Boundary value of $\prod_i (x-a_i)^{w_i}$ on the gap $(a_k,a_{k+1})$

Let $n\ge 2$, let $a_0\le a_1\le\cdots\le a_{n-1}$ be real numbers, and let $w_0,\ldots,w_{n-1}$ be real weights with $\sum_{i=0}^{n-1}w_i=1$. For a complex number $z$ and a real exponent $w$, let $z^{w}$ denote the principal power $\exp(w\operatorname{Log} z)$, whose branch cut is the negative real axis and which assigns argument $\pi$ to a negative real base.

Fix an index $k$ with $0\le k\le n-2$ and a real point $x$ strictly inside the gap, $a_k<x<a_{k+1}$. Then

In other words, the principal branch of $\prod_i(z-a_i)^{w_i}$, evaluated on the real segment $(a_k,a_{k+1})$, is the upper boundary value of the analytic continuation from the upper half plane: the factors with $i\le k$ have positive base and are real, while each factor with $i\ge k+1$ has negative base and contributes a phase $e^{i\pi w_i}$. The total phase is therefore $\pi\sum_{i>k}w_i=\pi\big(1-\sum_{i\le k}w_i\big)$, and $\sin\big(\pi(1-\sum_{i\le k}w_i)\big)=\sin\big(\pi\sum_{i\le k}w_i\big)$ gives the stated formula. This is the jump of the multivalued function $\prod_i(z-a_i)^{w_i}$ across its cut.