Analytic continuation along a path by admissible chains

Let $\mathrm{\Omega }\subseteq \mathbb{C}$ be a complex domain, let $\gamma :[0,1]\to \mathrm{\Omega }$ be a path, and let ${\xi }_0$ be a holomorphic germ at $\gamma (0)$.

An admissible continuation chain for ${\xi }_0$ along $\gamma$ consists of

1. a subdivision $0=t_0<t_1<\cdots <t_m=1,$
2. function elements $(f_j,U_j)$ for $0\le j\le m-1$,

such that

$\gamma ([t_j,t_{j+1}])\subseteq U_j(0\le j\le m-1),$

the initial germ of $(f_0,U_0)$ at $\gamma (0)$ is ${\xi }_0$, and for every $j<m-1$ the successive representatives agree at the joining point: $[f_j{]}_{\gamma (t_{j+1})}=[f_{j+1}{]}_{\gamma (t_{j+1})}\mathrm{.}$ Equivalently, $(f_{j+1},U_{j+1})$ is a direct analytic continuation of $(f_j,U_j)$ with the overlap point chosen to be $\gamma (t_{j+1})$ (Function elements and direct analytic continuation).

If such a chain exists, its terminal germ is the germ of $f_{m-1}$ at the endpoint $\gamma (1)$. We then say that ${\xi }_0$ admits analytic continuation along $\gamma$.