An algorithm for the evaluation of the complex Airy functions

Z. Schulten, D. G.M. Anderson, Roy G. Gordon

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)60-75
Number of pages16
JournalJournal of Computational Physics
Issue number1
StatePublished - Apr 1979
Externally publishedYes

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • Physics and Astronomy(all)
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics


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