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We present a new method to evaluate the α^' expansion of genus-one integrals over open-string punctures and unravel the structure of the elliptic multiple zeta values in its coefficients. This is done by obtaining a simple differential equation of Knizhnik-Zamolodchikov-Bernard-type satisfied by generating functions of such integrals, and solving it via Picard iteration. The initial condition involves the generating functions at the cusp τ→i∞ and can be reduced to genus-zero integrals.For strongly screened Coulomb interactions, quantum
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