Abstract
In the paper, by the Cauchy integral formula in the theory of complex functions, the authors establish an integral representation for the reciprocal of the weighted geometric mean of two positive numbers. By the integral representation, the authors derive that the reciprocal of the weighted geometric mean of two positive numbers is a Stieltjes function and consequently a (logarithmically) completely monotonic function, present several integral formulas for some kinds of improper integrals, deduce several integral representations for some functions related to the weighted geometric mean, find an equivalent relation between integral representations for the weighted geometric mean and its reciprocal, connect the integral formulas with the central Delannoy numbers, and apply some integral representations and some integral formulas to compute some concrete improper integrals appeared in the theory of complex functions.
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