Abstract

A new method of generating the Bessel functions and ratios of Bessel functions necessary for Mie calculations is presented. Accuracy is improved while eliminating the need for extended precision word lengths or large storage capability. The algorithm uses a new technique of evaluating continued fractions that starts at the beginning rather than the tail and has a built-in error check. The continued fraction representations for both spherical Bessel functions and ratios of Bessel functions of consecutive order are presented.

© 1976 Optical Society of America

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References

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  1. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (American Elsevier, New York, 1969), Chap. 2.
  2. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  3. J. Todd, in Handbook of Physics, E. U. Condon, H. Odishaw, Ed. (McGraw-Hill, New York, 1967), p. 1–93.
  4. J. J. Stephens, J. R. Gerhardt, J. Meterol. 18, 818 (1961).
    [CrossRef]
  5. F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill, New York, 1974), pp. 30–31.
  6. British Association for the Advancement of Science, Mathematical Tables. Vol. 10: Bessel Functions; Part 11; Functions of Positive Integer OrderCambridge (U.P., London, 1952).
  7. W. D. Ross, Appl. Opt. 11, 1919 (1972).
    [CrossRef] [PubMed]
  8. W. J. Lentz, “A New Method of Computing Spherical Bessel Functions of Complex Argument with Tables,” ECOM 5509 (1973), AD 767223.
  9. W. B. Jones, in Páde Approximants and Their Applications, P. R. Graves-Morris, Ed. (Academic, New York, 1973).
  10. G. Blanch, SIAM Rev. 6, No. 4 (1964).
    [CrossRef]
  11. W. Gautschi, SIAM Rev. 9, No. 1 (1967).
    [CrossRef]
  12. M. Abramowitz, F. W. J. Olver, H. A. Antosiewicz, in Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, Eds., Applied Mathematics Series 55 (U.S. Govt. Printing Office, Washington, D.C., 1965), Chaps. 3, 9, 10.

1973

W. J. Lentz, “A New Method of Computing Spherical Bessel Functions of Complex Argument with Tables,” ECOM 5509 (1973), AD 767223.

1972

1967

W. Gautschi, SIAM Rev. 9, No. 1 (1967).
[CrossRef]

1964

G. Blanch, SIAM Rev. 6, No. 4 (1964).
[CrossRef]

1961

J. J. Stephens, J. R. Gerhardt, J. Meterol. 18, 818 (1961).
[CrossRef]

Abramowitz, M.

M. Abramowitz, F. W. J. Olver, H. A. Antosiewicz, in Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, Eds., Applied Mathematics Series 55 (U.S. Govt. Printing Office, Washington, D.C., 1965), Chaps. 3, 9, 10.

Antosiewicz, H. A.

M. Abramowitz, F. W. J. Olver, H. A. Antosiewicz, in Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, Eds., Applied Mathematics Series 55 (U.S. Govt. Printing Office, Washington, D.C., 1965), Chaps. 3, 9, 10.

Blanch, G.

G. Blanch, SIAM Rev. 6, No. 4 (1964).
[CrossRef]

Deirmendjian, D.

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (American Elsevier, New York, 1969), Chap. 2.

Gautschi, W.

W. Gautschi, SIAM Rev. 9, No. 1 (1967).
[CrossRef]

Gerhardt, J. R.

J. J. Stephens, J. R. Gerhardt, J. Meterol. 18, 818 (1961).
[CrossRef]

Hildebrand, F. B.

F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill, New York, 1974), pp. 30–31.

Jones, W. B.

W. B. Jones, in Páde Approximants and Their Applications, P. R. Graves-Morris, Ed. (Academic, New York, 1973).

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

Lentz, W. J.

W. J. Lentz, “A New Method of Computing Spherical Bessel Functions of Complex Argument with Tables,” ECOM 5509 (1973), AD 767223.

Olver, F. W. J.

M. Abramowitz, F. W. J. Olver, H. A. Antosiewicz, in Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, Eds., Applied Mathematics Series 55 (U.S. Govt. Printing Office, Washington, D.C., 1965), Chaps. 3, 9, 10.

Ross, W. D.

Stephens, J. J.

J. J. Stephens, J. R. Gerhardt, J. Meterol. 18, 818 (1961).
[CrossRef]

Todd, J.

J. Todd, in Handbook of Physics, E. U. Condon, H. Odishaw, Ed. (McGraw-Hill, New York, 1967), p. 1–93.

Appl. Opt.

ECOM 5509

W. J. Lentz, “A New Method of Computing Spherical Bessel Functions of Complex Argument with Tables,” ECOM 5509 (1973), AD 767223.

J. Meterol.

J. J. Stephens, J. R. Gerhardt, J. Meterol. 18, 818 (1961).
[CrossRef]

SIAM Rev

G. Blanch, SIAM Rev. 6, No. 4 (1964).
[CrossRef]

W. Gautschi, SIAM Rev. 9, No. 1 (1967).
[CrossRef]

Other

M. Abramowitz, F. W. J. Olver, H. A. Antosiewicz, in Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, Eds., Applied Mathematics Series 55 (U.S. Govt. Printing Office, Washington, D.C., 1965), Chaps. 3, 9, 10.

W. B. Jones, in Páde Approximants and Their Applications, P. R. Graves-Morris, Ed. (Academic, New York, 1973).

F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill, New York, 1974), pp. 30–31.

British Association for the Advancement of Science, Mathematical Tables. Vol. 10: Bessel Functions; Part 11; Functions of Positive Integer OrderCambridge (U.P., London, 1952).

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (American Elsevier, New York, 1969), Chap. 2.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

J. Todd, in Handbook of Physics, E. U. Condon, H. Odishaw, Ed. (McGraw-Hill, New York, 1967), p. 1–93.

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Figures (1)

Fig. 1
Fig. 1

Comparison of forward recursions in single and double precision1 with a new continued fraction technique for generating An(z) as a function of increasing order n.

Equations (19)

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Ψ n ( z ) = ( π z / 2 ) 1 / 2 J n + 1 / 2 ( z ) ,
ζ n ( z ) = ( π z / 2 ) 1 / 2 [ J n + 1 / 2 ( z ) + ( - 1 ) n i J - n - 1 / 2 ( z ) ] ,
A n ( z ) = Ψ n ( z ) Ψ n ( z ) = - n z + J n - 1 / 2 ( z ) J n + 1 / 2 ( z ) ,
j n - 2 ( z ) j n - 1 ( z ) = 2 n - 1 z - j n ( z ) j n - 1 ( z )
f = a 1 + 1 a 2 + 1 a 3 + 1 a 4 + 1 a 5 + = a 1 + 1 a 2 + 1 a 3 + 1 a 4 + 1 a 5 + .
J ν - 1 ( z ) J ν ( z ) = 2 ν z - 1 + 1 - 2 ( ν + 1 ) z - 1 + 1 2 ( ν + 2 ) z - 1 + 1 - 2 ( ν + 3 ) z - 1 + .
f ( x ) = [ a 1 , a 2 , ] = a 1 + 1 a 2 + 1 a 3 + 1 a 4 +
f n ( x ) = [ a 1 , a 2 , , a n ] = a 1 + 1 a 2 + + 1 a n .
f n ( x ) = [ a 1 ] .. [ a n - 1 , , a 1 ] [ a n , , a 1 ] [ a 2 ] .. [ a n - 1 , , a 2 ] [ a n , , a 2 ] .
a n = ( - 1 ) n + 1 2 ( ν + n - 1 ) x - 1             n = 1 , 2 , 3 , [ a 1 ] = 19             [ a 2 ] = - 21             [ a 2 , a 1 ] = a 2 + 1 a 1 = - 20.94736842 J ν - 1 J ν = ( 19 ) ( - 20.94736842 ) ( 22.95226131 ) ( - 24.95643131 ) ( 26.95993017 ) ( - 21 )             ( 22.95238095 ) ( - 24.95643154 ) ( 26.95993017 ) = 18.95228198.
[ a m - 1 , , a 1 ] = α .
[ a m , , α 1 ] = β = a m + ( 1 / α ) 0 ,
[ a m + 1 , , a 1 ] = γ = a m + 1 + 1 β - , [ a m + 2 , , a 1 ] = Z = a m + 2 + 1 γ .
ξ = β γ = β a m + 1 β + 1 β = a m + 1 β + 1 ,
Z = a m + 2 γ + 1 γ = a m + 2 ( a m + 1 β + 1 ) + β a m + 1 β + 1 = a m + 2 ξ + β ξ .
j n ( z ) = ( π 2 z ) 1 / 2 ( z 2 ) ν / Γ ( ν + 1 ) 1 + z 2 / ( 4 · 1 ( ν + 1 ) ) 1 - z 2 / ( 4 · 1 ( ν + 1 ) ) + z 2 / ( 4 k ( ν + k ) ) 1 - z 2 / ( 4 k ( ν + k ) ) +
j n ( z ) = ( z ) n / ( 2 n + 1 ) ! ! 1 + z 2 / ( 4 · 1 ( ν + 1 ) ) 1 - z 2 / ( 4 · 1 ( ν + 1 ) ) + z 2 / ( 4 k ( ν + k ) ) 1 - z 2 / ( 4 k ( ν + k ) ) +
f n = b 0 + c 1 a 1 c 1 b 1 + c 1 c 2 a 2 c 2 b 2 + c 2 c 3 a 3 c 3 b 3 + c n - 1 c n a n c n b n ,
z n ( 2 n + 1 ) ! ! j n ( z ) = 1 + 1 A 1 z 2 - B 1 + 1 C 1 - z 2 D 1 + 1 A k z 2 - B k + 1 C k - z 2 D k ,

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