Bessel functions Jn(z) and Yn(z) of integer order and complex argument

Toit, C.F. du

doi:10.1016/0010-4655(93)90153-4

Cited by 9 publications

(5 citation statements)

References 8 publications

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“…This is expressed by the condition (10). Furthermore, using the property (2) of J n (x, y), we can perform the sum over s in (49), with the result…”

Section: A Physical Origin Of the Recurrence Relationmentioning

confidence: 99%

“…It follows from the expression (49), that a quantum Volkov state (we consider the spinless case for simplicity) can be regarded as a superposition of an infinite number of plane waves with definite, discrete, four-momenta q µ + nk µ . The amplitude to find the particle with fourmomentum q µ + nk µ is given by J n (x, y), with x and y as in Eq.…”

Section: B Classical-quantum Correspondence Of Volkov Statesmentioning

confidence: 99%

“…Remarkably, numerical values of J n (x) can be computed by using only the recurrence relation and the normalization condition, and not a single initial value is needed [e.g., one might otherwise imagine J 0 (x) to be calculated by a series expansion]. Miller's algorithm has subsequently been refined and the error propagation analyzed by several authors [35,45,46,47], and also implemented [48,49,50].…”

Section: Miller-type Algorithm For Generalized Bessel Functions a Rec...mentioning

confidence: 99%

“…For the wave function (49) constructed from the solution to the recurrence relation (11) to be finite, we must demand J n (x, y) to be normalizable. This is expressed by the condition (10).…”

Section: A Physical Origin Of the Recurrence Relationmentioning

confidence: 99%

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Recursive algorithm for arrays of generalized Bessel functions: Numerical access to Dirac-Volkov solutions

Lötstedt

2009

Phys. Rev. E

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In the relativistic and the nonrelativistic theoretical treatment of moderate and high-power lasermatter interaction, the generalized Bessel function occurs naturally when a Schrödinger-Volkov and Dirac-Volkov solution is expanded into plane waves. For the evaluation of cross sections of quantum electrodynamic processes in a linearly polarized laser field, it is often necessary to evaluate large arrays of generalized Bessel functions, of arbitrary index but with fixed arguments. We show that the generalized Bessel function can be evaluated, in a numerically stable way, by utilizing a recurrence relation and a normalization condition only, without having to compute any initial value. We demonstrate the utility of the method by illustrating the quantum-classical correspondence of the Dirac-Volkov solutions via numerical calculations.

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“…This is expressed by the condition (10). Furthermore, using the property (2) of J n (x, y), we can perform the sum over s in (49), with the result…”

Section: A Physical Origin Of the Recurrence Relationmentioning

confidence: 99%

Section: B Classical-quantum Correspondence Of Volkov Statesmentioning

confidence: 99%

Section: Miller-type Algorithm For Generalized Bessel Functions a Rec...mentioning

confidence: 99%

Section: A Physical Origin Of the Recurrence Relationmentioning

confidence: 99%

See 2 more Smart Citations

Recursive algorithm for arrays of generalized Bessel functions: Numerical access to Dirac-Volkov solutions

Lötstedt

2009

Phys. Rev. E

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“…The author's program, BESCJY, which is based on the algorithm described in references [12,18,19 Chapter 71, was compared to Bessel-function routines of MA TLAB 3.5 by Mathworks, Inc., and MATHEMATICA versions 1 2 and 2 0 by Wolfram Research, Inc.…”

Section: Errors In Commercial Software For the Computation Of < ( Z )mentioning

confidence: 99%

Evaluation of some algorithms and programs for the computation of integer-order Bessel functions of the first and second kind with complex arguments

Toit

1993

IEEE Antennas Propag. Mag.

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The purpose of this article is to provide some insight on the efficiency and accuracy of a few available algorithms for the computation of integer-order Bessel functions First, the computation of integer-order Bessel hnctions of the first kind (J,,(z)), using the Fast Fourier Transform (FFT) algorithm as opposed to recurrence techniques, is investigated It is shown that recurrence techniques are superior to the FFT technique, both in accuracy and speed efficiency Second, an erroneous algorithm, suggested in the literature and used by commercially available software, specifically MA TLAR 3.5 and MATHFNA AKA 1.2, for computing integer-order Bessel functions of the second kind ( < , ( z ) ) , is revealed by comparing these routines with an algorithm developed by the author Catastrophic errors result from the use of the erroneous algorithm, for the computation of large orders with non-real arguments IntroductionPhysical problems, formulated in a cylindrical-or sphericalcoordinate system, have prompted the development of various methods for the computation of the Bessel hnctions J , , ( z ) and T,(z), for integer order, I?, and a complex or real argument, z. To be able to use them successfully, it is important to know the numerical limitations of these algorithms. In terms of flexibility and efficiency, the methods may be divided roughly into two groups. The first group describes methods where, due to numerical problems, severe restrictions are placed on the magnitude of the argument and the maximum order [ I , 2, 3; 4, Appendix 11. The second group describes methods which allow for the computation of a range of orders, using either recurrence techniques [5, p. 385; 6, 7, 8, 9, 10, 1 1 , 121 or alternative methods, of which the use of the FFT algorithm [13, p. 2901, in particular, is an interesting example [14, IS].In Section 3, the accuracy and efficiency of the FFT method is compared to recurrence techniques. The problems of determining the starting point of backward recurrence, and the minimum number of sample points required by the FFT method, are also discussed.Finally, the problem of computing Y,,(z) for complex arguments is addressed in Section 4, and the accuracy of a few commercial mathematical routines, namely MATLAB 3.5 and MA THE-MATICA versions 1.2 and 2.0, is investigated. Forward recurrence is usually recommended [5,6,7, 81, to compute all orders of E, from U, and E;. Recurrence is stable in the direction of increasing function values only, hence this procedure is valid for real arguments. For complex arguments, however, forward recurrence becomes unstable, since I~, ( z ) l decreases initially with increasing ii, before a turning point is reached (see Figure 4), and it increases only for larger orders beyond the turning point. Apparently, this erroneous algorithm is employed by MATZAB 3.5 and MATHE-MATICA I . 2 . Since the answers are presumably accepted in good faith by the scientific community, it is evidently quite a serious problem. Comparison between the FFT method and recurrence technique...

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Calculation of generalized Lorenz-Mie theory based on the localized beam models

Jia

Shen

2017

Journal of Quantitative Spectroscopy and Radiative Transfer

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Bessel functions Jn(z) and Yn(z) of integer order and complex argument

Cited by 9 publications

References 8 publications

Recursive algorithm for arrays of generalized Bessel functions: Numerical access to Dirac-Volkov solutions

Recursive algorithm for arrays of generalized Bessel functions: Numerical access to Dirac-Volkov solutions

Evaluation of some algorithms and programs for the computation of integer-order Bessel functions of the first and second kind with complex arguments

Calculation of generalized Lorenz-Mie theory based on the localized beam models

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