Let $X$ be a compact Riemann surface, $u$ a meromorphic function on $X$ with divisor supported on a set of points $\{P_1, ..., P_n\}$, and $f$ a meromorphic function on $X$ such that $f$ has no pole or zeros on $\{P_1, ..., P_n\}$. Let $d\log(u)$ be the logarithmic differential form of $u$. Do we have the following equality in $\mathbf{C}^{\times}$: $$\exp(\int_{div(f)} d\log(u)) = \prod_{i=1}^n f(P_i)^{\text{ord}_{P_i}(u)}\text{ ?}$$ Here, $\text{div}(f)$ is the divisor of $f$, and the integral is well-defined (independant of the choice of the path) up to $2\pi i \mathbf{Z}$.

This identity is true if for all $i$ we have $f(P_i)=1$, as it relates to the analytic Abel-Jacobi description of the generalized Jacobian of $X$ with respect to the divisor $(P_1)+...+(P_n)$.