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Let $f,g:U\to {𝔸}^1$ be two regular functions from the smooth affine complex variety $U$ to the affine line. The associated exponential Gauß-Manin systems on the affine line are defined to be the cohomology sheaves of the direct image of the exponential differential system ${𝒪}_U{e}^g$ with respect to $f$. We prove that its holomorphic solutions admit representations in terms of period integrals over topological chains with possibly closed support and with rapid decay condition.