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

Dawson’s Integral is u(y)≡exp(-y2)∫0yexp(z2)dz. We show that by solving the differential equation du/dy+2yu=1du/dy+2yu=1 using the orthogonal rational Chebyshev functions of the second kind, SB2n(y;L)SB2n(y;L), which generates a pentadiagonal Petrov–Galerkin matrix, one can obtain an accuracy of roughly (3/8)N(3/8)N digits where NN 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 NN-term approximation can be found in only O(N)O(N) operations, the new algorithm is the best arbitrary-precision strategy for computing Dawson’s Integral.