Continuous Complex Multipole Expansions with Observer-and Resolution-scale Dependence

Building on a prior derivation of the generalized multipole expansion via analytic continuation and the Sommerfeld-Watson transformation, we extend the formulation to include observer and resolution-dependent weighting of the effective multipole order. This enables continuous interpolation between multipole orders and controlled, differentiable variation of the generalized potential with respect to an external scale parameter. The resulting framework provides a mathematically rigorous generalization of the classical multipole expansion, facilitating systematic analysis of continuous multipole variations while building on the foundational derivation.