The uniform asymptotic swallowtail approximation: practical methods for oscillating integrals with four coalescing saddle points

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Abstract
Practical methods for the numerical implementation of the uniform swallowtail approximation have been developed. This approximation arises in the uniform asymptotic evaluation of oscillating integrals with four coalescing saddle points. A complex contour quadrature technique has been used to evaluate the swallowtail canonical integral S(x,y,z) and its partial derivatives delta S/ delta x, delta S/ delta y, delta S/ delta z. This method has the advantage that it is straightforward to implement on a computer and results of high accuracy are readily obtained. A comparison is made with other methods that have been reported in the literature for the evaluation of S(x,y,z). Isometric plots of mod S(x,y,z) mod , mod delta S/ delta x mod , mod delta S/ delta y mod , mod delta S/ delta z mod are presented and some properties of the zeros of S(0,y,z) that lie on the line y=0 are also discussed. Two methods for the evaluation of the mapping parameters (x,y,z) are described: an iterative method that is valid when (x,y,z) is not close to the swallowtail caustic and an algebraic method valid for (x,y,z) on the caustic and for y=0. Symbolic algebraic computer programs have been used to carry out the necessary algebraic manipulations. In practice both methods for determining (x,y,z) are complementary. An application of the uniform swallowtail approximation to the butterfly canonical integral has been made. The uniform asymptotic swallowtail approximation can now be regarded as a practical tool for the evaluation of oscillating integrals with four coalescing saddle points.