Abstract

On the basis of an approximate relation, the Mie series are replaced by new ones, which can be summed exactly with the aid of various modified and generalized forms of the addition theorem for cylindrical functions. The sums obtained simultaneously simplify the initial expressions for the scattering characteristics and preserve their analytical nature. The conventional approximations for the amplitude functions and the efficiency factors rigorously follow from the new approaches if the optical parameters are properly restricted. The acceptable domains of these approaches contain the long wavelength region and are extensive enough to study the various (including inverse) scattering problems for the real disperse systems, in which the particles are suspended in a medium with similar optical properties.

© 1991 Optical Society of America

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References

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  1. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  2. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  3. K. S. Shifrin, Introduction to Ocean Optics (Gidrometeoizdat, Leningrad, 1983).
  4. A. Y. Perelman, “An Application of Mie’s Series to Soft Particles,” Pageoph 116, 1077–1088 (1978).
    [Crossref]
  5. A. Y. Perelman, “Extinction Efficiency Factor for Suspended Oceanic Particles,” Izv. Akad Nauk S.S.S.R. Fiz. Atmos. Okeana 22, 242–250 (1986).
  6. A. Y. Perelman, “The Scattering of Light by a Translucent Sphere Described in Soft-Particle Approximation,” Dokl. Akad Nauk S.S.S.R. 281, 51–54 (1985).
  7. K. S. Shifrin, Scattering of Light in a Turbid Medium (Gostechizdat, Moscow, 1951), Chap. 7.
  8. A. N. Tichonov, V. Y. Arsenin, Solution of Ill-Posed Problems (Winston-Wiley, New York, 1977).
  9. S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (American Elsevier, New York, 1977).
  10. D. L. Phillips, “A Technique for the Numerical Solution of Certain Integral Equations of the First Kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
    [Crossref]
  11. S. Twomey, “On the Numerical Solution of Fredholm Integral Equations of the First Kind by the Inversion of the Linear System Produced by Quadrature,” J. Assoc. Comput. Mach. 10, 97–108 (1963).
    [Crossref]
  12. A. Y. Perelman, “Solution of Integral Equations of the First Kind with Kernel Depending on the Product,” Sov. J. Vychisl. Math. Math. Phys. 7, 94–112 (1967).
  13. K. S. Shifrin, V. F. Raskin, “Spectral Transparency and Inverse Problem of the Scattering Theory,” Opt. Spectrosc. 11, 268–271 (1961).
  14. K. S. Shifrin, A. Y. Perelman, “Determination of the Particle Spectrum of a Dispersed System from Data of Its Transparency,” Opt. Spectrosc. 15, 533–542 (1963).
  15. K. S. Shifrin, A. Y. Perelman, “Calculation of Particle Distribution by the Data on Spectral Transparency,” Pageoph58, 208–220 (1964).
  16. M. A. Box, B. H. J. McKellar, “Further Relations Between Analytic Inversion Formulas for Multispectral Extinction Data,” Appl. Opt. 20, 3829–3831 (1981).
    [Crossref] [PubMed]
  17. G. Viera, M. A. Box, “Information Content Analysis of Aerosol Remote-Sensing Experiments Using an Analytic Eigen-function Theory: Anomalous Diffraction Approximation,” Appl. Opt. 24, 4525–4533 (1985).
    [Crossref] [PubMed]
  18. C. B. Smith, “Inversion of the Anomalous Diffraction Approximation for Variable Complex Index of Refraction Near Unity,” Appl. Opt. 21, 3363–3366 (1982).
    [Crossref] [PubMed]
  19. A. L. Fymat, “Remote Monitoring of Environmental Particulate Pollution: a Problem in Inversion of First-Kind Integral Equations,” Appl. Math. Comput. 1, 131–185 (1975).
    [Crossref]
  20. M. Bertero, C. D. Mol, E. R. Pike, “Particle Size Distribution from Spectral Turbidity: a Singular-System Analysis,” Inverse Problems 2, 247–258 (1986).
    [Crossref]
  21. J. D. Klett, “Anomalous Diffraction Model for Inversion of Multispectral Extinction Data Including Absorbing Effects,” Appl. Opt. 23, 4499–4508 (1984).
    [Crossref] [PubMed]
  22. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1965), Chapt. 9.

1986 (2)

A. Y. Perelman, “Extinction Efficiency Factor for Suspended Oceanic Particles,” Izv. Akad Nauk S.S.S.R. Fiz. Atmos. Okeana 22, 242–250 (1986).

M. Bertero, C. D. Mol, E. R. Pike, “Particle Size Distribution from Spectral Turbidity: a Singular-System Analysis,” Inverse Problems 2, 247–258 (1986).
[Crossref]

1985 (2)

G. Viera, M. A. Box, “Information Content Analysis of Aerosol Remote-Sensing Experiments Using an Analytic Eigen-function Theory: Anomalous Diffraction Approximation,” Appl. Opt. 24, 4525–4533 (1985).
[Crossref] [PubMed]

A. Y. Perelman, “The Scattering of Light by a Translucent Sphere Described in Soft-Particle Approximation,” Dokl. Akad Nauk S.S.S.R. 281, 51–54 (1985).

1984 (1)

1982 (1)

1981 (1)

1978 (1)

A. Y. Perelman, “An Application of Mie’s Series to Soft Particles,” Pageoph 116, 1077–1088 (1978).
[Crossref]

1975 (1)

A. L. Fymat, “Remote Monitoring of Environmental Particulate Pollution: a Problem in Inversion of First-Kind Integral Equations,” Appl. Math. Comput. 1, 131–185 (1975).
[Crossref]

1967 (1)

A. Y. Perelman, “Solution of Integral Equations of the First Kind with Kernel Depending on the Product,” Sov. J. Vychisl. Math. Math. Phys. 7, 94–112 (1967).

1964 (1)

K. S. Shifrin, A. Y. Perelman, “Calculation of Particle Distribution by the Data on Spectral Transparency,” Pageoph58, 208–220 (1964).

1963 (2)

K. S. Shifrin, A. Y. Perelman, “Determination of the Particle Spectrum of a Dispersed System from Data of Its Transparency,” Opt. Spectrosc. 15, 533–542 (1963).

S. Twomey, “On the Numerical Solution of Fredholm Integral Equations of the First Kind by the Inversion of the Linear System Produced by Quadrature,” J. Assoc. Comput. Mach. 10, 97–108 (1963).
[Crossref]

1962 (1)

D. L. Phillips, “A Technique for the Numerical Solution of Certain Integral Equations of the First Kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
[Crossref]

1961 (1)

K. S. Shifrin, V. F. Raskin, “Spectral Transparency and Inverse Problem of the Scattering Theory,” Opt. Spectrosc. 11, 268–271 (1961).

Arsenin, V. Y.

A. N. Tichonov, V. Y. Arsenin, Solution of Ill-Posed Problems (Winston-Wiley, New York, 1977).

Bertero, M.

M. Bertero, C. D. Mol, E. R. Pike, “Particle Size Distribution from Spectral Turbidity: a Singular-System Analysis,” Inverse Problems 2, 247–258 (1986).
[Crossref]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Box, M. A.

Fymat, A. L.

A. L. Fymat, “Remote Monitoring of Environmental Particulate Pollution: a Problem in Inversion of First-Kind Integral Equations,” Appl. Math. Comput. 1, 131–185 (1975).
[Crossref]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Klett, J. D.

McKellar, B. H. J.

Mol, C. D.

M. Bertero, C. D. Mol, E. R. Pike, “Particle Size Distribution from Spectral Turbidity: a Singular-System Analysis,” Inverse Problems 2, 247–258 (1986).
[Crossref]

Perelman, A. Y.

A. Y. Perelman, “Extinction Efficiency Factor for Suspended Oceanic Particles,” Izv. Akad Nauk S.S.S.R. Fiz. Atmos. Okeana 22, 242–250 (1986).

A. Y. Perelman, “The Scattering of Light by a Translucent Sphere Described in Soft-Particle Approximation,” Dokl. Akad Nauk S.S.S.R. 281, 51–54 (1985).

A. Y. Perelman, “An Application of Mie’s Series to Soft Particles,” Pageoph 116, 1077–1088 (1978).
[Crossref]

A. Y. Perelman, “Solution of Integral Equations of the First Kind with Kernel Depending on the Product,” Sov. J. Vychisl. Math. Math. Phys. 7, 94–112 (1967).

K. S. Shifrin, A. Y. Perelman, “Calculation of Particle Distribution by the Data on Spectral Transparency,” Pageoph58, 208–220 (1964).

K. S. Shifrin, A. Y. Perelman, “Determination of the Particle Spectrum of a Dispersed System from Data of Its Transparency,” Opt. Spectrosc. 15, 533–542 (1963).

Phillips, D. L.

D. L. Phillips, “A Technique for the Numerical Solution of Certain Integral Equations of the First Kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
[Crossref]

Pike, E. R.

M. Bertero, C. D. Mol, E. R. Pike, “Particle Size Distribution from Spectral Turbidity: a Singular-System Analysis,” Inverse Problems 2, 247–258 (1986).
[Crossref]

Raskin, V. F.

K. S. Shifrin, V. F. Raskin, “Spectral Transparency and Inverse Problem of the Scattering Theory,” Opt. Spectrosc. 11, 268–271 (1961).

Shifrin, K. S.

K. S. Shifrin, A. Y. Perelman, “Calculation of Particle Distribution by the Data on Spectral Transparency,” Pageoph58, 208–220 (1964).

K. S. Shifrin, A. Y. Perelman, “Determination of the Particle Spectrum of a Dispersed System from Data of Its Transparency,” Opt. Spectrosc. 15, 533–542 (1963).

K. S. Shifrin, V. F. Raskin, “Spectral Transparency and Inverse Problem of the Scattering Theory,” Opt. Spectrosc. 11, 268–271 (1961).

K. S. Shifrin, Introduction to Ocean Optics (Gidrometeoizdat, Leningrad, 1983).

K. S. Shifrin, Scattering of Light in a Turbid Medium (Gostechizdat, Moscow, 1951), Chap. 7.

Smith, C. B.

Tichonov, A. N.

A. N. Tichonov, V. Y. Arsenin, Solution of Ill-Posed Problems (Winston-Wiley, New York, 1977).

Twomey, S.

S. Twomey, “On the Numerical Solution of Fredholm Integral Equations of the First Kind by the Inversion of the Linear System Produced by Quadrature,” J. Assoc. Comput. Mach. 10, 97–108 (1963).
[Crossref]

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (American Elsevier, New York, 1977).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Viera, G.

Appl. Math. Comput. (1)

A. L. Fymat, “Remote Monitoring of Environmental Particulate Pollution: a Problem in Inversion of First-Kind Integral Equations,” Appl. Math. Comput. 1, 131–185 (1975).
[Crossref]

Appl. Opt. (4)

Dokl. Akad Nauk S.S.S.R. (1)

A. Y. Perelman, “The Scattering of Light by a Translucent Sphere Described in Soft-Particle Approximation,” Dokl. Akad Nauk S.S.S.R. 281, 51–54 (1985).

Inverse Problems (1)

M. Bertero, C. D. Mol, E. R. Pike, “Particle Size Distribution from Spectral Turbidity: a Singular-System Analysis,” Inverse Problems 2, 247–258 (1986).
[Crossref]

Izv. Akad Nauk S.S.S.R. Fiz. Atmos. Okeana (1)

A. Y. Perelman, “Extinction Efficiency Factor for Suspended Oceanic Particles,” Izv. Akad Nauk S.S.S.R. Fiz. Atmos. Okeana 22, 242–250 (1986).

J. Assoc. Comput. Mach. (2)

D. L. Phillips, “A Technique for the Numerical Solution of Certain Integral Equations of the First Kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
[Crossref]

S. Twomey, “On the Numerical Solution of Fredholm Integral Equations of the First Kind by the Inversion of the Linear System Produced by Quadrature,” J. Assoc. Comput. Mach. 10, 97–108 (1963).
[Crossref]

Opt. Spectrosc. (2)

K. S. Shifrin, V. F. Raskin, “Spectral Transparency and Inverse Problem of the Scattering Theory,” Opt. Spectrosc. 11, 268–271 (1961).

K. S. Shifrin, A. Y. Perelman, “Determination of the Particle Spectrum of a Dispersed System from Data of Its Transparency,” Opt. Spectrosc. 15, 533–542 (1963).

Pageoph (2)

K. S. Shifrin, A. Y. Perelman, “Calculation of Particle Distribution by the Data on Spectral Transparency,” Pageoph58, 208–220 (1964).

A. Y. Perelman, “An Application of Mie’s Series to Soft Particles,” Pageoph 116, 1077–1088 (1978).
[Crossref]

Sov. J. Vychisl. Math. Math. Phys. (1)

A. Y. Perelman, “Solution of Integral Equations of the First Kind with Kernel Depending on the Product,” Sov. J. Vychisl. Math. Math. Phys. 7, 94–112 (1967).

Other (7)

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

K. S. Shifrin, Introduction to Ocean Optics (Gidrometeoizdat, Leningrad, 1983).

K. S. Shifrin, Scattering of Light in a Turbid Medium (Gostechizdat, Moscow, 1951), Chap. 7.

A. N. Tichonov, V. Y. Arsenin, Solution of Ill-Posed Problems (Winston-Wiley, New York, 1977).

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (American Elsevier, New York, 1977).

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1965), Chapt. 9.

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Tables (2)

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Table I Values of ρ(m)

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Table II Values of ρj

Equations (95)

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{ m = m 1 m 2 1 = m i m ( m 2 > 0 ) , μ = cos θ , x = 2 π m 2 a λ 1 , y = m x .
Q = Q S C + Q a ,
Q = 2 x 2 n = 1 ( 2 n + 1 ) Re ( a 1 n + a 2 n ) ,
Q S C = 2 x 2 n = 1 ( 2 n + 1 ) ( | a 1 n | 2 + | a 2 n | 2 ) ,
Q S C = x 2 0 π [ | S 1 ( μ ) | 2 + | S 2 ( μ ) | 2 ] sin θ d θ
{ S 1 ( μ ) = n = 1 ( 2 n + 1 ) n ¯ 1 [ a 1 n π n ( μ ) + a 2 n τ n ( μ ) ] , S 2 ( μ ) = n = 1 ( 2 n + 1 ) n ¯ 1 [ a 1 n τ n ( μ ) + a 2 n π n ( μ ) ] ,
k = 1 , 2 and n ¯ = n ( n + 1 ) .
Re a k n = | a k n | 2 ( m = Re m ) .
Q = 4 x 2 Re S ( 1 ) , S ( 1 ) = S k ( 1 )
a k n = a k n ( m , x ) = h k n h k n + i h k + 2 n
= h k n 2 i h k n h k + 2 n h k n 2 + h k + 2 n 2 ,
{ h 1 n = x ψ n ( y ) ψ n ( x ) y ψ n ( y ) ψ n ( x ) , h 2 n = y ψ n ( y ) ψ n ( x ) x ψ n ( y ) ψ n ( x ) ,
{ h 3 n = x ψ n ( y ) χ n ( x ) y ψ n ( y ) χ n ( x ) , h 4 n = y ψ n ( y ) χ n ( x ) x ψ n ( y ) χ n ( x ) .
n ¯ z ( x ) = x 2 [ z ( x ) + z ( x ) ] ,
{ ψ n ( x ) = 1 ( 2 n + 1 ) ! ! x n + 1 1 2 ( 2 n + 3 ) ! ! x n + 3 + O ( x n + 5 ) , χ n ( x ) = ( 2 n 1 ) ! ! x n + O ( x n + 2 ) ,
{ h 1 n = O ( x 2 n + 2 ) , h 2 n = O ( x 2 n + 4 ) , h 3 n , h 4 n = O ( x ) ( x 0 )
S k R ( μ ) = i ( m 2 + 2 ) 1 ( m 2 1 ) x 3 μ k 1 ,
S k R G ( μ ) = 2 i ( m 1 ) u 2 ψ 1 ( u ) x 3 μ k 1 , u = 2 x sin θ / 2 .
π n ( μ ) = P n ( μ ) , τ n ( μ ) = n ¯ P n ( μ ) μ π n ( μ ) ,
P n ( μ ) = F ( n , n + 1 , 1 , 1 μ 2 ) ,
{ π n ( μ ) = n ¯ 2 + m = 1 n ( 1 ) m m + 1 ( n ¯ ) ( 2 m + 2 ) ! ! m ! ( 1 μ ) m , τ n ( μ ) = n ¯ π n ( μ ) + m = 1 n ( 1 ) m ( n ¯ m ) m ( n ¯ ) ( 2 m ) ! ! m ! ( 1 μ ) m ,
m ( z ) = n = 0 m 1 ( z n ¯ ) .
{ π n ( μ ) = n ¯ 2 n ¯ ( n ¯ 2 ) 8 ( 1 μ ) , τ n ( μ ) = n ¯ 2 n ¯ ( 3 n ¯ 2 ) 8 ( 1 μ ) ,
A k ( μ ) = H 11 + H 21 ( 1 μ ) H k + O ( θ 4 ) ( θ 0 ) ,
H k = 0 . 25 [ ( 2 k 1 ) H 12 + ( 2 k + 5 ) H 22 ] 0 . 5 ( H 11 + H 21 ) ,
H k r = n = 1 ( n + 0 . 5 ) n ¯ r 1 a k n .
a k n l h k n , l = i x 1 | m | 1 / 2 .
h k n 2 + h k + 2 n 2 x 2 | m | ,
a k n x 2 | m | 1 ( h k n 2 i h k n h k + 2 n ) .
H 1 r = l ( y F r x x F r y ) , H 2 r = l ( x F r x y F r y ) ,
F r = n = 1 ( n + 0 . 5 ) n ¯ r 1 ψ n ( x ) ψ n ( y ) .
{ F 1 = 0 . 5 [ υ ψ 0 ( z ) ψ 0 ( x ) ψ 0 ( y ) ] , F 2 = υ 2 ψ 1 ( z ) ,
{ H 11 + H 21 = l ( x + y ) υ ψ 1 ( z ) , H 12 = l ( x + y ) [ υ 2 ψ 2 ( z ) + 2 υ ψ 1 ( z ) ] , H 22 = l ( x + y ) υ 2 ψ 2 ( z ) .
{ H 1 = i ( m + 1 ) m 2 | m | 1 / 2 x 4 z 2 ψ 2 ( z ) , H 2 = i ( m + 1 ) m 2 | m | 1 / 2 x 4 [ z 2 ψ 2 ( z ) + m 1 x 2 z 1 ψ 1 ( z ) ] ,
A k ( μ ) = i ( m + 1 ) m | m | 1 / 2 x 2 [ μ k 1 z 1 ψ 1 ( z ) ( 1 μ ) m x 2 z 2 ψ 2 ( z ) ] , z = ( m 1 ) x ,
z 1 ψ 1 ( z ) = z / 3 + O ( z 3 ) , z 2 ψ 2 ( z ) = z / 15 + O ( z 3 ) .
A k ( μ ) = 3 1 i m | m | 1 / 2 ( m 2 1 ) x 3 μ k 1 ( x 1 ) .
A k ( μ ) = 3 1 i m | m | 1 / 2 ( m 2 1 ) x 3 [ μ k 1 5 1 m x 2 ( 1 μ ) ] + O ( θ 4 ) , ( x | m 1 ) 1 ) .
ψ 1 ( u ) u 2 = n = 1 ( 1 ) n x 2 n n ! ( 2 n + 3 ) ! ! ( 1 μ ) n , u = 2 x sin θ / 2 ,
S k R G ( μ ) = 2 3 i ( m 1 ) x 3 [ μ k 1 x 2 5 ( 1 μ ) ] + O ( θ 4 ) .
a k n = O ( x 4 n + 4 k 2 ) + i O ( x 2 n + 2 k 1 ) .
A ( 1 ) = x 2 | m | 1 n = 1 ( n + 0 . 5 ) × [ h 1 n 2 + h 2 n 2 i ( h 1 n h 3 n + h 2 n h 4 n ) ]
A ( 1 ) = x 2 8 | m | [ ( m 2 + 1 ) 2 + ω ( m , ρ ) ω ( m , R ) 2 m ] ,
{ ω ( m , z ) = [ a ( m ) + a 0 ( m ) z 2 ] e i ( z ) + i a 1 ( m ) e 1 ( z ) + a 2 ( m ) e 2 ( z ) , a ( m ) = ( m 2 1 ) 2 ( m 2 + 1 ) , a 0 ( m ) = 2 ( m 2 1 ) 2 ( m 1 ) 2 , a 1 ( m ) = ( m + 1 ) 2 ( m 4 2 m 3 2 m 2 2 m + 1 ) , a 2 ( m ) = a 0 ( m ) a 1 ( m ) , e i ( z ) = 0 z 1 exp ( i t ) t d t , e 1 ( z ) = exp ( i z ) z , e 2 ( z ) = 1 exp ( i z ) z 2 ,
ρ = 2 ( m 1 ) x , R = 2 ( m + 1 ) x .
S k ( μ ) A k ( μ ) = A ( 1 ) ( 1 μ ) H k , μ = cos θ ,
Q S = 1 2 | m | Re [ ( m 2 + 1 ) 2 + ω ( m , ρ ) ω ( m , R ) 2 m ] ,
lim Q S = Q H ,
Q H = 4 Re K ( i ρ ) , K ( w ) = 1 2 + exp ( w ) w + exp ( w ) 1 w 2
Q S = Q R if x 1 , | m 1 | 1 , Q S = Q R G if x | m 1 | 1 , | m 1 | 1 .
Q S = Q S + + Q S ,
{ Q S + = 1 2 | m | Re [ ( m 2 + 1 ) 2 + ω + ( m , ρ ) ω + ( m , R ) 2 m ] , ω + ( m , z ) = [ a ( m ) + a 0 ( m ) z 2 ] c i ( z ) + a 1 ( m ) s ( z ) + a 2 ( m ) c ( z ) ,
{ Q S = Re [ i ω ( m , ρ ) + ω ( m , R ) 4 | m | m ] , ω ( m , z ) = [ a ( m ) + a 0 ( m ) z 2 ] s i ( z ) + a 1 ( m ) σ ( z ) + a 2 ( m ) γ ( z ) .
c i ( z ) = 0 z t 1 ( 1 cos t ) d t , s i ( z ) = 0 z t 1 sin t d t , s ( z ) = z 1 sin z , σ ( z ) = z 1 cos z , c ( z ) = z 2 ( 1 cos z ) , γ ( z ) = z 1 sin z ,
0 x x ( m ) , m = 1 + M ( M > 1 ) ,
ρ ( m ) = 2 ( m 1 ) x ( m ) .
ρ j = 2 ( m 1 ) x j ( m ) , O < ρ 1 < ρ 2 <
{ Q S + = Re ξ + ( m , ρ ) ξ + ( m , R ) 16 | m | m , ξ + ( m , z ) = 4 a ( m ) c i ( z ) + a 0 ( m ) H i ( z ) a 1 ( m ) H ( z ) ,
{ Q S = Im ξ ( m , ρ ) + ξ ( m , R ) 16 | m | m , ξ ( m , z ) = 4 a ( m ) s i ( z ) + a 0 ( m ) h i ( z ) a 1 ( m ) h ( z ) ,
{ H ( z ) = 2 4 s ( z ) + 4 c ( z ) , h ( z ) = 4 σ ( z ) + 4 γ ( z ) , H i ( z ) = 0 1 t H ( z t ) d t , h i ( z ) = 0 1 t h ( z t ) d t .
{ H i ( z ) = 1 4 c ( z ) + 4 z 2 c i ( z ) , h i ( z ) = 4 γ ( z ) + 4 z 2 s i ( z ) ,
{ Q S + = x 4 Re [ u 1 s ( m ) ( m 2 1 ) 2 ] + x 6 Re [ u 2 s ( m ) ( m 2 1 ) 2 ] + , Q S = x Im [ υ 1 s ( m ) ( m 2 1 ) ] + x 3 Im [ υ 2 s ( m ) ( m 2 1 ) 2 ] + ,
{ u 1 s ( m ) = 8 27 | m | m 2 , u 2 s ( m ) = 8 185 | m | m 2 ( m 2 + 1 ) , υ 1 s ( m ) = 4 9 | m | m 2 ( m 2 + 2 ) , υ 2 s ( m ) = 8 225 t | m | m 2 ( m 2 + 10 ) .
{ Q S C = x 4 Re [ u 1 ( m ) ( m 2 1 ) 2 ] + x 6 Re [ u 2 ( m ) ( m 2 1 ) 2 ] + , Q a = x Im [ υ 1 ( m ) ( m 2 1 ) ] + x 3 Im [ υ 2 ( m ) ( m 2 1 ) 2 ] + ,
{ u 1 ( m ) = 8 3 ( m 2 + 2 ) 2 , u 2 ( m ) = 16 ( m 2 2 ) 5 ( m 2 + 2 ) 3 , υ 1 ( m ) = 4 m 2 + 2 , υ 2 ( m ) = 4 ( m 4 + 27 m 2 + 38 ) 15 ( m 2 + 2 ) 2 ( 2 m 2 + 3 ) .
u 1 s ( 1 ) = u 1 ( 1 ) = 8 / 27 , u 2 s ( 1 ) = u 2 ( 1 ) = 16 / 135 , υ 1 s ( 1 ) = υ 1 ( 1 ) = 4 / 3 , υ 2 s ( 1 ) = υ 2 ( 1 ) = 88 / 225 ,
{ Q H = Q H S C + Q H a , Q H S C = x 2 u 1 H ( m ) + , Q H a = x υ 1 H ( m ) + x 2 υ 2 H ( m ) + ,
{ u 1 H ( m ) = 2 ( m 1 ) 2 ( 1 + tan 2 β ) , υ 1 H ( m ) = 8 3 ( m 1 ) tan β , υ 2 H ( m ) = 4 ( m 1 ) 2 tan 2 β ,
T ( m , λ ) = 0 Q f ( a ) d a , f ( a ) = π a 2 p ( a ) ,
Q S = ξ + ( m , ρ ) ξ + ( m , R ) 16 m 2 ( m = Re m ) .
M ξ + ( m , t x ) = 4 t p Γ c ( p ) [ a ( m ) p a 0 ( m ) ( p 2 ) 2 a 1 ( m ) p 2 ]
f ( a ) = 1 2 π i c i c + i G ( a , p ) L ( m , p ) d p ( 2 < c < 0 ) ,
{ g ( m , ν ) = 16 m 2 T ( m , λ ) , v = 4 π λ 1 , G ( a , p ) = 0 g ( m , ν ) ( a ν ) p d ν , L ( m , p ) = ( 2 π ) 1 ( 1 p 2 ) A ( m , p ) , A ( m , p ) = ( p + 1 ) Γ c ( p ) [ l ( m , p ) l ( m , p ) ] 1 , l ( m , p ) = | m 1 | p 1 [ a ( m ) ( p + 1 ) 2 + a 0 ( m ) ( p 1 ) a 1 ( m ) ( p 2 1 ) ] .
{ G ( a , p ) G τ ( a , p ) + τ [ c 0 ( m ) + c 2 ( m ) ν 2 ] ( a ν ) p d ν , G τ ( a , p ) 0 < ν j < τ g ( m , ν j ) ( a ν j ) p Δ ν j .
2 π f ( a ) = O < ν j < τ g ( m , ν j ) ϕ ( a ν j ) Δ ν j + c 0 ( m ) τ ϕ 0 ( a τ ) + c 2 ( m ) τ 1 ϕ 2 ( a τ ) ,
{ ϕ ( x ) = 1 2 π i c i c + i ( p 2 1 ) A ( m , p ) x p d p , ϕ 0 ( x ) = 1 2 π i c i c + i ( p + 1 ) A ( m , p ) x p d p , ϕ 2 ( x ) = 1 2 π i c i c + i ( p 1 ) A ( m , p ) x p d p ,
{ ϕ ( x ) = n = 1 ( 4 n 2 1 ) a ( m , n ) x 2 n , ϕ 0 ( x ) = n = 1 ( 2 n 1 ) a ( m , n ) x 2 n , ϕ 2 ( x ) = n = 1 ( 2 n + 1 ) a ( m , n ) x 2 n ,
Δ ( m , x ) = | Q S Q | Q [ x > x ( m ) , m = Re m ]
{ n = 0 ( 2 n + 1 ) ψ n ( x i ) ψ n ( x j ) P n ( μ ) = x i x j s ( ω i j ) , n = 0 ( 2 n + 1 ) ψ n ( x i ) χ n ( x j ) P n ( μ ) = x i x j σ ( ω i j ) , | x i | < x j .
{ n = 0 ( n + 0 . 5 ) r = 1 4 ψ n ( x r ) x r = 1 4 1 1 sin ω 12 sin ω 34 ω 12 ω 34 d μ , n = 0 ( n + 0 . 5 ) r = 1 3 ψ n ( x r ) x r χ n ( x 4 ) x 4 = 1 4 1 1 sin ω 12 cos ω 34 ω 12 ω 34 d μ .
n = 0 ( n + 0 . 5 ) ψ n 2 ( β ) ψ n 2 ( α ) = α 128 β { ( 4 + 2 R ρ ) [ c i ( R ) c i ( ρ ) ] + R 2 ρ 2 + ( R 2 + 4 R ρ + ρ 2 ) [ s ( R ) s ( ρ ) ] + ( 3 R 2 + 4 R ρ ρ 2 ) c ( R ) ( R 2 + 4 R ρ + 3 ρ 2 ) c ( ρ ) } ,
n = 0 ( n + 0 . 5 ) ψ n ( β ) ψ n ( β ) ψ n ( α ) ψ n ( α ) = 1 128 { 4 [ c i ( R ) c i ( ρ ) ] + ( R 2 ρ 2 ) [ s ( R ) + s ( ρ ) ] + ( 3 R 2 + ρ 2 ) c ( R ) ( R 2 + 3 ρ 2 ) c ( ρ ) } ,
n = 0 ( n + 0 . 5 ) ψ n 2 ( β ) ψ n ( α ) χ n ( α ) = α 128 β { ( 4 + 2 R ρ ) [ s i ( R ) + s i ( ρ ) ] + ( R 2 + 4 R ρ + ρ 2 ) [ σ ( R ) + σ ( ρ ) ] + ( 3 R 2 + 4 R ρ ρ 2 ) γ ( R ) + ( R 2 + 4 R ρ + 3 ρ 2 ) γ ( ρ ) } ,
n = 0 ( n + 0 . 5 ) ψ n 2 ( β ) ψ n ( α ) χ n ( α ) = β 128 α { ( 4 2 R p ) [ s i ( R ) + s i ( ρ ) ] + ( R 2 4 R ρ + ρ 2 ) [ σ ( R ) + σ ( ρ ) ] + ( 3 R 2 4 R ρ ρ 2 ) γ ( R ) + ( R 2 4 R ρ + 3 p 2 ) γ ( ρ ) } ,
n = 0 ( n + 0 . 5 ) ψ n ( β ) ψ n ( β ) [ ψ n ( α ) χ n ( α ) + ψ n ( α ) χ n ( α ) ] = 1 64 { 4 [ s i ( R ) + s i ( ρ ) ] + ( R 2 ρ 2 ) [ σ ( R ) σ ( ρ ) ] + ( 3 R 2 + ρ 2 ) γ ( R ) + ( R 2 + 3 ρ 2 ) γ ( ρ ) } .
{ J ν ( ν / cosh γ ) ( 2 π ν tanh γ ) 1 / 2 exp [ ν ( γ tanh γ ) ] , Y ν ( ν / cosh γ ) ( 0 . 5 π ν tanh γ ) 1 / 2 exp [ ν ( γ tanh γ ) ] ,
H ( z ) = 2 z 0 1 ( 1 t 2 ) sin ( z t ) d t , H i ( z ) = 2 z 0 1 d t 0 1 u 2 ( 1 t 2 ) sin ( z t u ) d u , c i ( z ) = z 0 1 d t 0 1 sin ( z t u ) d u , }
h ( z ) = 2 z 0 1 ( 1 t 2 ) cos ( z t ) d t , h i ( z ) = 2 z 0 1 d t 0 1 u 2 ( 1 t 2 ) cos ( z t u ) d u , s i ( z ) = z 0 1 d t 0 1 cos ( z t u ) d u }
M f ( x ) = 0 f ( x ) x P 1 d x
M ( x sin x ) = p Γ c ( p ) ,
Γ c ( p ) = Γ ( p ) cos π p 2 .
M H ( x ) = 4 Γ c ( p ) p 2 , M H i ( x ) = 4 Γ c ( p ) ( p 2 ) 2 , M c i ( x ) = Γ c ( p ) p .
2 < Re p < 0 .
2 Γ c ( p ) Γ c ( 1 p ) = π
lim ρ 2 n ( p + 2 n ) Γ c ( p ) = ( 1 ) n ( 2 n ) ! , lim p 2 n 1 ( p + 2 n + 1 ) Γ c ( p ) = 0 .

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