Positive-definiteness and integral representations for special functions

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We characterize a holomorphic positive definite function f defined on a horizontal strip of the complex plane as the Fourier-Laplace transform of a unique exponentially finite measure on R. The classical theorems of Bochner on positive definite functions and of Widder on exponentially convex functions become special cases of this characterization: they are respectively the real and pure imaginary sections of the complex integral representation. We apply this representation to special cases, including the Γ, ζ and Bessel functions, and construct explicitly the corresponding measures, thus providing new insight into the nature of complex positive and co-positive definite functions: in the case of the zeta function this process leads to a new proof of an integral representation on the critical strip.