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The field scattered by a scattering body or by an aperture in the free space (or in an unbounded homogenous medium) can be described in terms of a double integral. In this paper we show how a canonical integral on a polygonal domain, with a constant amplitude function and a quadratic phase variation, can be exactly expressed in terms of special functions, namely Fresnel integrals and generalized Fresnel integrals. This exact reduction represents a paradigm for deriving a new asymptotic evaluation for a more general integral. This new asymptotic uniform integral evaluation is expressed in the format of the uniform geometrical theory of diffraction which is convenient for numerical computations.