TY - JOUR

T1 - An algorithm for the evaluation of the complex Airy functions

AU - Schulten, Z.

AU - Anderson, D. G.M.

AU - Gordon, Roy G.

N1 - Funding Information:
Recently interest in Airy functions of a complex argument has arisen in the quantum mechanical study of physical problems with complex transition points which occur in the calculation of wave functions when the energy curves have an avoided crossing * Previous address: Committee Applied Mathematics, Harvard University, Cambridge, Mass. supported in part by the National Science Foundation under grants NSF GP-3472 b and NSF MPS.

PY - 1979/4

Y1 - 1979/4

N2 - The evaluation of complex Airy functions is required in the approximation of certain second-order linear differential equations arising in the treatment of multiple turning-point and energy curve-crossing problems in quantum mechanics. Pairs of numerically linearly independent solutions throughout the z-plane can be constructed from the fundamental solutions to the complex Airy equation, Ai(z), Bi(z), and Ai(z e±2πi/3). Integral representations for these complex functions and their derivatives are given, and being of the Stieltjes type, the integrals are evaluated using the generalized Gaussian quadrature method of Shohat and Tamarkin as implemented by Gordon. These integral representations, employed together with the Taylor series for small z and the appropriate connection formulas, allow the creation of an accurate and efficient algorithm to evaluate the complex functions over the entire z-plane. The algorithm is presented in detail at the end of this article.

AB - The evaluation of complex Airy functions is required in the approximation of certain second-order linear differential equations arising in the treatment of multiple turning-point and energy curve-crossing problems in quantum mechanics. Pairs of numerically linearly independent solutions throughout the z-plane can be constructed from the fundamental solutions to the complex Airy equation, Ai(z), Bi(z), and Ai(z e±2πi/3). Integral representations for these complex functions and their derivatives are given, and being of the Stieltjes type, the integrals are evaluated using the generalized Gaussian quadrature method of Shohat and Tamarkin as implemented by Gordon. These integral representations, employed together with the Taylor series for small z and the appropriate connection formulas, allow the creation of an accurate and efficient algorithm to evaluate the complex functions over the entire z-plane. The algorithm is presented in detail at the end of this article.

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U2 - 10.1016/0021-9991(79)90062-7

DO - 10.1016/0021-9991(79)90062-7

M3 - Article

AN - SCOPUS:0007561355

VL - 31

SP - 60

EP - 75

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 1

ER -