The inverse Mellin transform via analytic continuation

We present a method to calculate the $x$--space expressions of massless or massive operatormatrix elements in QCD and QED containing local composite operator insertions, depending onthe discrete Mellin index $N$, directly without computing the Mellin--space expressions inexplicit form analytically. Here $N$ belongs either to the even or odd positive integers. Themethod is based on the resummation of the operators into effective propagators and relies onan analytic continuation between two continuous variables. We apply it to iterative integralsas well as to the case of iterative non--iterative integrals, generalizing the former ones.The $x$--space expressions are needed to derive the small--$x$ behaviour of the respectivequantities, which can usually not be accessed in $N$--space. We illustrate the method fordifferent (iterative) alphabets, including non--iterative $_2F_1$ and elliptic structures,as examples, occurring in different massless and massive three--loop calculations. Likewisethe method applies even to the closed analytic solutions of more general cases of differentialequations not factorizing in first order