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.

翻译：摘要：我们提出一个通用积分框架，用于在显式正则条件下从矩生成函数(MGF)计算分数阶、复、绝对及对数矩。通过沿垂直路径评估MGF的复延拓，我们得到无需显式概率密度函数与高阶导数的精确积分表达式。我们利用对称柯西主值建立了负分数阶矩的存在条件，包括要求分布不存在中心点处的点质量。通过正态逆高斯分布及半连续复合泊松-伽马分布的实际应用，我们展示了该框架的理论覆盖范围与计算实用性。针对后者情形，该框架通过计算条件分数阶矩处理边界点质量。