Evaluating of Dawson's Integral by solving its differential equation using orthogonal rational Chebyshev functions

Dawson's Integral is uðyÞ expðÀy 2 Þ R y 0 expðz 2 Þdz. We show that by solving the differential equation du=dy þ 2yu ¼ 1 using the orthogonal rational Chebyshev functions of the second kind, SB 2n ðy; LÞ, which generates a pentadiagonal Petrov–Galerkin matrix, one can obtain an accuracy of roughly ð3=8ÞN digits where N is the number of terms in the spectral series. The SB series is not as efficient as previously known approximations for low to moderate accuracy. However, because the N-term approximation can be found in only OðNÞ operations, the new algorithm is the best arbitrary-precision strategy for computing Dawson's Integral.