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.
