Abstract

This paper is the second of two focusing on the analytical solutions for light transport in infinite homogeneous tissue with an azimuth-dependent (m-dependent) anisotropic scattering kernel by two approaches, Case’s singular eigenfuncions (CSEs) expansion and Fourier transform, and proving the consistence of the two solutions theoretically. In this paper, the analytical solution for the m-dependent truncated scattering kernel was derived via the Fourier transform and inversion, and expanded with the m-dependent generalized singular eigenfuncions (GSEs). Two kinds of GSEs that are defined by Ganapol in the case m=0 are extended to arbitrary azimuthal orders and proven to be consistent with CSEs both in expression forms and in intrinsic behaviors. By applying the Fourier transform inversion on the solution for the three-term recurrences, the Green’s function of radiance distributions is obtained successfully, and it conforms perfectly to the CSEs solution in the limit, which has already been discussed in our first accompanying paper. Meanwhile, as a byproduct, a series of identities about the m-dependent Chandrasekhar orthogonal polynomials were presented and will be greatly helpful for further studies.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Light and temperature distribution in laser irradiated tissue: the influence of anisotropic scattering and refractive index

Massoud Motamedi, Sohi Rastegar, Gerald LeCarpentier, and Ashley J. Welch
Appl. Opt. 28(12) 2230-2237 (1989)

References

  • View by:
  • |
  • |
  • |

  1. A. D. Kim and J. B. Keller, “Light propagation in biological tissue,” J. Opt. Soc. Am. A 20(1), 92–98 (2003).
    [Crossref] [PubMed]
  2. A. D. Kim, “Transport theory for light propagation in tissues,” in Biomedical Topical Meeting, OSA Technical Digest (Optical Society of America, 2004), paper SD4.
  3. L. V. Wang and H. I. Wu, Biomedical Optics: Principles and Imaging, (Wiley-Interscience, 2007).
  4. A. J. Welch and M. J. C. V. Gemert, Optical-Thermal Response of Laser-Irradiated Tissue, (Springer Netherlands, 2011).
  5. A. Liemert and A. Kienle, “Light transport in three-dimensional semi-infinite scattering media,” J. Opt. Soc. Am. A 29(7), 1475–1481 (2012).
    [Crossref] [PubMed]
  6. V. V. Tuchin, “Light scattering study of tissues,” Phys-USP 40(5), 495–515 (1997).
    [Crossref]
  7. M. Machida, “Singular eigenfunctions for the three-dimensional radiative transport equation,” J. Opt. Soc. Am. A 31(1), 67–74 (2014).
    [Crossref] [PubMed]
  8. A. D. Klose and E. W. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. 220(1), 441–470 (2006).
    [Crossref]
  9. P. González-Rodríguez and A. D. Kim, “Light propagation in tissues with forward-peaked and large-angle scattering,” Appl. Opt. 47(14), 2599–2609 (2008).
    [Crossref] [PubMed]
  10. K. M. Case and R. D. Hazeltine, “Fourier transform methods in linear transport theory,” J. Math. Phys. 12(9), 1970–1980 (1971).
    [Crossref]
  11. K. Kobayashi, H. Kikuchi, and K. Tsutsuguchi, “Solution of multigroup transport equation in x-y-z geometry by the spherical harmonics method using finite fourier transformation,” J. Nucl. Sci. Technol. 30(1), 31–47 (1993).
    [Crossref]
  12. B. D. Ganapol, “A consistent theory of neutral particle transport in an infinite medium,” Transp. Theory Stat. Phys. 29(1–2), 43–68 (2000).
    [Crossref]
  13. B. D. Ganapol, “Fourier transform transport solutions in spherical geometry,” Transp. Theory Stat. Phys. 32(5–7), 587–605 (2003).
    [Crossref]
  14. B. D. Ganapol, “Chandrasekhar polynomials and the solution to the transport equation in an infinite medium,” J. Comput. Theor. Trans. 41 (1–7), 433–473 (2014).
    [Crossref]
  15. B. D. Ganapol, “The Fourier transform solution for the Green’s function of monoenergetic neutron transport theory,” in 2014 Hawaii University International Conferences on Science, Technology, Engineering, Math & Education (2014).
  16. B. D. Ganapol, “The infinite medium Green’s function of monoenergetic neutron transport theory via Fourier transform,” Nucl. Sci. Eng. 180(2), 224–246 (2015).
    [Crossref]
  17. M. Machida, “The Green’s function for the three-dimensional linear Boltzmann equation via Fourier transform,” J. Phys. A Math. Theor. 49(17), 175001 (2016).
    [Crossref]
  18. Inönü and Erdal, “Orthogonality of a set of polynomials encountered in neutron transport and radiative transfer theories,” J. Math. Phys. 11(2), 568–577 (1970).
    [Crossref]
  19. R. D. M. Garcia and C. E. Siewert, “On the dispersion function in particle transport theory,” Zeitschrift Für Angewandte Mathematik Und Physik Zamp 33(6), 801–806 (1994).
    [Crossref]
  20. D. Zwillinger and V. Moll, Table of Integrals, Series, and Products (Eighth Edition), (Academic Press, 2015)
  21. Z. X. Wang and D. R. Guo, Special Functions, (World Scientific, 1989).
  22. S. J. Zhang and J. M. Jin, and R. E. Crandall, Computation of Special Functions (John Wiley & Sons Inc., 1996).
  23. N. I. Muskhelishvili, Singular Integral Equations: Boundary Problems of Function Theory and Their Application to Mathematical Physics (Dover Publications, 2011)
  24. E. M. Stein and R. Shakarchi, Complex Analysis (Princeton University Press, 2003)
  25. K. M. Case, “Scattering theory, orthogonal polynomials, and the transport equation,” J. Math. Phys. 15(7), 974–983 (1974).
    [Crossref]
  26. C. E. Siewert and J. R. T. Jr, “A particular solution for the PN, method in spherical geometry,” J. Quant. Spectrosc. Radiat. Transf. 46(5), 459–462 (1991).
    [Crossref]
  27. R. D. M. Garcia, “A PN particular solution for the radiative transfer equation in spherical geometry,” J. Quant. Spectrosc. Radiat. Transf. 196, 155–158 (2017).
    [Crossref]

2017 (1)

R. D. M. Garcia, “A PN particular solution for the radiative transfer equation in spherical geometry,” J. Quant. Spectrosc. Radiat. Transf. 196, 155–158 (2017).
[Crossref]

2016 (1)

M. Machida, “The Green’s function for the three-dimensional linear Boltzmann equation via Fourier transform,” J. Phys. A Math. Theor. 49(17), 175001 (2016).
[Crossref]

2015 (1)

B. D. Ganapol, “The infinite medium Green’s function of monoenergetic neutron transport theory via Fourier transform,” Nucl. Sci. Eng. 180(2), 224–246 (2015).
[Crossref]

2014 (2)

B. D. Ganapol, “Chandrasekhar polynomials and the solution to the transport equation in an infinite medium,” J. Comput. Theor. Trans. 41 (1–7), 433–473 (2014).
[Crossref]

M. Machida, “Singular eigenfunctions for the three-dimensional radiative transport equation,” J. Opt. Soc. Am. A 31(1), 67–74 (2014).
[Crossref] [PubMed]

2012 (1)

2008 (1)

2006 (1)

A. D. Klose and E. W. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. 220(1), 441–470 (2006).
[Crossref]

2003 (2)

A. D. Kim and J. B. Keller, “Light propagation in biological tissue,” J. Opt. Soc. Am. A 20(1), 92–98 (2003).
[Crossref] [PubMed]

B. D. Ganapol, “Fourier transform transport solutions in spherical geometry,” Transp. Theory Stat. Phys. 32(5–7), 587–605 (2003).
[Crossref]

2000 (1)

B. D. Ganapol, “A consistent theory of neutral particle transport in an infinite medium,” Transp. Theory Stat. Phys. 29(1–2), 43–68 (2000).
[Crossref]

1997 (1)

V. V. Tuchin, “Light scattering study of tissues,” Phys-USP 40(5), 495–515 (1997).
[Crossref]

1994 (1)

R. D. M. Garcia and C. E. Siewert, “On the dispersion function in particle transport theory,” Zeitschrift Für Angewandte Mathematik Und Physik Zamp 33(6), 801–806 (1994).
[Crossref]

1993 (1)

K. Kobayashi, H. Kikuchi, and K. Tsutsuguchi, “Solution of multigroup transport equation in x-y-z geometry by the spherical harmonics method using finite fourier transformation,” J. Nucl. Sci. Technol. 30(1), 31–47 (1993).
[Crossref]

1991 (1)

C. E. Siewert and J. R. T. Jr, “A particular solution for the PN, method in spherical geometry,” J. Quant. Spectrosc. Radiat. Transf. 46(5), 459–462 (1991).
[Crossref]

1974 (1)

K. M. Case, “Scattering theory, orthogonal polynomials, and the transport equation,” J. Math. Phys. 15(7), 974–983 (1974).
[Crossref]

1971 (1)

K. M. Case and R. D. Hazeltine, “Fourier transform methods in linear transport theory,” J. Math. Phys. 12(9), 1970–1980 (1971).
[Crossref]

1970 (1)

Inönü and Erdal, “Orthogonality of a set of polynomials encountered in neutron transport and radiative transfer theories,” J. Math. Phys. 11(2), 568–577 (1970).
[Crossref]

Case, K. M.

K. M. Case, “Scattering theory, orthogonal polynomials, and the transport equation,” J. Math. Phys. 15(7), 974–983 (1974).
[Crossref]

K. M. Case and R. D. Hazeltine, “Fourier transform methods in linear transport theory,” J. Math. Phys. 12(9), 1970–1980 (1971).
[Crossref]

Erdal,

Inönü and Erdal, “Orthogonality of a set of polynomials encountered in neutron transport and radiative transfer theories,” J. Math. Phys. 11(2), 568–577 (1970).
[Crossref]

Ganapol, B. D.

B. D. Ganapol, “The infinite medium Green’s function of monoenergetic neutron transport theory via Fourier transform,” Nucl. Sci. Eng. 180(2), 224–246 (2015).
[Crossref]

B. D. Ganapol, “Chandrasekhar polynomials and the solution to the transport equation in an infinite medium,” J. Comput. Theor. Trans. 41 (1–7), 433–473 (2014).
[Crossref]

B. D. Ganapol, “Fourier transform transport solutions in spherical geometry,” Transp. Theory Stat. Phys. 32(5–7), 587–605 (2003).
[Crossref]

B. D. Ganapol, “A consistent theory of neutral particle transport in an infinite medium,” Transp. Theory Stat. Phys. 29(1–2), 43–68 (2000).
[Crossref]

Garcia, R. D. M.

R. D. M. Garcia, “A PN particular solution for the radiative transfer equation in spherical geometry,” J. Quant. Spectrosc. Radiat. Transf. 196, 155–158 (2017).
[Crossref]

R. D. M. Garcia and C. E. Siewert, “On the dispersion function in particle transport theory,” Zeitschrift Für Angewandte Mathematik Und Physik Zamp 33(6), 801–806 (1994).
[Crossref]

González-Rodríguez, P.

Hazeltine, R. D.

K. M. Case and R. D. Hazeltine, “Fourier transform methods in linear transport theory,” J. Math. Phys. 12(9), 1970–1980 (1971).
[Crossref]

Inönü,

Inönü and Erdal, “Orthogonality of a set of polynomials encountered in neutron transport and radiative transfer theories,” J. Math. Phys. 11(2), 568–577 (1970).
[Crossref]

Jr, J. R. T.

C. E. Siewert and J. R. T. Jr, “A particular solution for the PN, method in spherical geometry,” J. Quant. Spectrosc. Radiat. Transf. 46(5), 459–462 (1991).
[Crossref]

Keller, J. B.

Kienle, A.

Kikuchi, H.

K. Kobayashi, H. Kikuchi, and K. Tsutsuguchi, “Solution of multigroup transport equation in x-y-z geometry by the spherical harmonics method using finite fourier transformation,” J. Nucl. Sci. Technol. 30(1), 31–47 (1993).
[Crossref]

Kim, A. D.

Klose, A. D.

A. D. Klose and E. W. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. 220(1), 441–470 (2006).
[Crossref]

Kobayashi, K.

K. Kobayashi, H. Kikuchi, and K. Tsutsuguchi, “Solution of multigroup transport equation in x-y-z geometry by the spherical harmonics method using finite fourier transformation,” J. Nucl. Sci. Technol. 30(1), 31–47 (1993).
[Crossref]

Larsen, E. W.

A. D. Klose and E. W. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. 220(1), 441–470 (2006).
[Crossref]

Liemert, A.

Machida, M.

M. Machida, “The Green’s function for the three-dimensional linear Boltzmann equation via Fourier transform,” J. Phys. A Math. Theor. 49(17), 175001 (2016).
[Crossref]

M. Machida, “Singular eigenfunctions for the three-dimensional radiative transport equation,” J. Opt. Soc. Am. A 31(1), 67–74 (2014).
[Crossref] [PubMed]

Siewert, C. E.

R. D. M. Garcia and C. E. Siewert, “On the dispersion function in particle transport theory,” Zeitschrift Für Angewandte Mathematik Und Physik Zamp 33(6), 801–806 (1994).
[Crossref]

C. E. Siewert and J. R. T. Jr, “A particular solution for the PN, method in spherical geometry,” J. Quant. Spectrosc. Radiat. Transf. 46(5), 459–462 (1991).
[Crossref]

Tsutsuguchi, K.

K. Kobayashi, H. Kikuchi, and K. Tsutsuguchi, “Solution of multigroup transport equation in x-y-z geometry by the spherical harmonics method using finite fourier transformation,” J. Nucl. Sci. Technol. 30(1), 31–47 (1993).
[Crossref]

Tuchin, V. V.

V. V. Tuchin, “Light scattering study of tissues,” Phys-USP 40(5), 495–515 (1997).
[Crossref]

Appl. Opt. (1)

J. Comput. Phys. (1)

A. D. Klose and E. W. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. 220(1), 441–470 (2006).
[Crossref]

J. Comput. Theor. Trans. (1)

B. D. Ganapol, “Chandrasekhar polynomials and the solution to the transport equation in an infinite medium,” J. Comput. Theor. Trans. 41 (1–7), 433–473 (2014).
[Crossref]

J. Math. Phys. (3)

K. M. Case and R. D. Hazeltine, “Fourier transform methods in linear transport theory,” J. Math. Phys. 12(9), 1970–1980 (1971).
[Crossref]

Inönü and Erdal, “Orthogonality of a set of polynomials encountered in neutron transport and radiative transfer theories,” J. Math. Phys. 11(2), 568–577 (1970).
[Crossref]

K. M. Case, “Scattering theory, orthogonal polynomials, and the transport equation,” J. Math. Phys. 15(7), 974–983 (1974).
[Crossref]

J. Nucl. Sci. Technol. (1)

K. Kobayashi, H. Kikuchi, and K. Tsutsuguchi, “Solution of multigroup transport equation in x-y-z geometry by the spherical harmonics method using finite fourier transformation,” J. Nucl. Sci. Technol. 30(1), 31–47 (1993).
[Crossref]

J. Opt. Soc. Am. A (3)

J. Phys. A Math. Theor. (1)

M. Machida, “The Green’s function for the three-dimensional linear Boltzmann equation via Fourier transform,” J. Phys. A Math. Theor. 49(17), 175001 (2016).
[Crossref]

J. Quant. Spectrosc. Radiat. Transf. (2)

C. E. Siewert and J. R. T. Jr, “A particular solution for the PN, method in spherical geometry,” J. Quant. Spectrosc. Radiat. Transf. 46(5), 459–462 (1991).
[Crossref]

R. D. M. Garcia, “A PN particular solution for the radiative transfer equation in spherical geometry,” J. Quant. Spectrosc. Radiat. Transf. 196, 155–158 (2017).
[Crossref]

Nucl. Sci. Eng. (1)

B. D. Ganapol, “The infinite medium Green’s function of monoenergetic neutron transport theory via Fourier transform,” Nucl. Sci. Eng. 180(2), 224–246 (2015).
[Crossref]

Phys-USP (1)

V. V. Tuchin, “Light scattering study of tissues,” Phys-USP 40(5), 495–515 (1997).
[Crossref]

Transp. Theory Stat. Phys. (2)

B. D. Ganapol, “A consistent theory of neutral particle transport in an infinite medium,” Transp. Theory Stat. Phys. 29(1–2), 43–68 (2000).
[Crossref]

B. D. Ganapol, “Fourier transform transport solutions in spherical geometry,” Transp. Theory Stat. Phys. 32(5–7), 587–605 (2003).
[Crossref]

Zeitschrift Für Angewandte Mathematik Und Physik Zamp (1)

R. D. M. Garcia and C. E. Siewert, “On the dispersion function in particle transport theory,” Zeitschrift Für Angewandte Mathematik Und Physik Zamp 33(6), 801–806 (1994).
[Crossref]

Other (9)

D. Zwillinger and V. Moll, Table of Integrals, Series, and Products (Eighth Edition), (Academic Press, 2015)

Z. X. Wang and D. R. Guo, Special Functions, (World Scientific, 1989).

S. J. Zhang and J. M. Jin, and R. E. Crandall, Computation of Special Functions (John Wiley & Sons Inc., 1996).

N. I. Muskhelishvili, Singular Integral Equations: Boundary Problems of Function Theory and Their Application to Mathematical Physics (Dover Publications, 2011)

E. M. Stein and R. Shakarchi, Complex Analysis (Princeton University Press, 2003)

B. D. Ganapol, “The Fourier transform solution for the Green’s function of monoenergetic neutron transport theory,” in 2014 Hawaii University International Conferences on Science, Technology, Engineering, Math & Education (2014).

A. D. Kim, “Transport theory for light propagation in tissues,” in Biomedical Topical Meeting, OSA Technical Digest (Optical Society of America, 2004), paper SD4.

L. V. Wang and H. I. Wu, Biomedical Optics: Principles and Imaging, (Wiley-Interscience, 2007).

A. J. Welch and M. J. C. V. Gemert, Optical-Thermal Response of Laser-Irradiated Tissue, (Springer Netherlands, 2011).

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 (1)

Fig. 1
Fig. 1 Complex z-plane and k-plane

Equations (142)

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

P l m ( μ )= ( 1 ) m ( 1 μ 2 ) m/2 d m P l ( μ ) d μ m
μ( 2l+1 ) P l m ( μ )( lm+1 ) P l+1 m ( μ )( l+m ) P l1 m ( μ )=0
P l m ( z )= ( z 2 1 ) m/2 d m P l ( z ) d z m
P m m ( z )=( 2m1 )!! ( z 2 1 ) m/2 , P m+1 m ( z )=( 2m+1 )z P m m ( z )
Q l ( z )= 1 2 1 1 P l ( v ) zv dv
Q l m ( z )= ( 1 ) m 2 ( z 2 1 ) m/2 1 1 ( 1 μ 2 ) m/2 P l m ( μ ) zμ dμ
z( 2l+1 ) Q l m ( z )( lm+1 ) Q l+1 m ( z )( l+m ) Q l1 m ( z )= ( 2m )! ( 2m1 )!! ( z 2 1 ) m/2 δ lm
Q m m ( z )= [ ( 2m1 )!! ] 2 2 P m m ( z ) 1 1 ( 1 μ 2 ) m zμ dμ, Q m+1 m ( z )=( 2m+1 )z Q m m ( z ) ( 2m )! P m m ( z )
z h l g l m ( z )( lm+1 ) g l+1 m ( z )( l+m ) g l1 m ( z )=0
z h l ρ l m ( z )( lm+1 ) ρ l+1 m ( z )( l+m ) ρ l1 m ( z )= ρ m+1 m ( z ) δ lm
g m m ( z )= ( 2m )! 2 m m! =( 2m1 )!!, g m+1 m =z h m g m m ( z )
ρ m m ( z )=0, ρ m+1 m = z ( 2m1 )!!
l=m N ( lm )! ( l+m )! ( 2l+1 ) g l m ( z ) P l m ( μ ) = ϖz zμ l=m N ( lm )! ( l+m )! ( 2l+1 ) f l g l m ( z ) P l m ( μ ) + ( N+1m )! ( N+m )! g N+1 m ( z ) P N m ( μ ) g N m ( z ) P N+1 m ( μ ) zμ
l=m N ( lm )! ( l+m )! ( 2l+1 ) ρ l m ( z ) P l m ( μ ) = ϖz zμ l=m N ( lm )! ( l+m )! ( 2l+1 ) f l ρ l m ( z ) P l m ( μ ) + ( N+1m )! ( N+m )! ρ N+1 m ( z ) P N m ( μ ) ρ N m ( z ) P N+1 m ( μ ) zμ z zμ ( 1 μ 2 ) m/2 ( 2m )!
ϖz l=m L ( lm )! ( l+m )! ( 2l+1 ) f l g l m ( z ) P l m ( z ) = ( L+1m )! ( L+m )! [ g L m ( z ) P L+1 m ( z ) g L+1 m ( z ) P L m ( z ) ]
ϖz l=m L ( lm )! ( l+m )! ( 2l+1 ) f l g l m ( z ) Q l m ( z ) = ( L+1m )! ( L+m )! [ g L m ( z ) Q L+1 m ( z ) g L+1 m ( z ) Q L m ( z ) ] + ( z 2 1 ) m/2
ϖz l=m L ( lm )! ( l+m )! ( 2l+1 ) f l ρ l m ( z ) P l m ( z ) = ( L+1m )! ( L+m )! [ ρ L m ( z ) P L+1 m ( z ) ρ L+1 m ( z ) P L m ( z ) ] + z ( 1 z 2 ) m/2 ( 2m )!
ϖz l=m L ( lm )! ( l+m )! ( 2l+1 ) f l ρ l m ( z ) Q l m ( z ) = ( L+1m )! ( L+m )! [ ρ L m ( z ) Q L+1 m ( z ) ρ L+1 m ( z ) Q L m ( z ) ] + ρ m+1 m ( z ) Q m m ( z ) ( 2m )!
l=m N ( lm )! ( l+m )! 2l+1 2 P l m ( z ) P l m ( μ ) = 1 2 ( N+1m )! ( N+m )! P N+1 m ( z ) P N m ( μ ) P N m ( z ) P N+1 m ( μ ) zμ
l=m N ( lm )! ( l+m )! 2l+1 2 Q l m ( z ) P l m ( μ ) = 1 2 ( N+1m )! ( N+m )! Q N+1 m ( z ) P N m ( μ ) Q N m ( z ) P N+1 m ( μ ) zμ + 1 2 ( 1 μ 2 ) m/2 ( z 2 1 ) m/2 zμ
g l1 m ( z ) ρ l m ( z ) g l m ( z ) ρ l1 m ( z )= ( l+m1 )! ( lm )! z ( 2m )!
P l+1 m ( z ) Q l m ( z ) P l m ( z ) Q l+1 m ( z )= ( l+m )! ( l+1m )!
μ ψ m ( τ,μ; μ 0 ) τ + ψ m ( τ,μ; μ 0 )= ϖ 2 l=| m | L β l m P l m ( μ ) 1 +1 P l m ( μ ) ψ m ( τ, μ ; μ 0 )d μ +δ( τ )δ( μ μ 0 ) e im φ 0
ψ ˜ m ( k,μ; μ 0 )= ψ m ( τ,μ; μ 0 ) e ikτ dτ
ψ m ( τ,μ; μ 0 )= 1 2π ψ ˜ m ( k,μ; μ 0 ) e ikτ dk
( 1+ikμ ) ψ ˜ m ( k,μ; μ 0 )= ϖ 2 l=m L β l m P l m ( μ ) 1 +1 P l m ( μ ) ψ ˜ m ( k, μ ; μ 0 )d μ +δ( μ μ 0 ) e im φ 0
ψ ˜ m ( k,μ; μ 0 )= l=m ( lm )! ( l+m )! 2l+1 2 ψ ˜ l m ( k; μ 0 ) P l m ( μ )
ψ ˜ j m ( k; μ 0 )=ϖ l=m L β l m L j,l m ( z ) ψ ˜ l m ( k; μ 0 ) + z z μ 0 P j m ( μ 0 ) e im φ 0
L j,l m ( z )= z 2 1 1 P j m ( μ ) P l m ( μ ) zμ dμ ={ z P l m ( z ) Q j m ( z ),lj z Q l m ( z ) P j m ( z ),l>j
l=m L [ δ jl ϖ β l m L j,l m ( z ) ] ψ ˜ L,l m ( k; μ 0 )= z z μ 0 P j m ( μ 0 ) e im φ 0
[ ψ ˜ m ] l = ψ ˜ L,l m ( k; μ 0 )
[ L m ] j,l = L j,l m ( z )
[ β m ] j,l = β l m δ jl
[ P m ] l = P l m ( μ 0 )
[ Iϖ L m β m ] ψ ˜ m = z z μ 0 e im φ 0 P m
ψ ˜ m = z z μ 0 e im φ 0 [ Iϖ L m β m ] 1 P m
z h l ψ ˜ l m ( k; μ 0 )( lm+1 ) ψ ˜ l+1 m ( k; μ 0 )( l+m ) ψ ˜ l1 m ( k; μ 0 )=z S l m ( μ 0 )
ψ ˜ l m ( k; μ 0 )= a m ( z; μ 0 ) g l m ( z )+ b m ( z; μ 0 ) ρ l m ( z )+z j=m l η l,j m ( z ) S j ( μ 0 )
ψ ˜ l m ( k; μ 0 )= ψ ˜ m m ( k; μ 0 ) g m m ( z ) g l m ( z )+z j=m l η l,j m ( z ) S j ( μ 0 )
z h l η l,j m ( z )( lm+1 ) η l+1,j m ( z )( l+m ) η l1,j m ( z )=0
η l,l m =0
η l+1,l m ( z )=1/ ( lm+1 )
η l,j m ( z )= c j m ( z ) g l m ( z )+ d j m ( z ) ρ l m ( z )
c j m ( z )= ρ j m ( z ) jm+1 1 ρ j m ( z ) g j+1 m ( z ) g j m ( z ) ρ j+1 m ( z )
d j m ( z )= g j m ( z ) jm+1 1 g j m ( z ) ρ j+1 m ( z ) ρ j m ( z ) g j+1 m ( z )
η l,j m ( z )= ( jm )! ( j+m )! ( 2m )! z [ ρ j m ( z ) g l m ( z ) g j m ( z ) ρ l m ( z ) ]
ψ ˜ l m ( k; μ 0 )= ψ ˜ m m ( k; μ 0 ) g m m ( z ) g l m ( z )+ χ l m ( z, μ 0 )
χ l m ( z,μ )( 2m )! j=m l ( jm )! ( j+m )! [ ρ j m ( z ) g l m ( z ) g j m ( z ) ρ l m ( z ) ] S j ( μ )
L m,l m ( z )= z 2 1 1 P m m ( μ ) P l m ( μ ) zμ dμ =z P m m ( z ) Q l m ( z )
ψ ˜ m m ( k; μ 0 )= 1 Λ * m ( z ) [ z z μ 0 P m m ( μ 0 ) e im φ 0 +ϖz P m m ( z ) l=m L β l m Q l m ( z ) χ l m ( z, μ 0 ) ]
Λ * m ( z )1ϖz P m m ( z ) g m m ( z ) l=m L β l m g l m ( z ) Q l m ( z )
Λ m ( z )1 ϖz 2 1 1 g( z,μ ) zμ ( 1 μ 2 ) m/2 dμ
g m ( z,μ ) l=m L β l m g l m ( z ) P l m ( μ )
ϕ L m ( z,μ ) l=m ( lm )! ( l+m )! 2l+1 2 g l m ( z ) P l m ( μ )
ϕ L m ( z,μ;N ) l=m N ( lm )! ( l+m )! 2l+1 2 g l m ( z ) P l m ( μ )
ϕ L m ( z,μ;N )= ϖz 2 φ N m ( z,μ ) zμ + 1 2 ( Nm+1 )! ( N+m )! g N+1 m ( z ) P N m ( μ ) g N m ( z ) P N+1 m ( μ ) zμ
φ N m ( z,μ ) l=m N ( lm )! ( l+m )! ( 2l+1 ) f l g l m ( z ) P l m ( μ )
φ N m ( z,μ )= φ L m ( z,μ )= l=m L ( lm )! ( l+m )! ( 2l+1 ) f l g l m ( z ) P l m ( μ )
z( 2l+1 ) g l m ( z )( lm+1 ) g l+1 m ( z )( l+m ) g l1 m ( z )=0
g l m ( z )=α( z ) P l m ( z )+β( z ) Q l m ( z )
{ g L m ( z )=α( z ) P L m ( z )+β( z ) Q L m ( z ) g L+1 m ( z )=α( z ) P L+1 m ( z )+β( z ) Q L+1 m ( z )
{ α( z )= Λ L m ( z )= ( Lm+1 )! ( L+m )! [ g L+1 m ( z ) Q L m ( z ) g L m ( z ) Q L+1 m ( z ) ] β( z )= ψ L m ( z )= ( Lm+1 )! ( L+m )! [ g L+1 m ( z ) P L m ( z ) g L m ( z ) P L+1 m ( z ) ]
g l m ( z )= Λ L m ( z ) P l m ( z )+ ψ L m ( z ) Q l m ( z )
g l m ( z )= ( Lm+1 )! ( L+m )! { g L m ( z )[ P L+1 m ( z ) Q l m ( z ) Q L+1 m ( z ) P l m ( z ) ] + g L+1 m ( z )[ P l m ( z ) Q L m ( z ) P L m ( z ) Q l m ( z ) ] }
P l m ( z ) Q L m ( z ) P L m ( z ) Q l m ( z ) = ( 1 ) m 2( 2m1 )!! 1 1 [ d m P l ( z ) d z m P L m ( μ ) d m P L ( z ) d z m P l m ( μ ) ] P m m ( μ ) zμ dμ
ϕ L m ( z,μ;N )= ϖz 2 φ L m ( z,μ ) zμ + Λ L m ( z ) δ N m ( z,μ )+ ψ L m ( z ) q N m ( z,μ )
δ N m ( z,μ )= 1 2 ( N+1m )! ( N+m )! P N+1 m ( z ) P N m ( μ ) P N m ( z ) P N+1 m ( μ ) zμ
q N m ( z,μ )= 1 2 ( N+1m )! ( N+m )! Q N+1 m ( z ) P N m ( μ ) Q N m ( z ) P N+1 m ( μ ) zμ
δ( vμ )= l=m ( lm )! ( l+m )! 2l+1 2 P l m ( v ) P l m ( μ )
δ N m ( v,μ )= l=m N ( lm )! ( l+m )! 2l+1 2 P l m ( v ) P l m ( μ )
lim N δ N m ( v,μ )=δ( vμ )
1 2 ( 1 μ 2 ) m/2 ( v 2 1 ) m/2 vμ = l=m ( lm )! ( lm )! 2l+1 2 Q l m ( v ) P l m ( μ )
q N m ( v,μ )= l=m N ( lm )! ( l+m )! 2l+1 2 Q l m ( v ) P l m ( μ ) 1 2 ( 1 μ 2 ) m/2 ( v 2 1 ) m/2 vμ
lim N q N m ( v,μ )=0
ϕ L m ( v,μ )= lim N ϕ L m ( v,μ )= ϖv 2 φ L m ( v,μ ) vμ
[ ϕ L m ( v,μ;N ) ] ± = ϖv 2 φ L m ( v,μ ) [ 1 zμ ] ± + [ Λ L m ( v ) ] ± [ δ N m ( v,μ ) ] ± + ψ L m ( v ) [ q N m ( v,μ ) ] ±
[ 1 zμ ] ± =P 1 vμ iπδ( vμ )
[ Q l m ( v ) ] ± =P Q l m ( v ) iπ 2 P l m ( v )
[ Λ L m ( v ) ] ± = ( Lm+1 )! ( L+m )! { g L+1 m ( v ) [ Q L m ( v ) ] ± g L m ( v ) [ Q L+1 m ( v ) ] ± } =P Λ L m ( v )± iπ 2 ψ L m ( v )
[ δ N m ( v,μ ) ] ± = 1 2 ( N+1m )! ( N+m )! [ P N+1 m ( v ) P N m ( μ ) P N m ( v ) P N+1 m ( μ ) ] [ 1 zμ ] ± =P δ N m ( v,μ )
[ q N m ( v,μ ) ] ± = 1 2 ( N+1m )! ( N+m )! { [ Q N+1 m ( v ) ] ± P N m ( μ ) [ Q N m ( v ) ] ± P N+1 m ( μ ) } [ 1 zμ ] ± =P q N m ( v,μ ) iπ 2 [ P δ N m ( v,μ )δ( vμ ) ]
[ ϕ L m ( v,μ;N ) ] ± = ϖv 2 φ L m ( v,μ )P 1 vμ +P Λ L m ( v )P δ N m ( v,μ )+ ψ L m ( v )P q N m ( v,μ ) iπ ϖv 2 φ L m ( v,v )δ( vμ )± iπ 2 ψ L m ( v )P δ N m ( v,μ ) iπ 2 ψ L m ( v )[ P δ N m ( v,μ )δ( vμ ) ]
ϖv φ L m ( v,v )= ψ L m ( v )
[ ϕ L m ( v,μ;N ) ] ± = ϖv 2 φ L m ( v,μ )P 1 vμ +P Λ L m ( v )P δ N m ( v,μ )+ ψ L m ( v )P q N m ( v,μ )
ϕ L m ( v,μ )= ϖv 2 φ L m ( v,μ )P 1 vμ +P Λ L m ( v )δ( vμ )
λ( v )=1 ϖv 2 P 1 1 g m ( v,μ ) vμ ( 1 μ 2 ) m/2 dμ =1 ϖv 2 l=m L ( lm )! ( l+m )! ( 2l+1 ) f l g l m ( v ) P 1 1 P l m ( μ ) ( 1 μ 2 ) m/2 vμ dμ =1ϖv l=m L ( lm )! ( l+m )! ( 2l+1 ) f l g l m ( v )P[ ( 1 v 2 ) m/2 Q l m ( v ) ]
Λ L m ( z )= ( z 2 1 ) m/2 ϖz l=m L ( lm )! ( l+m )! ( 2l+1 ) f l g l m ( z ) Q l m ( z )
P Λ L m ( v )= λ( v ) ( 1 v 2 ) m/2
Θ L m ( z,μ ) l=m ( lm )! ( l+m )! 2l+1 2 ρ l m ( z ) P l m ( μ )
Θ L m ( z,μ;N ) l=m N ( lm )! ( l+m )! 2l+1 2 ρ l m ( z ) P l m ( μ )
Θ L m ( z,μ;N )= 1 2 z zμ [ ϖ h L m ( z,μ ) ( 1 μ 2 ) m/2 ( 2m )! ] 1 2 1 zμ ( N+1m )! ( N+m )! [ ρ N m ( z ) P N+1 m ( μ ) ρ N+1 m ( z ) P N m ( μ ) ]
h N m ( z,μ ) h L m ( z,μ )= l=m L ( lm )! ( l+m )! ( 2l+1 ) f l ρ l m ( z ) P l m ( μ )
ρ l m ( z )= γ L m ( z ) P l m ( z )+ θ L m ( z ) Q l m ( z )
{ γ L m ( z )= ( Lm+1 )! ( L+m )! [ ρ L+1 m ( z ) Q L m ( z ) ρ L m ( z ) Q L+1 m ( z ) ] θ L m ( z )= ( Lm+1 )! ( L+m )! [ ρ L+1 m ( z ) P L m ( z ) ρ L m ( z ) P L+1 m ( z ) ]
Θ L m ( z,μ;N )= 1 2 z zμ [ ϖ h L m ( z,μ ) ( 1 μ 2 ) m/2 ( 2m )! ]+ γ L m ( z ) δ N m ( z,μ )+ θ L m ( z ) q N m ( z,μ )
Θ L m ( v,μ )= 1 2 v vμ [ ϖ h L m ( v,μ ) ( 1 μ 2 ) m/2 ( 2m )! ]
ϖv h L m ( v,v ) v ( 1 v 2 ) m/2 ( 2m )! = θ L m ( v )
[ Θ L m ( v,μ;N ) ] ± = v 2 [ ϖ h L m ( v,μ ) ( 1 μ 2 ) m/2 ( 2m )! ]P 1 vμ +P γ L m ( v )P δ N m ( v,μ )+ θ L m ( v )P q N m ( v,μ )
Θ L m ( v,μ )= v 2 [ ϖ h L m ( v,μ ) ( 1 μ 2 ) m/2 ( 2m )! ]P 1 vμ +P γ L m ( v )δ( vμ )
Λ L m ( z ) θ L m ( z ) ψ L m ( z ) γ L m ( z )= z ( 2m )!
Λ L m ( z )[ ϖ h L m ( z,z ) ( 1 z 2 ) m/2 ( 2m )! ]ϖ φ N m ( z,z ) γ L m ( z )= 1 ( 2m )!
γ L m ( z ) Λ L m ( z ) = 1 ϖ φ L m ( z,z ) [ 1 ( 2m )! 1 Λ L m ( z ) +ϖ h L m ( z,z ) ( 1 z 2 ) m/2 ( 2m )! ]
ψ ˜ m ( k,μ; μ 0 )= l=m ( lm )! ( l+m )! 2l+1 2 ψ ˜ l m ( k; μ 0 ) P l m ( μ ) = ϕ L m ( z,μ ) ψ ˜ m m ( k; μ 0 ) g m m ( z ) +T( z,μ; μ 0 )
T( z,μ; μ 0 ) l=m ( lm )! ( l+m )! 2l+1 2 χ l m ( z, μ 0 ) P l m ( μ )
ψ ˜ L m ( k,μ; μ 0 ;N )= ϕ L m ( z,μ;N ) ψ ˜ m m ( k; μ 0 ) g m m ( z ) +T( z,μ; μ 0 ;N )
T( z,μ; μ 0 ;N ) l=m N ( lm )! ( l+m )! 2l+1 2 χ l m ( z, μ 0 ) P l m ( μ )
S( z, μ 0 )ϖz P m m ( z ) l=m L β l m Q l m ( z ) χ l m ( z, μ 0 )
S( z, μ 0 )=( 2m )!ϖz P m m ( z ) × l=m L β l m Q l m ( z ) j=m l ( jm )! ( j+m )! [ ρ j m ( z ) g l m ( z ) g j m ( z ) ρ l m ( z ) ] S j ( μ 0 )
S( z, μ 0 )=( 2m )! P m m ( z ) j=m L ( jm )! ( j+m )! S j ( μ 0 ) ×{ ρ j m ( z )[ ϖz l=m L β l m g l m ( z ) Q l m ( z ) ϖz l=m j1 β l m g l m ( z ) Q l m ( z ) ] g j m ( z )[ ϖz l=m L β l m ρ l m ( z ) Q l m ( z ) ϖz l=m j1 β l m ρ l m ( z ) Q l m ( z ) ] }
S( z, μ 0 )=( 2m )! P m m ( z ) × j=m L ( jm )! ( j+m )! { ρ j m ( z )[ Λ j1 m ( z ) Λ L m ( z ) ] g j m ( z )[ γ j1 m ( z ) γ L m ( z ) ] } S j ( μ 0 )
S( z, μ 0 )=( 2m )! P m m ( z ) e im φ 0 ×{ j=m L ( jm )! ( j+m )! ( 2j+1 ) P j m ( μ 0 )[ ρ j m ( z ) Λ j1 m ( z ) g j m ( z ) γ j1 m ( z ) ] Λ L m ( z ) j=m L ( jm )! ( j+m )! ( 2j+1 ) ρ j m ( z ) P j m ( μ 0 ) + γ L m ( z ) j=m L ( jm )! ( j+m )! ( 2j+1 ) g j m ( z ) P j m ( μ 0 ) }
ρ j m ( z ) Λ j1 m ( z ) g j m ( z ) γ j1 m ( z )= z ( 2m )! Q j m ( z )
j=m L ( jm )! ( j+m )! ( 2j+1 ) P j m ( μ 0 )[ ρ j m ( z ) Λ j1 m ( z ) g j m ( z ) γ j1 m ( z ) ] = 2z ( 2m )! j=m L ( jm )! ( j+m )! 2j+1 2 P j m ( μ 0 ) Q j m ( z ) = 2z ( 2m )! [ q L m ( z, μ 0 )+ 1 2 ( 1 μ 0 2 ) m/2 ( z 2 1 ) m/2 z μ 0 ]
ϕ L ( z,μ;L )= ϕ L ( z,μ;N ) + Λ L m ( z )[ δ L m ( z,μ ) δ N m ( z,μ ) ]+ ψ L m ( z )[ q L m ( z,μ ) q N m ( z,μ ) ]
Θ L ( z,μ;L )= Θ L ( z,μ;N ) + γ L m ( z )[ δ L m ( z,μ ) δ N m ( z,μ ) ]+ θ L m ( z )[ q L m ( z,μ ) q N m ( z,μ ) ]
S( z, μ 0 )=( 2m )! P m m ( z ) e im φ 0 ×{ 2[ γ L m ( z ) ϕ L m ( z, μ 0 ;N ) Λ L m ( z ) Θ L m ( z, μ 0 ;N ) z ( 2m )! q N m ( z, μ 0 ) ] z ( 2m )! ( 1 μ 0 2 ) m/2 ( z 2 1 ) m/2 z μ 0 }
ψ ˜ m m ( k; μ 0 )= 2( 2m )! P m m ( z ) e im φ 0 Λ * m ( z ) ×[ γ L m ( z ) ϕ L m ( z, μ 0 ;N ) Λ L m ( z ) Θ L m ( z, μ 0 ;N ) z ( 2m )! q N m ( z, μ 0 ) ]
Λ * m ( z )= Λ L m ( z ) ( z 2 1 ) m/2
ψ ˜ m m ( k; μ 0 )=2( 2m )!( 2m1 )!! e im φ 0 ×[ γ L m ( z ) Λ L m ( z ) ϕ L m ( z, μ 0 ;N ) Θ L m ( z, μ 0 ;N ) 1 Λ L m ( z ) z ( 2m )! q N m ( z, μ 0 ) ]
ψ ˜ L m ( k,μ; μ 0 ;N )=2( 2m )! e im φ 0 ×{ ϕ L m ( z, μ 0 ;N ) ϕ L m ( z,μ;N ) ϖ φ L m ( z,z ) [ 1 ( 2m )! 1 Λ L m ( z ) +ϖ h L m ( z,z ) ( 1 z 2 ) m/2 ( 2m )! ] 1 Λ L m ( z ) z ( 2m )! q N m ( z, μ 0 ) ϕ L m ( z,μ;N ) } H( z,μ; μ 0 ;N )
H( z,μ; μ 0 ;N )2( 2m )! e im φ 0 ϕ L m ( z,μ;N ) Θ L m ( z, μ 0 ;N )T( z,μ; μ 0 ;N )
I L m ( τ,μ; μ 0 ;N )= lim R ε0 1 2π C ψ ˜ L m ( k,μ; μ 0 ;N ) e ikτ dk
C Γ R + C R + + Γ ε + + C ε + Γ ε ++ C R
ψ m ( τ,μ; μ 0 ;N )= I L m ( τ,μ; μ 0 ;N ) lim R ε0 1 2π C Γ R ψ ˜ L m ( k,μ; μ 0 ;N ) e ikτ dk
Λ L m ( ± v j m )=0
I L m ( τ,μ; μ 0 ;N )=2πi j=1 M L m Res[ 1 2π ψ ˜ L m ( k,μ; μ 0 ;N );k= k j m ] e i k j m τ
I L,j m ( τ,μ; μ 0 ;N )=Res[ ψ ˜ L m ( k,μ; μ 0 ;N );k= k j m ] e i k j m τ = lim k k j m ( k k j m ) ψ ˜ L m ( k,μ; μ 0 ;N ) e i k j m τ
I L,j m ( τ,μ; μ 0 ;N )=i ϕ L m ( v j m , μ 0 ;N ) ϕ L m ( v j m ,μ;N ) N ˜ 0 m ( v j m ) e im φ 0 e τ/ v j m +i q N m ( v j m , μ 0 ) ϕ L m ( v j m ,μ;N ) E 0 m ( v j m ) e im φ 0 e τ/ v j m
N ˜ 0 m ( v j m )= 1 2 ϖ ( v j m ) 2 φ L m ( v j m , v j m ) d Λ L m ( z ) dz | z= v j m
E 0 m ( v j m )= 1 2 v j m d Λ L m ( z ) dz | z= v j m
I L m ( τ,μ; μ 0 ;N )= j=1 M L m ϕ L m ( v j m , μ 0 ;N ) ϕ L m ( v j m ,μ;N ) N ˜ 0 m ( v j m ) e im φ 0 e τ/ v j m j=1 M L m q N m ( v j m , μ 0 ) ϕ L m ( v j m ,μ;N ) E 0 m ( v j m ) e im φ 0 e τ/ v j m
z ± = i k ± = y±iε ε 2 + y 2 =v±iε+O( ε 2 ),v[ 0,1 ]
I L,branch m ( τ,μ; μ 0 ;N )= i 2π lim ε0 { iε 1+iε 1 z + 2 ψ ˜ L m ( z + ,μ; μ 0 ;N ) e τ/ z + d z + iε 1iε 1 z 2 ψ ˜ L m ( z ,μ; μ 0 ;N ) e τ/ z d z } = i 2π 0 1 e τ/v v 2 [ ψ ˜ L m ( z + ,μ; μ 0 ;N ) ψ ˜ L m ( z + ,μ; μ 0 ;N ) ]dv
ψ ˜ L m ( z + ,μ; μ 0 ;N ) ψ ˜ L m ( z + ,μ; μ 0 ;N ) =2 e im φ 0 ϕ L m ( v, μ 0 ;N ) ϕ L m ( v,μ;N ) ϖ φ L m ( v,v ) { 1 [ Λ L m ( v ) ] + 1 [ Λ L m ( v ) ] } 2 e im φ 0 v ϕ L m ( v,μ;N ){ [ q N m ( v, μ 0 ) ] + [ Λ L m ( v ) ] + [ q N m ( v, μ 0 ) ] [ Λ L m ( v ) ] }
1 [ Λ L m ( v ) ] + 1 [ Λ L m ( v ) ] = iπϖv φ N m ( v,v ) [ Λ L m ( v ) ] + [ Λ L m ( v ) ]
[ q N m ( v, μ 0 ) ] + [ Λ L m ( v ) ] + [ q N m ( v, μ 0 ) ] [ Λ L m ( v ) ] = iπ F L m ( v,μ ) [ Λ L m ( v ) ] + [ Λ L m ( v ) ]
F L m ( v,μ )=[ P δ N m ( v,μ )δ( vμ ) ]P Λ L m ( v ) ψ L m ( v )P q N m ( v,μ )
ψ ˜ L m ( z + ,μ; μ 0 ;N ) ψ ˜ L m ( z + ,μ; μ 0 ;N ) = 2iπv ϕ L m ( v, μ 0 ;N ) ϕ L m ( v,μ;N ) [ Λ L m ( v ) ] + [ Λ L m ( v ) ] e im φ 0 2iπv ϕ L m ( v,μ;N ) F L m ( v,μ ) [ Λ L m ( v ) ] + [ Λ L m ( v ) ] e im φ 0
I L,branch m ( τ,μ; μ 0 ;N ) = 0 1 [ ϕ L m ( v, μ 0 ;N ) ϕ L m ( v,μ;N ) N ˜ m ( v ) + 2iπv ϕ L m ( v,μ;N ) F L m ( v,μ ) v N ˜ m ( v ) ] e τ/v e im φ 0 dv
N ˜ m ( v )=v [ Λ L m ( v ) ] + [ Λ L m ( v ) ]
ψ m ( τ,μ; μ 0 ;N )= j=1 M L m ϕ L m ( v j m , μ 0 ;N ) ϕ L m ( v j m ,μ;N ) N ˜ 0 m ( v j m ) e τ/ v j m e im φ 0 j=1 M L m q N m ( v j m , μ 0 ) ϕ L m ( v j m ,μ;N ) E 0 m ( v j m ) e τ/ v j m e im φ 0 + 0 1 [ ϕ L m ( v, μ 0 ;N ) ϕ L m ( v,μ;N ) N ˜ m ( v ) + 2iπv ϕ L m ( v,μ;N ) F L m ( v,μ ) v N ˜ m ( v ) ] e τ/v e im φ 0 dv
ψ m ( τ,μ; μ 0 )= j=1 M L m ϕ L m ( v j m , μ 0 ) ϕ L m ( v j m ,μ ) N ˜ 0 m ( v j m ) e τ/ v j m e im φ 0 + 0 1 ϕ L m ( v, μ 0 ) ϕ L m ( v,μ ) N ˜ m ( v ) e τ/v e im φ 0 dv

Metrics