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

摘要

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