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The paper accepts the “functional paradigm” in the formative process of the image of an object. It holds that the used formulae must be mathematically demonstrated. As for many systems with memory, the frequently met formula is a defined integral. Its inferior limit is considered “the initial instant”. In this paper we show that: (1) The integral representation formula, that gives the image η (t) as a functional on the history ξ =ξ (τ),τ≤q t, of the evolutions both of the observed object and the observer, is only an approximate one. There exists the possibility to appreciate the order of size of the error. (2) The inferior limit of the integral may be mathematically determined and its choice requires some discussions. (3) A more precise formula of η (t) contains also a double integral upon a quadratic form of ξ(τ). The mathematical model is based on the differential calculus on the linear topological locally convex spaces.