On the asymptotic expansions of various quantum invariants I: the colored Jones polynomial of twist knots at the root of unity $e^{\frac{2\pi \sqrt{-1}}{N+\frac{1}{2}}}$

This is the first article in a series devoted to the study of the asymptotic expansions of various quantum invariants related to the twist knots. In this paper, by using the saddle point method developed by Ohtsuki, we obtain an asymptotic expansion formula for the colored Jones polynomial of twist knots ${\mathcal{K}}_p$ with $p\ge 6$ at the root of unity $e^{\frac{2\pi \sqrt{-1}}{N+\frac{1}{2}}}$.