Asymptotics for the Error Function

Abstract

In this chapter the asymptotic forms for the error function and the related function u(a) over the principal branch of the complex plane are regularised via Borel summation. The resulting equations are expressed in terms of a Stokes multiplier, which toggles between -1/2 and 1/2 for the different Stokes sectors. Numerical studies are then conducted for large and small values of the magnitude of the variable, viz. |z|, over the entire principal branch. For the large values of |z| the truncated series is the dominant contribution which is consistent with standard asymptotics. Although the truncated series dominates for small values of|z|, so does the regularised value of its remainder in the opposite sense. Hence, when both contributions are combined, the remaining contribution with the Stokes multiplier can become substantial. Nevertheless, in each case where all the contributions are summed, one always obtains the exact values of the error function. Then an expression for the Stokes multiplier is obtained. By carrying out an extensive numerical analysis in the vicinity of the Stokes line along the positive real axis, it is found that irrespective of the value of variable, the Stokes multiplier is discontinuous and not smooth as implied by the leading order term.