ABSTRACT
The basic problem to which we now turn our attention is the solution of the equation
(3.1)
for primitives F(x|ω) given . This constitutes a generalization of the corresponding problem of the integral calculus, namely the discovery of primitives F(x) satisfying
(3.2)
Progress in the integral calculus was impeded until a constructive definition was framed providing one of the primitives of (3.2). This definition-the Riemann integral-formed the foundation for the theory of integration. Its properties allowed a fruitful theory to be developed. Similarly, one would like a constructive definition of a particular primitive of (3.1) that would possess rich analytic properties permitting a useful theory to be developed. It should provide simple representations of important functions and have means of ready asymptotic computation and approximation. For example, certainly F(x|ω) corresponding to being a polynomial should also be a polynomial; such a primitive exists, as can be seen from the Newton expansion (1.8). The unique determination of F(x|ω) should rest on its value at a single point rather than a
specification throughout an interval, and one would also like to reduce to a
solution of (3.2) because whenever Df(x) exists. All these properties are provided by the formulation of Nörlund [17], which will now
be studied.
