacci Sequences and Binet-Fibonacci Curves doi: null url: https://arxiv.org/abs/1707.09151v1 claim: Section 5.1 extends the Binet formula to a real-parameter complex curve after selecting the +pi logarithm branch. strata_touched: [] license: citation-only triage: anchor

Generalized Fibonacci sequences and Binet-Fibonacci curves

The primary source is the fixed arXiv v1 PDF, §5.1, printed pages 13–14. It writes $\log (-1)=i(2n+1)\pi$ and then selects $n=0$. Its real-parameter curve is

B_+(t)=\frac{e^{at}-e^{-at}e^{i\pi t}}{\varphi+\varphi^{-1}},
\qquad a=\log\varphi.

The displayed real and imaginary components on printed page 14 are

\Re B_+(t)=\frac{e^{at}-e^{-at}\cos(\pi t)}{\sqrt5},\qquad
\Im B_+(t)=-\frac{e^{-at}\sin(\pi t)}{\sqrt5}.

Thus the negative-frequency branch used in FIB hyperbolic geometry and phase boundary, §§一、六 is $\overline{B_{+}(t)}$ for real $t$. All integer samples agree. The source attests the complex Binet continuation and its branch choice. Its later area, curvature, spiral and natural-form comparisons are not consumed. The consumer’s autonomous embedding, observation fibers and native continuing-response comparison are not attributed to this paper.