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Accurate bessel functions Jn(z), Yn(z), Hn(1)(z) and Hn(2)(z) of integer order and complex argument
C-559 Computer Physics Communications 17 (1979) 321-336 © North-Holland Publishing Company
ACCURATE BESSEL FUNCTIONS AND COMPLEX ARGUMENT *
Jn(z), Yn(z), H(I)(z) AND//gn2)(z) OF INTEGER ORDER
R.W.B. A R D I L L a n d K.J.M. M O R I A R T Y
Department o f Mathematics, Royal Holloway College, Englefield Green, Egham, Surrey TW20 OEX, UK Received 6 June 1978; in revised form 15 January 1979
PROGRAM SUMMARY
Title o f program: BESJYH Catalogue number: ACYQ Computer: CDC 6600, CDC 7600; Installation: University of London Computer Centre
Operating system: CDC NOS/BE, SCOPE Programming language used: FORTRAN IV High speed storage required: 6 kwords (BESJYH: 3043, CJ: 498, CY:883, CH:l112)
Number o f bits in a word: 60 Overlay structure: none Number o f magnetic tapes required: none Other peripherals used: card reader, line printer Number o f cards in combined program and test deck: 1521 Keywords: general purpose, atomic, molecular, nuclear, expansion, representation, eikonal, impact parameter, helicity, partial wave, Bessel functions, Kelvin functions, Neumann
* Part of the Royal Holloway College Special Functions Mathematics Project.
functions, Weber's function, Hankel functions, complex functions
Nature o f physical problem The Bessel function appears in a wide range of physical applications, and in particular where there is axial symmetry. The package contains complex function routines to calculate Jn(z), Yn(z), H(l)(z) and H(n2)(z) for integer n and complex z.
Method o f solution The method of solution is based on the ascending series representations and asymptotic forms of the Bessel functions Jn(z) and Yn(z) and asymptotic forms of the modified Bessel functions ln(z) and Kn(z); all of these are given in ref. [ I ].
Restrictions on the complexity o f the problem The program will return results for all values of Izl up to machine overflows in the Bessel functions. The size of the order should not be too large (say, Inl < 15) or accuracy will be lost. For large Inl, the user should incorporate into the program the formulae given in ref. [ 1 ] for Debye's asymptotic expansions or better, the uniform asymptotic expansions. The value of the relative accuracy parameter, EPS, should not be set below about 10 - I 1. For the asymptotic region, the accuracy EPS may not always be achieved (since the asymptotic series may have to be truncated at their lower terms), in which case the output parameter ISET will indicate this and an estimation of the relative error is also produced. The functions Yn(z), H(nl)(z) and H(n2)(z) have a branch point at the origin, together with a cut along the negative real axis.
Typical running time The test run output at the end of the Long Write Up took about 4.2 s.
References [ 1 ] M. Abramowitz and I.A. Stegun, Handbook of mathematical functions with formulas, graphs and mathematical tables (Dover Publications, New York, 1968).
ACCURATE BESSEL FUNCTIONS AND COMPLEX ARGUMENT *
Jn(z), Yn(z), H(I)(z) AND//gn2)(z) OF INTEGER ORDER
R.W.B. A R D I L L a n d K.J.M. M O R I A R T Y
Department o f Mathematics, Royal Holloway College, Englefield Green, Egham, Surrey TW20 OEX, UK Received 6 June 1978; in revised form 15 January 1979
PROGRAM SUMMARY
Title o f program: BESJYH Catalogue number: ACYQ Computer: CDC 6600, CDC 7600; Installation: University of London Computer Centre
Operating system: CDC NOS/BE, SCOPE Programming language used: FORTRAN IV High speed storage required: 6 kwords (BESJYH: 3043, CJ: 498, CY:883, CH:l112)
Number o f bits in a word: 60 Overlay structure: none Number o f magnetic tapes required: none Other peripherals used: card reader, line printer Number o f cards in combined program and test deck: 1521 Keywords: general purpose, atomic, molecular, nuclear, expansion, representation, eikonal, impact parameter, helicity, partial wave, Bessel functions, Kelvin functions, Neumann
* Part of the Royal Holloway College Special Functions Mathematics Project.
functions, Weber's function, Hankel functions, complex functions
Nature o f physical problem The Bessel function appears in a wide range of physical applications, and in particular where there is axial symmetry. The package contains complex function routines to calculate Jn(z), Yn(z), H(l)(z) and H(n2)(z) for integer n and complex z.
Method o f solution The method of solution is based on the ascending series representations and asymptotic forms of the Bessel functions Jn(z) and Yn(z) and asymptotic forms of the modified Bessel functions ln(z) and Kn(z); all of these are given in ref. [ I ].
Restrictions on the complexity o f the problem The program will return results for all values of Izl up to machine overflows in the Bessel functions. The size of the order should not be too large (say, Inl < 15) or accuracy will be lost. For large Inl, the user should incorporate into the program the formulae given in ref. [ 1 ] for Debye's asymptotic expansions or better, the uniform asymptotic expansions. The value of the relative accuracy parameter, EPS, should not be set below about 10 - I 1. For the asymptotic region, the accuracy EPS may not always be achieved (since the asymptotic series may have to be truncated at their lower terms), in which case the output parameter ISET will indicate this and an estimation of the relative error is also produced. The functions Yn(z), H(nl)(z) and H(n2)(z) have a branch point at the origin, together with a cut along the negative real axis.
Typical running time The test run output at the end of the Long Write Up took about 4.2 s.
References [ 1 ] M. Abramowitz and I.A. Stegun, Handbook of mathematical functions with formulas, graphs and mathematical tables (Dover Publications, New York, 1968).