6-ary mekso/mathematical operator: Heaviside function/step/Theta function of a, of order b, in structure c, using distribution d, within approximated limit e, with value f_b at 0

Explanation: A sequence/family of (strictly monotonically increasingly, ordered) indexed function distributions (d), each of which is of/with respect to/in variable/indeterminant a, is considered; they must have the property that, as the index increases, they converge to some value (described later and determined by b, e, and f_b) for all a. This happens on structure c, which defines what various values and limits mean. b is the 'order' of the result of this convergence (so that it is the bth-order integral (equivalently, -bth-order derivative) of the Heaviside step/Theta function with respect to a so that b = -1 yields the Dirac delta 'function' centered/with interesting point at 0, b = 0 yields the Heaviside step/Theta function with respect to a centered at/with interesting point 0, etc.). Natural numbers (or 0) for b yield a 'special' polynomial (let us call it p) of that order in indeterminant a for all values of a greater than 0 and 0 for all values of a less than 0 so that p(a) = a^b Theta(a). Input e is for when the particular functions in d are of interest and only e of them have been computed (so that, for finite e, the output is not exactly the bth order Heaviside function); in other words, the limit of the family indices is being taken to e; this is of particular importance for negative values of b: |b| > 1, since they will not be identically 0 in some sense (else, all information is lost about the special structure/nature of this function). The contextless defaults are: b = 0, c is the field of real numbers (with absolute value norm and ordering due to signed absolute value), d is bth-order integral of ((1 + y + z * erf(k a / (2^(1/2)) ) ) / 2)-like functions that are properly normalized and are of proper height, e is infinity (countable). Heaviside(0,0,...,f_0) = 1/2 = f_0 is typical and is a contextless default for b = 0 (however, it may in particular be set to 0 or 1 so that the function is left-/right-continuous at a = 0; for values of b other than 0, f_b = 0 is the contextless default; note that f_(-1) must be infinity 'of the proper magnitude' (it cannot be specified to be otherwise). See also: zdeltadirake