Abstract

The mean upward-scattering coefficient of the downward-traveling photons and the mean downward-scattering coefficient of the upward-traveling photons are two factors needed for the two-stream approximation to the radiative-transfer equation. Numerical values of each shape factor just beneath the surface and at asymptotic depths give an indication of the range of values at intermediate depths in spatially uniform waters with no sources and are used to obtain an approximate depth-dependent model for each shape factor. The shape factors are computed for different surface-illumination conditions, wavelengths, and chlorophyll concentrations.

© 1995 Optical Society of America

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References

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  1. E. Aas, “Two-stream irradiance model for deep waters,” Appl. Opt. 26, 2095–2101 (1987).
    [CrossRef] [PubMed]
  2. R. W. Preisendorfer, C. D. Mobley, “Direct and inverse irradiance models in hydrologic optics,” Limnol. Oceanogr. 29, 903–929 (1984).
    [CrossRef]
  3. R. H. Stavn, A. D. Weidemann, “Shape factors, two-flow models, and the problem of irradiance inversion in estimating optical parameters,” Limnol. Oceanogr. 34, 1426–1441 (1989).
    [CrossRef]
  4. Z. Tao, N. J. McCormick, R. Sanchez, “Ocean source and optical property estimation from explicit and implicit algorithms,” Appl. Opt. 33, 3265–3275 (1994).
    [CrossRef] [PubMed]
  5. J. R. V. Zaneveld, “An asymptotic closure theory for irradiance in the sea and its inversion to obtain the inherent optical properties,” Limnol. Oceanogr. 34, 1442–1452 (1989).
    [CrossRef]
  6. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), Secs. 46–48.
  7. N. J. McCormick, “Asymptotic optical attenuation,” Limnol. Oceanogr. 37, 1570–1578 (1992).
    [CrossRef]
  8. W. J. Wiscombe, G. W. Grams, “The backscattered fraction in two-stream approximations,” J. Atmos. Sci. 33, 2440–2451 (1976).
    [CrossRef]
  9. L. C. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [CrossRef]
  10. R. D. M. Garcia, C. E. Siewert, “Benchmark results in radiative transfer,” Transport Theory Stat. Phys. 14, 437–483 (1985).
    [CrossRef]
  11. R. D. M. Garcia, “A review of the Facile (FN) method in particle transport theory,” Transport Theory Stat. Phys. 14, 391–435 (1985).
    [CrossRef]
  12. C. Devaux, C. E. Siewert, Y. C. Yuan, “The complete solution for radiative transfer problems with reflecting boundaries and internal sources,” Astrophys. J. 253, 773–784 (1982).
    [CrossRef]
  13. I. Kuisščer, N. J. McCormick, “Some analytical results for radiative transfer in thick atmospheres,” Transport Theory Stat. Phys. 20, 351–381 (1991).
    [CrossRef]
  14. R. C. Smith, K. S. Baker, “Optical properties of the clearest natural waters,” Appl. Opt. 20, 177–184 (1981).
    [CrossRef] [PubMed]
  15. L. Prieur, S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific absorption of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26, 671–689 (1981).
    [CrossRef]
  16. C. D. Mobley, Light and Water, Radiative Transfer in Natural Waters (Academic, New York, 1994), Sec. 3.8.
  17. P. W. Francisco, N. J. McCormick, “Chlorophyll concentration effects on asymptotic optical attenuation,” Limnol. Oceanogr. 39, 1195–1205 (1994).
    [CrossRef]
  18. R. H. Stavn, A. D. Weidemann, “Effects of Raman scattering across the visible spectrum in clear ocean water: a Monte Carlo study,” Appl. Opt. 32, 6853–6863 (1993).
    [CrossRef] [PubMed]
  19. N. J. McCormick, “Mathematical models for the mean cosine of irradiance and the diffuse attenuation coefficient,” Limnol. Oceanogr. (to be published).

1994

P. W. Francisco, N. J. McCormick, “Chlorophyll concentration effects on asymptotic optical attenuation,” Limnol. Oceanogr. 39, 1195–1205 (1994).
[CrossRef]

Z. Tao, N. J. McCormick, R. Sanchez, “Ocean source and optical property estimation from explicit and implicit algorithms,” Appl. Opt. 33, 3265–3275 (1994).
[CrossRef] [PubMed]

1993

1992

N. J. McCormick, “Asymptotic optical attenuation,” Limnol. Oceanogr. 37, 1570–1578 (1992).
[CrossRef]

1991

I. Kuisščer, N. J. McCormick, “Some analytical results for radiative transfer in thick atmospheres,” Transport Theory Stat. Phys. 20, 351–381 (1991).
[CrossRef]

1989

R. H. Stavn, A. D. Weidemann, “Shape factors, two-flow models, and the problem of irradiance inversion in estimating optical parameters,” Limnol. Oceanogr. 34, 1426–1441 (1989).
[CrossRef]

J. R. V. Zaneveld, “An asymptotic closure theory for irradiance in the sea and its inversion to obtain the inherent optical properties,” Limnol. Oceanogr. 34, 1442–1452 (1989).
[CrossRef]

1987

1985

R. D. M. Garcia, C. E. Siewert, “Benchmark results in radiative transfer,” Transport Theory Stat. Phys. 14, 437–483 (1985).
[CrossRef]

R. D. M. Garcia, “A review of the Facile (FN) method in particle transport theory,” Transport Theory Stat. Phys. 14, 391–435 (1985).
[CrossRef]

1984

R. W. Preisendorfer, C. D. Mobley, “Direct and inverse irradiance models in hydrologic optics,” Limnol. Oceanogr. 29, 903–929 (1984).
[CrossRef]

1982

C. Devaux, C. E. Siewert, Y. C. Yuan, “The complete solution for radiative transfer problems with reflecting boundaries and internal sources,” Astrophys. J. 253, 773–784 (1982).
[CrossRef]

1981

L. Prieur, S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific absorption of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26, 671–689 (1981).
[CrossRef]

R. C. Smith, K. S. Baker, “Optical properties of the clearest natural waters,” Appl. Opt. 20, 177–184 (1981).
[CrossRef] [PubMed]

1976

W. J. Wiscombe, G. W. Grams, “The backscattered fraction in two-stream approximations,” J. Atmos. Sci. 33, 2440–2451 (1976).
[CrossRef]

1941

L. C. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Aas, E.

Baker, K. S.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), Secs. 46–48.

Devaux, C.

C. Devaux, C. E. Siewert, Y. C. Yuan, “The complete solution for radiative transfer problems with reflecting boundaries and internal sources,” Astrophys. J. 253, 773–784 (1982).
[CrossRef]

Francisco, P. W.

P. W. Francisco, N. J. McCormick, “Chlorophyll concentration effects on asymptotic optical attenuation,” Limnol. Oceanogr. 39, 1195–1205 (1994).
[CrossRef]

Garcia, R. D. M.

R. D. M. Garcia, “A review of the Facile (FN) method in particle transport theory,” Transport Theory Stat. Phys. 14, 391–435 (1985).
[CrossRef]

R. D. M. Garcia, C. E. Siewert, “Benchmark results in radiative transfer,” Transport Theory Stat. Phys. 14, 437–483 (1985).
[CrossRef]

Grams, G. W.

W. J. Wiscombe, G. W. Grams, “The backscattered fraction in two-stream approximations,” J. Atmos. Sci. 33, 2440–2451 (1976).
[CrossRef]

Greenstein, J. L.

L. C. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Henyey, L. C.

L. C. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Kuisšcer, I.

I. Kuisščer, N. J. McCormick, “Some analytical results for radiative transfer in thick atmospheres,” Transport Theory Stat. Phys. 20, 351–381 (1991).
[CrossRef]

McCormick, N. J.

Z. Tao, N. J. McCormick, R. Sanchez, “Ocean source and optical property estimation from explicit and implicit algorithms,” Appl. Opt. 33, 3265–3275 (1994).
[CrossRef] [PubMed]

P. W. Francisco, N. J. McCormick, “Chlorophyll concentration effects on asymptotic optical attenuation,” Limnol. Oceanogr. 39, 1195–1205 (1994).
[CrossRef]

N. J. McCormick, “Asymptotic optical attenuation,” Limnol. Oceanogr. 37, 1570–1578 (1992).
[CrossRef]

I. Kuisščer, N. J. McCormick, “Some analytical results for radiative transfer in thick atmospheres,” Transport Theory Stat. Phys. 20, 351–381 (1991).
[CrossRef]

N. J. McCormick, “Mathematical models for the mean cosine of irradiance and the diffuse attenuation coefficient,” Limnol. Oceanogr. (to be published).

Mobley, C. D.

R. W. Preisendorfer, C. D. Mobley, “Direct and inverse irradiance models in hydrologic optics,” Limnol. Oceanogr. 29, 903–929 (1984).
[CrossRef]

C. D. Mobley, Light and Water, Radiative Transfer in Natural Waters (Academic, New York, 1994), Sec. 3.8.

Preisendorfer, R. W.

R. W. Preisendorfer, C. D. Mobley, “Direct and inverse irradiance models in hydrologic optics,” Limnol. Oceanogr. 29, 903–929 (1984).
[CrossRef]

Prieur, L.

L. Prieur, S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific absorption of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26, 671–689 (1981).
[CrossRef]

Sanchez, R.

Sathyendranath, S.

L. Prieur, S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific absorption of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26, 671–689 (1981).
[CrossRef]

Siewert, C. E.

R. D. M. Garcia, C. E. Siewert, “Benchmark results in radiative transfer,” Transport Theory Stat. Phys. 14, 437–483 (1985).
[CrossRef]

C. Devaux, C. E. Siewert, Y. C. Yuan, “The complete solution for radiative transfer problems with reflecting boundaries and internal sources,” Astrophys. J. 253, 773–784 (1982).
[CrossRef]

Smith, R. C.

Stavn, R. H.

R. H. Stavn, A. D. Weidemann, “Effects of Raman scattering across the visible spectrum in clear ocean water: a Monte Carlo study,” Appl. Opt. 32, 6853–6863 (1993).
[CrossRef] [PubMed]

R. H. Stavn, A. D. Weidemann, “Shape factors, two-flow models, and the problem of irradiance inversion in estimating optical parameters,” Limnol. Oceanogr. 34, 1426–1441 (1989).
[CrossRef]

Tao, Z.

Weidemann, A. D.

R. H. Stavn, A. D. Weidemann, “Effects of Raman scattering across the visible spectrum in clear ocean water: a Monte Carlo study,” Appl. Opt. 32, 6853–6863 (1993).
[CrossRef] [PubMed]

R. H. Stavn, A. D. Weidemann, “Shape factors, two-flow models, and the problem of irradiance inversion in estimating optical parameters,” Limnol. Oceanogr. 34, 1426–1441 (1989).
[CrossRef]

Wiscombe, W. J.

W. J. Wiscombe, G. W. Grams, “The backscattered fraction in two-stream approximations,” J. Atmos. Sci. 33, 2440–2451 (1976).
[CrossRef]

Yuan, Y. C.

C. Devaux, C. E. Siewert, Y. C. Yuan, “The complete solution for radiative transfer problems with reflecting boundaries and internal sources,” Astrophys. J. 253, 773–784 (1982).
[CrossRef]

Zaneveld, J. R. V.

J. R. V. Zaneveld, “An asymptotic closure theory for irradiance in the sea and its inversion to obtain the inherent optical properties,” Limnol. Oceanogr. 34, 1442–1452 (1989).
[CrossRef]

Appl. Opt.

Astrophys. J.

L. C. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

C. Devaux, C. E. Siewert, Y. C. Yuan, “The complete solution for radiative transfer problems with reflecting boundaries and internal sources,” Astrophys. J. 253, 773–784 (1982).
[CrossRef]

J. Atmos. Sci.

W. J. Wiscombe, G. W. Grams, “The backscattered fraction in two-stream approximations,” J. Atmos. Sci. 33, 2440–2451 (1976).
[CrossRef]

Limnol. Oceanogr.

N. J. McCormick, “Asymptotic optical attenuation,” Limnol. Oceanogr. 37, 1570–1578 (1992).
[CrossRef]

J. R. V. Zaneveld, “An asymptotic closure theory for irradiance in the sea and its inversion to obtain the inherent optical properties,” Limnol. Oceanogr. 34, 1442–1452 (1989).
[CrossRef]

P. W. Francisco, N. J. McCormick, “Chlorophyll concentration effects on asymptotic optical attenuation,” Limnol. Oceanogr. 39, 1195–1205 (1994).
[CrossRef]

R. W. Preisendorfer, C. D. Mobley, “Direct and inverse irradiance models in hydrologic optics,” Limnol. Oceanogr. 29, 903–929 (1984).
[CrossRef]

R. H. Stavn, A. D. Weidemann, “Shape factors, two-flow models, and the problem of irradiance inversion in estimating optical parameters,” Limnol. Oceanogr. 34, 1426–1441 (1989).
[CrossRef]

L. Prieur, S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific absorption of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26, 671–689 (1981).
[CrossRef]

Transport Theory Stat. Phys.

I. Kuisščer, N. J. McCormick, “Some analytical results for radiative transfer in thick atmospheres,” Transport Theory Stat. Phys. 20, 351–381 (1991).
[CrossRef]

R. D. M. Garcia, C. E. Siewert, “Benchmark results in radiative transfer,” Transport Theory Stat. Phys. 14, 437–483 (1985).
[CrossRef]

R. D. M. Garcia, “A review of the Facile (FN) method in particle transport theory,” Transport Theory Stat. Phys. 14, 391–435 (1985).
[CrossRef]

Other

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), Secs. 46–48.

N. J. McCormick, “Mathematical models for the mean cosine of irradiance and the diffuse attenuation coefficient,” Limnol. Oceanogr. (to be published).

C. D. Mobley, Light and Water, Radiative Transfer in Natural Waters (Academic, New York, 1994), Sec. 3.8.

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Figures (3)

Fig. 1
Fig. 1

Phase function β ˜versus the scattering angle θ (with μ = cos θ) at different wavelengths λ (in nanometers) for two chlorophyll concentrations C (in milligrams per cubic meter).

Fig. 2
Fig. 2

Incident illumination ϕ(ν1, μ) for case 3 that causes apparent optical properties at some wavelengths λ (in nanometers) to be constant with depth. C = 0.01 mg m−3.

Fig. 3
Fig. 3

Incident illumination ϕ(ν1, μ) for case 3 that causes apparent optical properties at some wavelengths λ (in nanometers) to be constant with depth. C = 10 mg m−3.

Tables (4)

Tables Icon

Table 1 Inherent Optical Properties c, b, and bb (in Inverse Meters) at Various Wavelengths λ (in Nanometers) and Chlorophyll Concentrations C (in Milligrams per Cubic Meter)

Tables Icon

Table 2 Mean Cosines in the Downward and Upward Directions in Deep Water at Various Wavelengths λ (in Nanometers) for Two Chlorophyll Concentrations C (in Milligrams per Cubic Meter)

Tables Icon

Table 3 Two-Stream Shape Factors r±(∞) in Deep Water at Various Wavelengths λ (in Nanometers) for Two Chlorophyll Concentrations C (in Milligrams per Cubic Meter)

Tables Icon

Table 4 Two-Stream Shape Factors r±(0+) just beneath the Surface at Various Wavelengths λ (in Nanometers) for Two Chlorophyll Concentrations C (in Milligrams per Cubic Meter)

Equations (49)

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μ L ( z , μ ) / z + c L ( z , μ ) = b 1 1 β ˜ ( μ , μ ) L ( z , μ ) d μ = b 1 2 n = 0 N ( 2 n + 1 ) f n P n ( μ ) E n ( z ) ,
E n ( z ) = 1 1 P n ( μ ) L ( z , μ ) d μ = E n + ( z ) + E n ( z ) ,
E n + ( z ) = 0 1 P n ( μ ) L ( z , μ ) d μ ,
E n ( z ) = 1 0 P n ( μ ) L ( z , μ ) d μ ,
d E 1 ± ( z ) / d z = c ± ( z ) E 1 ± ( z ) ± b ( z ) E 1 ( z ) ,
a ± ( z ) = a ( z ) / μ ¯ ± ( z ) , b ± ( z ) = r ± ( z ) b b / μ ¯ ± ( z ) , c ± ( z ) = a ± ( z ) + b ± ( z ) ,
μ ¯ ± ( z ) = ± E 1 ± ( z ) / E 0 ± ( z ) ,
r + ( z ) b b b = 1 E 0 + ( z ) 0 1 d μ 1 0 d μ β ˜ ( μ , μ ) L ( z , μ ) ,
r ( z ) b b b = 1 E 0 ( z ) 1 0 d μ 0 1 d μ β ˜ ( μ , μ ) L ( z , μ ) .
b b / b = 1 0 β ˜ ( μ , 1 ) d μ = 1 2 ( 1 Λ b ) ,
Λ b = n 1 n odd α n ( 2 n + 1 ) f n ,
α n = 0 1 P n ( μ ) d μ
α n / α n 2 = ( n 2 ) / ( n + 1 ) , n odd , n 3 ,
r ± ( ) b b b = 1 2 [ 1 ( 3 f 1 2 + Λ ) μ ¯ ± ( ) ] ,
3 f 1 2 + Λ = n 1 n odd α n ( 2 n + 1 ) f n g n ( ν 1 ) g 1 ( ν 1 ) .
ω ν 1 2 1 1 n = 0 N ( 2 n + 1 ) f n g n ( μ ) P n ( μ ) 1 ν 1 μ d μ = 1 .
g n ( ν 1 ) = 1 1 P n ( μ ) ϕ ( ν 1 , μ ) d μ ,
ϕ ( ν 1 , μ ) = ω ν 1 2 ( ν 1 μ ) n = 0 N ( 2 n + 1 ) f n g n ( ν 1 ) P n ( μ ) .
( n + 1 ) g n + 1 ( ν 1 ) ( 2 n + 1 ) ( 1 ω f n ) ν 1 g n ( ν 1 ) + n g n 1 ( ν 1 ) = 0 , n 0 ,
r ± ( ) = 1 ( 3 f 1 2 + Λ ) μ ¯ ± ( ) 1 Λ b .
μ ¯ ± ( ) = g ˜ 1 ( ± ν 1 ) / g ˜ 0 ( ± ν 1 ) ,
g ˜ n ( ± ν 1 ) = 0 1 P n ( μ ) ϕ ( ± ν 1 , μ ) d μ .
L i ( 0 + , μ ) = 2 E d , 0 μ 1 , case 1 , = ( E d / μ 0 ) δ ( μ μ 0 ) , 0 μ 1 , case 2 , = E d ϕ ( ν 1 , μ ) , 0 μ 1 , case 3 ,
r + , i ( 0 + ) = 1 Λ + , i 1 Λ b , cases i = 1 , 2 , 3 .
Λ + , 1 = n 1 n odd α n 2 ( 2 n + 1 ) f n , case 1 , Λ + , 2 = n 1 n odd α n ( 2 n + 1 ) f n P n ( μ 0 ) , case 2 , Λ + , 3 = n 1 n odd α n ( 2 n + 1 ) f n [ g ˜ n ( ν 1 ) / g ˜ 0 ( ν 1 ) ] , case 3 .
L i ( 0 + , μ ) = k = 0 N A k P k ( 2 μ 1 ) , 0 μ 1 ,
r , i ( 0 + ) = 1 Λ , i 1 Λ b , cases i = 1 , 2 .
Λ , i = n 1 n odd α n ( 2 n + 1 ) f n k = 0 K ( A k / A 0 ) α ˜ n k , cases i = 1 , 2 ,
α ˜ n k = 0 1 P n ( μ ) P k ( 2 μ 1 ) d μ .
( 2 n + 2 ) ( 2 k + 1 ) α ˜ n + 1 , k ( 2 n + 1 ) ( 2 k + 1 ) α ˜ n k + 2 n ( 2 k + 1 ) α ˜ n 1 , k = ( 2 n + 1 ) ( k + 1 ) α ˜ n , k + 1 + ( 2 n + 1 ) k α ˜ n , k 1 , n 1 , k 1 ,
α ˜ n 0 = α n , α ˜ n 1 = α n , n odd , n 3 , = 2 n + 2 α n 1 , n even , n 2 , α ˜ 0 k = δ 0 k , α ˜ 1 k = δ 1 k / 6 , k odd , = δ 0 k / 2 , k even ,
L 3 ( 0 + , μ ) = E d ϕ ( ν 1 , μ ) = E d ϕ ( ν 1 , μ ) , 0 μ 1 .
Λ , 3 = n 1 n odd α n ( 2 n + 1 ) f n [ g ˜ n ( ν 1 ) / g ˜ 0 ( ν 1 ) ] , case 3 .
μ ¯ ± , 3 ( 0 + ) = g ˜ 1 ( ± ν 1 ) / g ˜ 0 ( ± ν 1 ) = μ ¯ ± ( ) ,
r ± , 3 ( 0 + ) = r ± ( ) .
R ( z ) = E ( z ) E + ( z ) = [ c + ( z ) + c ( z ) 2 b ( z ) ] { [ c + ( z ) + c ( z ) 2 b ( z ) ] 2 b + ( z ) b ( z ) } 1 / 2 ,
r + ( z ) 1 + μ ¯ + ( z ) / μ ¯ ( z ) b b / a 1 + r + ( z ) b b / a ,
R ( 0 + ) r + ( ) 1 + μ ¯ + ( ) / μ ¯ ( ) b b / a 1 + r + ( ) b b / a .
a p ( λ , C ) = 0 . 06 A p ( λ ) C 0 . 602 ,
b p ( λ , C ) = 0 . 30 ( 550 λ ) C 0 . 62 .
ω = ω p ( c p / c w ) + ω w ( c p / c w ) + 1 ,
ω β ˜ = ω p ( c p / c w ) β ˜ p + ω w β ˜ w ( c p / c w ) + 1 .
β ˜ ( μ , μ ) = γδ ( μ μ ) + ( 1 γ ) β ˜ ( μ , μ ) , = γδ ( μ μ ) + ( 1 γ ) 1 2 k = 0 K ( 2 n + 1 ) × f k P k ( μ ) P k ( μ ) .
r ± ( z ) = 1 Λ ± ( z ) 1 Λ b ,
Λ ± ( z ) = n 1 n odd α n ( 2 n + 1 ) f n σ+ C ( ν ) g ˜ n ( ± ν ) exp ( c z / ν ) d ν σ+ C ( ν ) g ˜ 0 ( ± ν ) exp ( c z / ν ) d ν .
σ+ C ( ν ) g ˜ n ( ± ν ) exp ( c z / ν ) d ν = j = 1 J C ( ν j ) g ˜ n ( ± ν j ) exp ( c z / ν j ) + 0 1 C ( ν ) g ˜ n ( ± ν ) exp ( c z / ν ) d ν
Λ ± ( z ) Λ ± , 3 + C ( ν 2 ) C ( ν 1 ) exp [ c z ( 1 ν 2 1 ν 1 ) ] n 1 n odd α n ( 2 n + 1 ) f n g ˜ n ( ± ν 2 ) g ˜ 0 ( ± ν 1 ) + 1 + exp [ c z ( 1 ν 2 1 ν 1 ) ] C ( ν 2 ) g ˜ 1 ( ± ν 2 ) C ( ν 1 ) g ˜ n ( ± ν 1 ) + ,
r ± ( z ) r ± ( ) + [ r ± ( 0 + ) r ± ( ) ] exp ( P z ) ,
P = c ( 1 ν 2 1 ν 1 ) .

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