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Zbl 1157.65331
Boyd, John P.
Evaluating of Dawson's integral by solving its differential equation using orthogonal rational Chebyshev functions. (English)
[J] Appl. Math. Comput. 204, No. 2, 914-919 (2008). ISSN 0096-3003

Summary: Dawson's integral is $u(y)\equiv \exp (-y^2) \int _0^y \exp (z^2)\text d z$. We show that by solving the differential equation d$u/$d$y+2y$u=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 2000:: *65D20 Computation of special functions
33B20 Incomplete beta and gamma functions
33F05 Numerical approximation of special functions

Keywords: Dawson's integral; complex error function; rational Chebyshev functions; spectral method; pentadiagonal Petrov-Galerkin matrix; algorithm

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