Arbitrarily Accurate Analytical Approximations for the Error Function

In this paper a spline based integral approximation is utilized to propose sequence of approximations the error function that converge at significantly faster manner than default Taylor series. The can be improved by utilizing erf(x) approximately equal one for x>>1. Two generalizations are possible, first on demarcating integration interval into m equally spaced sub-intervals. second, it larger fixed sub-interval, with known integral, and smaller sub-interval whose approximated. Both lead accuracy. Further, initial approximations, arising from generalization, as inputs custom dynamical system establish better convergence properties. Indicative results include those fourth order approximation, four sub-intervals, which leads relative bound 1.43 x 10-7 over positive real line. Various achieve set bounds 10-4, 10-6, 10-10 10-16, real, specified. Applications include, first, definition functions upper lower bounds, arbitrary accuracy, function. Second, new series Third, sequences exp(-x2) have higher properties approximation. Fourth, complementary demarcation eC(x) satisfies constraint eC(x)^2 + erf(x)^2 = 1. Fifth, arbitrarily accurate power harmonic distortion sinusoidal signal subject nonlinearity. Sixth, approximate expressions linear filtering step modelled