$\newcommand{\C}{\mathbb{C}}$ Suppose $f\colon \Omega\subset \C\to\C$ is a holomorphic function and $\gamma:[0,1]\to\Omega$ is a continuous path.
If $\Omega=\C\setminus\{0\}$, $\gamma(t):= e^{2\pi i t}$ and $f(z)=1/z$ then for $t\in[0,1]$ the formula $$f(\gamma(t))=f(\gamma(0))\cdot e^{\int_{\gamma|_{[0,t]}}f(z)dz}$$ holds true.
Is this true in general? Can a similar statement be made about vector fields on a manifold, for example using a spin structure?