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Evaluating of Dawson’s integral by solving its differential equation using orthogonal rational Chebyshev functions. (English) Zbl 1157.65331
Summary: Dawson’s integral is u(y)exp(-y 2 ) 0 y exp(z 2 )dz. We show that by solving the differential equation du/dy+2yu=1 using the orthogonal rational Chebyshev functions of the second kind, SB 2n (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.
MSC:
65D20Computation of special functions, construction of tables
33B20Incomplete beta and gamma functions
33F05Numerical approximation and evaluation of special functions