  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 $\mathbb{R}$. 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 $\Gamma$, $\zeta$ 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.
