# Extended Abel-Jacobi theorem

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 $$(P_1)+...+(P_n)$$.

This is true. Given two points $$a,b\in X$$ with $$u(a)$$ and $$u(b)\neq 0$$, one can choose a path from $$a$$ to $$b$$ and a determination of $$\log u$$ along that path, so that $$\exp\int^b_a d\log u=u(b)/u(a)$$. Thus, if $$\operatorname{div}(f)=\sum n_jQ_j$$, the left-hand side is $$\prod u(Q_j)^{n_j}$$.
Recall that if $$f,g$$ are two meromorphic functions on $$X$$, of order $$n$$ and $$m$$ at a point $$P$$, the tame symbol $$(f,g)_{P}$$ is $$(-1)^{nm}[g^{n}/f^m](P)$$. Thus, if $$\operatorname{div}(u)=\sum m_iP_i$$, we have $$(f,u)_{P_i}= f(P_i)^{-m_i}$$, $$(f,u)_{Q_j}= u(Q_j)^{n_j}$$, and $$(f,u)_{P}=1$$ at the other points. Now the product formula $$\prod (f,u)_P=1$$ gives your equality.