Evaluating of Dawson’s integral by solving its differential equation using orthogonal rational Chebyshev functions. (English) Zbl 1157.65331

Boyd, John P.

Appl. Math. Comput. 204, No. 2, 914-919 (2008).

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:

65D20	Computation of special functions and constants, construction of tables
33B20	Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals)
33F05	Numerical approximation and evaluation of special functions

Keywords:

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

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