Abstract

A set of continually factorized uniformly convergent series is introduced for optical refractivities. Upper and lower bounds on the refractivities are obtained by truncating two such adjacent series, which alternately converge from above and below, after finite numbers of terms. The extent to which the present method complements the Padé method for extrapolating the refractive index from infrared and visible to ultraviolet frequencies is demonstrated by calculations on heavy inert gases. With exactly the same original data, the present method sometimes helps to reduce the Padé bounds to one third. The present method also complements Wolfsohn’s method in obtaining bounded estimates of oscillator strengths from refractivity measurements. This is illustrated in the case of argon, for which the oscillator strengths of two close lines are separately determined. In the case of atomic hydrogen, the results of the present method for oscillator strengths and the refractivity, below as well as above the first resonance line, are compared with exact values. Finally, as an illustrative example of estimating the refractivity at frequencies above the first resonance line, we extend a Sellmeier dispersion equation for argon in the normal-dispersion region to frequencies between the first and third excitation levels.

© 1974 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. See, for example, E. R. Peck and D. J. Fisher, J. Opt. Soc. Am. 54, 1362 (1964); E. R. Peck and B. N. Khanna, J. Opt. Soc. Am. 52, 416 (1962); Z. G. Wilkinson, J. Opt. Soc. Am. 50, 1002 (1960).
    [Crossref]
  2. G. Wolfsohn, Z. Physik 85, 366 (1933).
    [Crossref]
  3. G. I. Chashchina, V. I. Gladushchak, and E. Ya. Shreider, Opt. Spectrosk. 24, 1008 (1968) [Opt. Spectrosc. 24, 542 (1968)].
  4. H. Padé, Ann. Sci. Ecole Normale Supérieure (Paris) Suppl. 91892.
  5. P. W. Langhoff and M. Karplus, J. Opt. Soc. Am. 59, 863 (1969).
    [Crossref]
  6. K. T. Tang, Phys. Rev. Lett. 23, 1271 (1969); J. Chem. Phys. 55, 1064 (1971).
    [Crossref]
  7. H. A. Lorentz, Theory of Electrons, 2nd ed. (Dover, New York, 1952), pp. 137 and 305.
  8. H. A. Kramers and W. C. Heisenberg, Z. Physik 31, 681 (1925). See also, J. C. Slater, Quantum Theory of Atomic Structure, Vol. I (McGraw–Hill, New York, 1960), p. 154.
    [Crossref]
  9. K. T. Tang, Phys. Rev. 177, 108 (1969); Phys. Rev. A 1, 1033 (1970); P. T. Gee and K. T. Tang, Phys. Rev. A 7, 1863 (1973).
    [Crossref]
  10. R. P. Saxon, J. Chem. Phys. 59, 1539 (1973).
    [Crossref]
  11. C. E. Moore, Atomic Energy Levels, Vols. I–III, Natl. Bur. Std. (U.S.) Circ. No. 467, 1949, 1952, 1958.
  12. A. E. Kingston, J. Opt. Soc. Am. 54, 1145 (1964).
    [Crossref]
  13. G. Starkschall and R. Gordon, J. Chem. Phys. 54, 663 (1971).
    [Crossref]
  14. G. I. Chashchina and E. Ya. Shreider, Opt. Spectrosk. 27, 161 (1969) [Opt. Spectrosc. 27, 79 (1969)].
  15. R. Abjean, A. Mehu, and A. Johannin-Gilles, Opt. Commun. 3, 45 (1971).
    [Crossref]
  16. J. A. Barker and P. J. Leonard, Phys. Lett. 13, 127 (1964).
  17. W. Kronjager, Z. Physik 98, 17 (1935).
    [Crossref]
  18. J. Kock, Kgl. Fysiograf. Sallskap. Lund. Forh. 19, 173 (1949).
  19. See, for example, P. W. Langhoff and M. Karplus, in The Padé Approximant in Theoretical Physics, edited by G. A. Baker and J. L. Gammel (Academic, New York, 1970).
  20. T. Larsen, Dissertation (Lund University, Sweden, 1939).
  21. P. W. Langhoff, Chem. Phys. Lett. 12, 217 (1971).
    [Crossref]

1973 (1)

R. P. Saxon, J. Chem. Phys. 59, 1539 (1973).
[Crossref]

1971 (3)

G. Starkschall and R. Gordon, J. Chem. Phys. 54, 663 (1971).
[Crossref]

R. Abjean, A. Mehu, and A. Johannin-Gilles, Opt. Commun. 3, 45 (1971).
[Crossref]

P. W. Langhoff, Chem. Phys. Lett. 12, 217 (1971).
[Crossref]

1969 (4)

P. W. Langhoff and M. Karplus, J. Opt. Soc. Am. 59, 863 (1969).
[Crossref]

G. I. Chashchina and E. Ya. Shreider, Opt. Spectrosk. 27, 161 (1969) [Opt. Spectrosc. 27, 79 (1969)].

K. T. Tang, Phys. Rev. 177, 108 (1969); Phys. Rev. A 1, 1033 (1970); P. T. Gee and K. T. Tang, Phys. Rev. A 7, 1863 (1973).
[Crossref]

K. T. Tang, Phys. Rev. Lett. 23, 1271 (1969); J. Chem. Phys. 55, 1064 (1971).
[Crossref]

1968 (1)

G. I. Chashchina, V. I. Gladushchak, and E. Ya. Shreider, Opt. Spectrosk. 24, 1008 (1968) [Opt. Spectrosc. 24, 542 (1968)].

1964 (3)

1949 (1)

J. Kock, Kgl. Fysiograf. Sallskap. Lund. Forh. 19, 173 (1949).

1935 (1)

W. Kronjager, Z. Physik 98, 17 (1935).
[Crossref]

1933 (1)

G. Wolfsohn, Z. Physik 85, 366 (1933).
[Crossref]

1925 (1)

H. A. Kramers and W. C. Heisenberg, Z. Physik 31, 681 (1925). See also, J. C. Slater, Quantum Theory of Atomic Structure, Vol. I (McGraw–Hill, New York, 1960), p. 154.
[Crossref]

1892 (1)

H. Padé, Ann. Sci. Ecole Normale Supérieure (Paris) Suppl. 91892.

Abjean, R.

R. Abjean, A. Mehu, and A. Johannin-Gilles, Opt. Commun. 3, 45 (1971).
[Crossref]

Barker, J. A.

J. A. Barker and P. J. Leonard, Phys. Lett. 13, 127 (1964).

Chashchina, G. I.

G. I. Chashchina and E. Ya. Shreider, Opt. Spectrosk. 27, 161 (1969) [Opt. Spectrosc. 27, 79 (1969)].

G. I. Chashchina, V. I. Gladushchak, and E. Ya. Shreider, Opt. Spectrosk. 24, 1008 (1968) [Opt. Spectrosc. 24, 542 (1968)].

Fisher, D. J.

Gladushchak, V. I.

G. I. Chashchina, V. I. Gladushchak, and E. Ya. Shreider, Opt. Spectrosk. 24, 1008 (1968) [Opt. Spectrosc. 24, 542 (1968)].

Gordon, R.

G. Starkschall and R. Gordon, J. Chem. Phys. 54, 663 (1971).
[Crossref]

Heisenberg, W. C.

H. A. Kramers and W. C. Heisenberg, Z. Physik 31, 681 (1925). See also, J. C. Slater, Quantum Theory of Atomic Structure, Vol. I (McGraw–Hill, New York, 1960), p. 154.
[Crossref]

Johannin-Gilles, A.

R. Abjean, A. Mehu, and A. Johannin-Gilles, Opt. Commun. 3, 45 (1971).
[Crossref]

Karplus, M.

P. W. Langhoff and M. Karplus, J. Opt. Soc. Am. 59, 863 (1969).
[Crossref]

See, for example, P. W. Langhoff and M. Karplus, in The Padé Approximant in Theoretical Physics, edited by G. A. Baker and J. L. Gammel (Academic, New York, 1970).

Kingston, A. E.

Kock, J.

J. Kock, Kgl. Fysiograf. Sallskap. Lund. Forh. 19, 173 (1949).

Kramers, H. A.

H. A. Kramers and W. C. Heisenberg, Z. Physik 31, 681 (1925). See also, J. C. Slater, Quantum Theory of Atomic Structure, Vol. I (McGraw–Hill, New York, 1960), p. 154.
[Crossref]

Kronjager, W.

W. Kronjager, Z. Physik 98, 17 (1935).
[Crossref]

Langhoff, P. W.

P. W. Langhoff, Chem. Phys. Lett. 12, 217 (1971).
[Crossref]

P. W. Langhoff and M. Karplus, J. Opt. Soc. Am. 59, 863 (1969).
[Crossref]

See, for example, P. W. Langhoff and M. Karplus, in The Padé Approximant in Theoretical Physics, edited by G. A. Baker and J. L. Gammel (Academic, New York, 1970).

Larsen, T.

T. Larsen, Dissertation (Lund University, Sweden, 1939).

Leonard, P. J.

J. A. Barker and P. J. Leonard, Phys. Lett. 13, 127 (1964).

Lorentz, H. A.

H. A. Lorentz, Theory of Electrons, 2nd ed. (Dover, New York, 1952), pp. 137 and 305.

Mehu, A.

R. Abjean, A. Mehu, and A. Johannin-Gilles, Opt. Commun. 3, 45 (1971).
[Crossref]

Moore, C. E.

C. E. Moore, Atomic Energy Levels, Vols. I–III, Natl. Bur. Std. (U.S.) Circ. No. 467, 1949, 1952, 1958.

Padé, H.

H. Padé, Ann. Sci. Ecole Normale Supérieure (Paris) Suppl. 91892.

Peck, E. R.

Saxon, R. P.

R. P. Saxon, J. Chem. Phys. 59, 1539 (1973).
[Crossref]

Shreider, E. Ya.

G. I. Chashchina and E. Ya. Shreider, Opt. Spectrosk. 27, 161 (1969) [Opt. Spectrosc. 27, 79 (1969)].

G. I. Chashchina, V. I. Gladushchak, and E. Ya. Shreider, Opt. Spectrosk. 24, 1008 (1968) [Opt. Spectrosc. 24, 542 (1968)].

Starkschall, G.

G. Starkschall and R. Gordon, J. Chem. Phys. 54, 663 (1971).
[Crossref]

Tang, K. T.

K. T. Tang, Phys. Rev. 177, 108 (1969); Phys. Rev. A 1, 1033 (1970); P. T. Gee and K. T. Tang, Phys. Rev. A 7, 1863 (1973).
[Crossref]

K. T. Tang, Phys. Rev. Lett. 23, 1271 (1969); J. Chem. Phys. 55, 1064 (1971).
[Crossref]

Wolfsohn, G.

G. Wolfsohn, Z. Physik 85, 366 (1933).
[Crossref]

Ann. Sci. Ecole Normale Supérieure (Paris) Suppl. (1)

H. Padé, Ann. Sci. Ecole Normale Supérieure (Paris) Suppl. 91892.

Chem. Phys. Lett. (1)

P. W. Langhoff, Chem. Phys. Lett. 12, 217 (1971).
[Crossref]

J. Chem. Phys. (2)

R. P. Saxon, J. Chem. Phys. 59, 1539 (1973).
[Crossref]

G. Starkschall and R. Gordon, J. Chem. Phys. 54, 663 (1971).
[Crossref]

J. Opt. Soc. Am. (3)

Kgl. Fysiograf. Sallskap. Lund. Forh. (1)

J. Kock, Kgl. Fysiograf. Sallskap. Lund. Forh. 19, 173 (1949).

Opt. Commun. (1)

R. Abjean, A. Mehu, and A. Johannin-Gilles, Opt. Commun. 3, 45 (1971).
[Crossref]

Opt. Spectrosk. (2)

G. I. Chashchina and E. Ya. Shreider, Opt. Spectrosk. 27, 161 (1969) [Opt. Spectrosc. 27, 79 (1969)].

G. I. Chashchina, V. I. Gladushchak, and E. Ya. Shreider, Opt. Spectrosk. 24, 1008 (1968) [Opt. Spectrosc. 24, 542 (1968)].

Phys. Lett. (1)

J. A. Barker and P. J. Leonard, Phys. Lett. 13, 127 (1964).

Phys. Rev. (1)

K. T. Tang, Phys. Rev. 177, 108 (1969); Phys. Rev. A 1, 1033 (1970); P. T. Gee and K. T. Tang, Phys. Rev. A 7, 1863 (1973).
[Crossref]

Phys. Rev. Lett. (1)

K. T. Tang, Phys. Rev. Lett. 23, 1271 (1969); J. Chem. Phys. 55, 1064 (1971).
[Crossref]

Z. Physik (3)

H. A. Kramers and W. C. Heisenberg, Z. Physik 31, 681 (1925). See also, J. C. Slater, Quantum Theory of Atomic Structure, Vol. I (McGraw–Hill, New York, 1960), p. 154.
[Crossref]

G. Wolfsohn, Z. Physik 85, 366 (1933).
[Crossref]

W. Kronjager, Z. Physik 98, 17 (1935).
[Crossref]

Other (4)

C. E. Moore, Atomic Energy Levels, Vols. I–III, Natl. Bur. Std. (U.S.) Circ. No. 467, 1949, 1952, 1958.

H. A. Lorentz, Theory of Electrons, 2nd ed. (Dover, New York, 1952), pp. 137 and 305.

See, for example, P. W. Langhoff and M. Karplus, in The Padé Approximant in Theoretical Physics, edited by G. A. Baker and J. L. Gammel (Academic, New York, 1970).

T. Larsen, Dissertation (Lund University, Sweden, 1939).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Refractivity of atomic hydrogen. —— Exact value; — · — [α1]1,1; ⋯ · [α]3,1; — · · — [α]10,1.

Fig. 2
Fig. 2

Refractivity of atomic hydrogen. —— Exact value; ⋯ · [α]2,2; — · — [α]4,2; — · · — [α]8,2.

Fig. 3
Fig. 3

Refractivity of argon. For ω < 1 value taken from Chashchina, Gladuschchak and Shreider (Ref. 3); for ω > 1, value shown here is the result to which Eq. (54) converges.

Fig. 4
Fig. 4

Bounds on the refractivity of argon from the continued-factor series in the normal dispersion region. — The result to which [α]n,1 and [α]n,2 converge; – – – – [α]1,1; ⋯ · [α]3,1; — · — [α]2,2; — · · — [α]4,2; — ⋯ — [α]7,2.

Fig. 5
Fig. 5

Bounds on the refractivity of argon from the continued-factor series in the region between 1 and 2. — The result to which [α]n,2 and [α]n,3 converge; – – – – [α]7,2; — — — [α]8,2; — · · — [α]9,2; — · — [α]9,3; ⋯ · [α]12,3.

Fig. 6
Fig. 6

Bounds on the refractivity of argon from the continued-factor series in the region between 2 and 3. — The result to which [α]n,3 and [α]n,4 converge; — · — [α]9,3; — · · — [α]12,3; – – – – [α]7,4; ⋯ · [α]9,4; — ⋯ — [α]13,4.

Tables (11)

Tables Icon

Table I Cauchy coefficientsa and first resonance frequency.b

Tables Icon

Table II Refractivities [(n − 1) × 104] of Ne.a

Tables Icon

Table III Refractivities [(n − 1) × 104] of Ar.a

Tables Icon

Table IV Refractivities [(n − 1) × 104] of Kr.a

Tables Icon

Table V Refractivities [(n − 1) × 104] of Xe.a

Tables Icon

Table VI Refractivities [(n − 1) × 104] of Xe.a

Tables Icon

Table VII Oscillator strength f1 of hydrogen atom.a

Tables Icon

Table VIII Oscillator strength f2 of hydrogen atom.a

Tables Icon

Table IX Cauchy coefficients for Ar.a

Tables Icon

Table X Oscillator strength f1 of Ar.a

Tables Icon

Table XI Oscillator strength f2 of Ar.a

Equations (62)

Equations on this page are rendered with MathJax. Learn more.

α ( ω ) = i = 1 f i i 2 - ω 2 ,
α ( ω ) = j = 0 μ j ω 2 j ,
μ j = S ( - 2 j - 2 ) = i = 1 f i i 2 j + 2 .
j = 0 n μ j ω 2 j < α ( ω )
[ α ( ω ) ] 0 , 1 = 1 2 μ 0 1 2 - ω 2
α ( ω ) - [ α ( ω ) ] 0 , 1 = - ω 2 1 2 - ω 2 i = 1 ( i 2 - 1 2 ) f i i 2 ( i 2 - ω 2 ) ,
α ( ω ) = 1 μ 0 1 2 - ω 2 - ω 2 1 2 - ω 2 α g ( 1 ) ( ω ) ,
α g ( 1 ) ( ω ) = i = 2 g i ( 1 ) i 2 - ω 2 ,
g i ( 1 ) = ( i 2 - 1 2 ) f i i 2 .
g i ( 1 ) 0 ;
α g ( 1 ) ( ω ) = i = 0 μ i ( 1 ) ω 2 i
μ j ( 1 ) = i = 2 g i ( 1 ) i 2 j + 2
[ α ( ω ) ] n , 1 = [ α ( ω ) ] 0 , 1 ω 2 1 2 - ω 2 ( j = 0 n - 1 μ j ( 1 ) ω 2 j ) ,
[ α ( ω ) ] n , 1 > α ( ω )             for             ω < 1 ;
< α ( ω )             for             1 < ω < 2 .
μ j ( 1 ) = S ( - 2 j - 2 ) - 1 2 S ( - 2 j - 4 ) .
[ α ( ω ) ] n , 1 = k = 0 n 0 C 2 k 1 ω 2 k / ( 1 2 - ω 2 ) ,
C 2 k I = { S ( - 2 ) 1 2 , k = 0 S ( - 2 k - 2 ) 1 2 - S ( - 2 k ) , k 1.
[ α ( ω ) ] n + 1 , 1 - [ α ( ω ) ] n , 1 = - μ n ( 1 ) ω 2 n + 2 1 2 - ω 2 .
[ α ( ω ) ] n + 1 , 1 < [ α ( ω ) ] n , 1             for             ω < 1
> [ α ( ω ) ] n , 1             for             1 < ω < 2 .
α ( ω ) = 1 2 μ 0 1 2 - ω 2 - ω 2 1 2 - ω 2 [ 2 2 μ 0 ( 1 ) 2 2 - ω 2 - ω 2 2 2 - ω 2 ] α g ( 2 ) ( ω ) ,
μ 0 ( 1 ) = i = 2 g i ( 1 ) i 2
α g ( 2 ) ( ω ) = i = 3 g i ( 2 ) i 2 - ω 2 ,
g i ( 2 ) = ( i 2 - 2 2 ) ( i 2 - 1 2 ) f i i 4 .
[ α ( ω ) ] 1 , 2 = 1 2 μ 0 1 2 - ω 2 - 2 2 μ 0 ( 1 ) ω 2 ( 1 2 - ω 2 ) ( 2 2 - ω 2 ) .
α g ( 2 ) ( ω ) = j = 0 μ j ( 2 ) ω 2 j ,
μ j ( 2 ) = i = 3 g i ( 2 ) i 2 j + 2 ,
[ α ( ω ) ] n , 2 = [ α ( ω ) ] 1 , 2 + ω 4 ( 1 2 - ω 2 ) ( 2 2 - ω 2 ) j = 0 n - 2 μ j ( 2 ) ω 2 j ,
[ α ( ω ) ] n , 2 < α ( ω )             for             ω < 1             and             2 < ω < 3
> α ( ω )             for             1 < ω < 2 .
[ α ( ω ) ] n , 2 = k = 0 n 1 C 2 k 2 ω 2 k / [ ( 1 2 - ω 2 ) ( 2 2 - ω 2 ) ]
C 2 k 2 = { S ( - 2 ) 1 2 2 2 , k = 0 S ( - 4 ) 1 2 2 2 - ( 1 2 + 2 2 ) S ( - 2 ) , k = 1 S ( - 2 k - 2 ) 1 2 2 2 - ( 1 2 + 2 2 ) S ( - 2 k ) + S ( - 2 k + 2 ) , k 2.
[ α ( ω ) ] n , m = k = 0 n m - 1 C 2 k m ω 2 k / ( i = 1 m ( i 2 - ω 2 ) ) ,
C 2 k m = ( - 1 ) m n = 0 m A n , k m S ( - 2 k + 2 n - 2 ) ,
A n , k m = { ( - 1 ) m - n ( m - n ) j 1 = 1 m j 2 = 1 m j m - n = 1 m ( l = 1 m - n j l 2 ) , n j , n < m 1 n j , n = m 0 n > j ,
[ α ( ω ) ] n , m > α ( ω )             for             k - 1 < ω < k ,             with             k = m , m - 2 ,
[ α ( ω ) ] n , m < α ( ω )             for             k < ω < k + 1 ,             with             k = m + 1 , m - 1 , .
C 2 k 3 = { S ( - 2 ) 1 2 2 2 3 2 , k = 0 S ( - 4 ) 1 2 2 2 3 2 - ( 1 2 2 2 + 2 2 3 2 + 3 2 1 2 ) S ( - 2 ) , k = 1 S ( - 6 ) 1 2 2 2 3 2 - ( 1 2 2 2 + 2 2 3 2 + 3 2 1 2 ) S ( - 4 ) + ( 1 2 + 2 2 + 3 2 ) × S ( - 2 ) , k = 2 S ( - 2 k - 2 ) 1 2 2 2 3 2 - ( 1 2 2 2 + 2 2 3 2 + 3 2 1 2 ) S ( - 2 k ) + ( 1 2 + 2 2 + 3 2 ) × S ( - 2 k + 2 ) - S ( - 2 k + 4 ) , k 3.
[ α ( ω ) ] n + 1 , m < [ α ( ω ) ] n , m ,             if             [ α ( ω ) ] n , m > α ( ω )
[ α ( ω ) ] n + 1 , m > [ α ( ω ) ] n , m ,             if             [ α ( ω ) ] n , m < α ( ω ) .
[ α ( ω ) ] n , m - [ α ( ω ) ] n - 1 , m = C 2 n m ω 2 n / i = 1 m ( i 2 - ω 2 ) .
C 2 n m < 0 for m odd C 2 n m > 0 for m even
S ( - 2 k - 2 ) 1 2 - S ( - 2 k ) < 0
S ( - 2 k - 2 ) 1 2 2 2 - ( 1 2 + 2 2 ) S ( - 2 k ) + S ( - 2 k + 2 ) > 0 ,
S ( - 2 k - 2 ) 1 2 2 2 3 2 - ( 1 2 2 2 + 2 2 3 2 + 3 2 1 2 ) S ( - 2 k ) + ( 1 2 + 2 2 + 3 2 ) S ( - 2 k + 2 ) - S ( - 2 k + 4 ) < 0
lim ω j ( j 2 - ω 2 ) α ( ω ) = f j .
[ f j ] n , m = lim ω j ( j 2 - ω 2 ) [ α ( ω ) ] n , m
[ f j ] n , k > [ f j ] n + 1 , k f j ,             k = 1 , 3 , 5 ,
[ f j ] n , k < [ f j ] n + 1 , k f j ,             k = 2 , 4 , 6 , .
[ f j ] n , m = k = 0 n m - 1 C 2 k m j 2 k / ( i j m ( i 2 - j 2 ) ) .
[ f 1 ] n , 1 = 1 2 n + 2 S ( - 2 n - 2 ) > f 1 ,
[ f 1 ] n , 2 = 1 2 n + 2 [ S ( - 2 n - 2 ) 2 2 - S ( - 2 n ) ] ( 2 2 - 1 2 ) < f 1 ,
[ f 1 ] n , 3 = 1 2 n + 2 [ S ( - 2 n - 2 ) 2 2 3 2 - ( 2 2 + 3 2 ) S ( - 2 n ) + S ( - 2 n + 2 ) ] ( 2 2 - 1 2 ) ( 3 2 - 1 2 ) > f 1 ,
[ f 2 ] n , 2 = 2 2 n + 1 [ S ( - 2 n - 2 ) 1 2 - S ( - 2 n ) ] ( 2 2 - 1 2 ) > f 2 ,
[ f 2 ] n , 3 = 2 2 n + 2 [ S ( - 2 n - 2 ) 1 2 3 2 - ( 1 2 + 3 2 ) S ( - 2 n ) + S ( - 2 n + 2 ) ] ( 1 2 - 2 2 ) ( 3 2 - 2 2 ) < f 3 .
n ( ω ) - 1 = 2 π N 0 α ( ω )
[ 2 , 1 ] α = A 0 + A 1 ω 2 1 + B 1 ω 2 + B 2 ω 4 ,
A 0 = S ( - 2 ) , A 1 = S ( - 4 ) + [ S ( - 2 ) S ( - 8 ) - S ( - 4 ) S ( - 6 ) ] [ ( S ( - 4 ) ) 2 / S ( - 2 ) - S ( - 6 ) ] , B 1 = [ S ( - 2 ) S ( - 8 ) - S ( - 4 ) S ( - 6 ) ] [ ( S ( - 4 ) ) 2 - S ( - 2 ) S ( - 6 ) ] , B 2 = [ ( S - 4 ) S ( - 8 ) - ( S ( - 6 ) ) 2 ] [ S ( - 2 ) S ( - 6 ) - ( S ( - 4 ) ) 2 ] .
α ( ω ) = α ( ω ) - f 1 1 2 - ω 2 - f 2 2 2 - ω 2 ;
S ( n ) = S ( n ) - f 1 1 n - f 2 2 n .
[ α ( ω ) ] n , m a = [ α ( ω ) ] n , m + f 1 1 2 - ω 2 + f 2 2 2 - ω 2 .