On solution to the simplest functional equation in curvilinear-band-type domain

We consider the functional equation $u(\xi + 1) - u(\xi) = d(\xi), \xi\in\Pi,$, in the region $\Pi\subset C$ of “curvilinear band” type. For sufficiently fast decreasing at infinity and holomorphic within the domain $\Pi$ functions $d(\xi)$ the existence of a holomorphic and bounded solution is shown, the uniqueness of the solutions is investigated. We also obtained precise estimates of the constructed solutions and its asymptotic behavior.