Moments by Integrating the Moment-Generating Function

We introduce a general integral framework for computing fractional, complex, absolute, and logarithmic moments from the moment-generating function (MGF) under explicit regularity conditions. By evaluating a complex extension of the MGF along a vertical contour, we obtain exact integral expressions that bypass the need for explicit probability densities and high-order derivatives. We establish conditions for negative fractional moments using the symmetric Cauchy principal value, including the requirement that the distribution have no point mass at the centering point. We demonstrate the theoretical scope and computational practicality of the framework through applications to the normal-inverse Gaussian distribution and a semicontinuous compound Poisson-Gamma distribution. In the latter case, the framework handles point masses at the boundary by evaluating conditional fractional moments.