Uniform Convergent Expansions of the Error Function in Terms of Elementary Functions

Abstract We derive a new analytic representation of the error function $${\text {erf}}\,\, z$$ erf z in form convergent series whose terms are exponential and rational functions. The expansion holds uniformly z double sector $$\vert \arg (\pm z)\vert <\pi /4$$ | arg ( ± ) < π / 4 . is accompanied by realistic bounds.