Abstract

We describe a new, multiply subtractive Kramers–Kronig (MSKK) method to find the optical constants of a material from a single transmittance or reflectance spectrum covering a small frequency domain. The MSKK method incorporates independent measurements of n and k at one or more reference wave-number values to minimize errors due to extrapolations of the data. An unexpected connection between the MSKK equations and the interpolation theory allows us to derive the equations from an interpolation theorem. We found that the locations of the reference points affect the accuracy of the values determined for the optical constants and that the optimal spacing of N reference data points is related to the zeros of a suitably transformed Chebychev polynomial of order N. We discuss our efforts to optimize both the number and the spacing of these reference points and apply our method to some test spectra.

© 1998 Optical Society of America

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References

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  1. R. H. Young, “Validity of the Kramers–Kronig transformation used in reflection spectroscopy,” J. Opt. Soc. Am. 67, 520–523 (1977).
    [CrossRef]
  2. J. S. Plaskett, P. N. Schatz, “On the Robinson and Price (Kramers–Kronig) method of interpreting reflection data taken through a transparent window,” J. Chem. Phys. 38, 612–617 (1963).
    [CrossRef]
  3. K. F. Palmer, M. Z. Williams, “Optical constant determination of thin films condensed on transmitting and reflecting surfaces,” Technical Report AEDC-TR-83-64 (AD-A140845) (Defense Technical Information Center, Fort Belvoir, Va., 1984).
  4. K. F. Palmer, M. Z. Williams, “Determination of the optical constants of a thin film from transmittance measurements of a single film thickness,” Appl. Opt. 24, 1788–1798 (1985).
    [CrossRef] [PubMed]
  5. S. Maeda, G. Thyagarajan, P. N. Schatz, “Absolute infrared intensity measurements in thin films: II. Solids deposited on halide plates,” J. Chem. Phys. 38, 3474–3481 (1963).
    [CrossRef]
  6. E. A. Lupaskho, V. K. Miloslavskii, I. N. Shklyarevskii, “Use of the Kramers–Kronig dispersion relations in determining the phase shift occurring upon reflection of light from thin dielectric layers,” Opt. Spektrosk. 24, 257–262 (1968) [Opt. Spectrosc. 24, 132–134 (1968)].
  7. E. A. Lupaskho, V. K. Miloslavskii, I. N. Shklyarevskii, “Use of the Kramers–Kronig dispersion relationships to calculate the phase of the wave reflected from thin dielectric layers,” Opt. Spektrosk. 29, 789–793 (1970) [Opt. Spectrosc. 29, 419–422 (1970)].
  8. J. S. Toll, “Causality and the dispersion relation: logical foundations,” Phys. Rev. 104, 1760–1770 (1956).
    [CrossRef]
  9. P.-O. Nilsson, “Determination of optical constants from intensity measurements at normal incidence,” Appl. Opt. 7, 435–442 (1968).
    [CrossRef] [PubMed]
  10. R. K. Ahrenkiel, “Modified Kramers–Kronig analysis of optical spectra,” J. Opt. Soc. Am. 61, 1651–1655 (1971).
    [CrossRef]
  11. H. M. Nussenzveig, Causality and Dispersion Relations (Academic, New York, 1972).
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    [CrossRef]
  13. P. J. Davis, Interpolation and Approximation (Dover, New York, 1975).
  14. L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1960).
  15. K.-E. Peiponen, E. M. Vartiainen, “Kramers–Kronig relations in optical data inversion,” Phys. Rev. B 44, 8301–8303 (1991).
    [CrossRef]
  16. E. M. Vartiainen, K. E. Peiponen, T. Asakura, “Maximum entropy model in reflection spectra analysis,” Opt. Commun. 89, 37–40 (1992).
    [CrossRef]
  17. E. M. Vartiainen, K.-E. Peiponen, T. Asakura, “Comparison between the optical constants obtained by the Kramers–Kronig analysis and the maximum entropy method: infrared optical properties of orthorhombic sulfur,” Appl. Opt. 32, 1126–1129 (1993).
    [CrossRef] [PubMed]
  18. K. F. Palmer, M. Z. Williams, B. A. Budde, W. T. Bertrand, “Optical analysis methods for material films condensed on cryogenic surfaces of spacecraft,” Technical Report AEDC-TR-94-3 (AD-A284014) (Defense Technical Information Center, Fort Belvoir, Va., 1994).
  19. G. D. Guenther, Modern Optics (Wiley, New York, 1990).
  20. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).
  21. A. V. Tikhonravov, P. W. Baumeister, K. V. Popov, “Phase properties of multilayers,” Appl. Opt. 36, 4382–4392 (1997).
    [CrossRef] [PubMed]
  22. K. F. Palmer, J. A. Roux, B. E. Wood, “The infrared optical properties of mixtures of molecular species at 20 K,” Technical Report AEDC-TR-80-30 (AD-A094214) (Defense Technical Information Center, Fort Belvoir, Va., 1981).
  23. R. P. Boas, Invitation to Complex Analysis (Random House, New York, 1987).

1997 (1)

1993 (1)

1992 (1)

E. M. Vartiainen, K. E. Peiponen, T. Asakura, “Maximum entropy model in reflection spectra analysis,” Opt. Commun. 89, 37–40 (1992).
[CrossRef]

1991 (1)

K.-E. Peiponen, E. M. Vartiainen, “Kramers–Kronig relations in optical data inversion,” Phys. Rev. B 44, 8301–8303 (1991).
[CrossRef]

1985 (1)

1980 (1)

1977 (1)

1971 (1)

1970 (1)

E. A. Lupaskho, V. K. Miloslavskii, I. N. Shklyarevskii, “Use of the Kramers–Kronig dispersion relationships to calculate the phase of the wave reflected from thin dielectric layers,” Opt. Spektrosk. 29, 789–793 (1970) [Opt. Spectrosc. 29, 419–422 (1970)].

1968 (2)

E. A. Lupaskho, V. K. Miloslavskii, I. N. Shklyarevskii, “Use of the Kramers–Kronig dispersion relations in determining the phase shift occurring upon reflection of light from thin dielectric layers,” Opt. Spektrosk. 24, 257–262 (1968) [Opt. Spectrosc. 24, 132–134 (1968)].

P.-O. Nilsson, “Determination of optical constants from intensity measurements at normal incidence,” Appl. Opt. 7, 435–442 (1968).
[CrossRef] [PubMed]

1963 (2)

J. S. Plaskett, P. N. Schatz, “On the Robinson and Price (Kramers–Kronig) method of interpreting reflection data taken through a transparent window,” J. Chem. Phys. 38, 612–617 (1963).
[CrossRef]

S. Maeda, G. Thyagarajan, P. N. Schatz, “Absolute infrared intensity measurements in thin films: II. Solids deposited on halide plates,” J. Chem. Phys. 38, 3474–3481 (1963).
[CrossRef]

1956 (1)

J. S. Toll, “Causality and the dispersion relation: logical foundations,” Phys. Rev. 104, 1760–1770 (1956).
[CrossRef]

Ahrenkiel, R. K.

Asakura, T.

Baumeister, P. W.

Bertrand, W. T.

K. F. Palmer, M. Z. Williams, B. A. Budde, W. T. Bertrand, “Optical analysis methods for material films condensed on cryogenic surfaces of spacecraft,” Technical Report AEDC-TR-94-3 (AD-A284014) (Defense Technical Information Center, Fort Belvoir, Va., 1994).

Boas, R. P.

R. P. Boas, Invitation to Complex Analysis (Random House, New York, 1987).

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

Budde, B. A.

K. F. Palmer, M. Z. Williams, B. A. Budde, W. T. Bertrand, “Optical analysis methods for material films condensed on cryogenic surfaces of spacecraft,” Technical Report AEDC-TR-94-3 (AD-A284014) (Defense Technical Information Center, Fort Belvoir, Va., 1994).

Cameron, D. G.

Davis, P. J.

P. J. Davis, Interpolation and Approximation (Dover, New York, 1975).

Goplen, T. G.

Guenther, G. D.

G. D. Guenther, Modern Optics (Wiley, New York, 1990).

Jones, R. N.

Landau, L. D.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1960).

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1960).

Lupaskho, E. A.

E. A. Lupaskho, V. K. Miloslavskii, I. N. Shklyarevskii, “Use of the Kramers–Kronig dispersion relationships to calculate the phase of the wave reflected from thin dielectric layers,” Opt. Spektrosk. 29, 789–793 (1970) [Opt. Spectrosc. 29, 419–422 (1970)].

E. A. Lupaskho, V. K. Miloslavskii, I. N. Shklyarevskii, “Use of the Kramers–Kronig dispersion relations in determining the phase shift occurring upon reflection of light from thin dielectric layers,” Opt. Spektrosk. 24, 257–262 (1968) [Opt. Spectrosc. 24, 132–134 (1968)].

Maeda, S.

S. Maeda, G. Thyagarajan, P. N. Schatz, “Absolute infrared intensity measurements in thin films: II. Solids deposited on halide plates,” J. Chem. Phys. 38, 3474–3481 (1963).
[CrossRef]

Miloslavskii, V. K.

E. A. Lupaskho, V. K. Miloslavskii, I. N. Shklyarevskii, “Use of the Kramers–Kronig dispersion relationships to calculate the phase of the wave reflected from thin dielectric layers,” Opt. Spektrosk. 29, 789–793 (1970) [Opt. Spectrosc. 29, 419–422 (1970)].

E. A. Lupaskho, V. K. Miloslavskii, I. N. Shklyarevskii, “Use of the Kramers–Kronig dispersion relations in determining the phase shift occurring upon reflection of light from thin dielectric layers,” Opt. Spektrosk. 24, 257–262 (1968) [Opt. Spectrosc. 24, 132–134 (1968)].

Nilsson, P.-O.

Nussenzveig, H. M.

H. M. Nussenzveig, Causality and Dispersion Relations (Academic, New York, 1972).

Palmer, K. F.

K. F. Palmer, M. Z. Williams, “Determination of the optical constants of a thin film from transmittance measurements of a single film thickness,” Appl. Opt. 24, 1788–1798 (1985).
[CrossRef] [PubMed]

K. F. Palmer, J. A. Roux, B. E. Wood, “The infrared optical properties of mixtures of molecular species at 20 K,” Technical Report AEDC-TR-80-30 (AD-A094214) (Defense Technical Information Center, Fort Belvoir, Va., 1981).

K. F. Palmer, M. Z. Williams, “Optical constant determination of thin films condensed on transmitting and reflecting surfaces,” Technical Report AEDC-TR-83-64 (AD-A140845) (Defense Technical Information Center, Fort Belvoir, Va., 1984).

K. F. Palmer, M. Z. Williams, B. A. Budde, W. T. Bertrand, “Optical analysis methods for material films condensed on cryogenic surfaces of spacecraft,” Technical Report AEDC-TR-94-3 (AD-A284014) (Defense Technical Information Center, Fort Belvoir, Va., 1994).

Peiponen, K. E.

E. M. Vartiainen, K. E. Peiponen, T. Asakura, “Maximum entropy model in reflection spectra analysis,” Opt. Commun. 89, 37–40 (1992).
[CrossRef]

Peiponen, K.-E.

Plaskett, J. S.

J. S. Plaskett, P. N. Schatz, “On the Robinson and Price (Kramers–Kronig) method of interpreting reflection data taken through a transparent window,” J. Chem. Phys. 38, 612–617 (1963).
[CrossRef]

Popov, K. V.

Roux, J. A.

K. F. Palmer, J. A. Roux, B. E. Wood, “The infrared optical properties of mixtures of molecular species at 20 K,” Technical Report AEDC-TR-80-30 (AD-A094214) (Defense Technical Information Center, Fort Belvoir, Va., 1981).

Schatz, P. N.

J. S. Plaskett, P. N. Schatz, “On the Robinson and Price (Kramers–Kronig) method of interpreting reflection data taken through a transparent window,” J. Chem. Phys. 38, 612–617 (1963).
[CrossRef]

S. Maeda, G. Thyagarajan, P. N. Schatz, “Absolute infrared intensity measurements in thin films: II. Solids deposited on halide plates,” J. Chem. Phys. 38, 3474–3481 (1963).
[CrossRef]

Shklyarevskii, I. N.

E. A. Lupaskho, V. K. Miloslavskii, I. N. Shklyarevskii, “Use of the Kramers–Kronig dispersion relationships to calculate the phase of the wave reflected from thin dielectric layers,” Opt. Spektrosk. 29, 789–793 (1970) [Opt. Spectrosc. 29, 419–422 (1970)].

E. A. Lupaskho, V. K. Miloslavskii, I. N. Shklyarevskii, “Use of the Kramers–Kronig dispersion relations in determining the phase shift occurring upon reflection of light from thin dielectric layers,” Opt. Spektrosk. 24, 257–262 (1968) [Opt. Spectrosc. 24, 132–134 (1968)].

Thyagarajan, G.

S. Maeda, G. Thyagarajan, P. N. Schatz, “Absolute infrared intensity measurements in thin films: II. Solids deposited on halide plates,” J. Chem. Phys. 38, 3474–3481 (1963).
[CrossRef]

Tikhonravov, A. V.

Toll, J. S.

J. S. Toll, “Causality and the dispersion relation: logical foundations,” Phys. Rev. 104, 1760–1770 (1956).
[CrossRef]

Vartiainen, E. M.

E. M. Vartiainen, K.-E. Peiponen, T. Asakura, “Comparison between the optical constants obtained by the Kramers–Kronig analysis and the maximum entropy method: infrared optical properties of orthorhombic sulfur,” Appl. Opt. 32, 1126–1129 (1993).
[CrossRef] [PubMed]

E. M. Vartiainen, K. E. Peiponen, T. Asakura, “Maximum entropy model in reflection spectra analysis,” Opt. Commun. 89, 37–40 (1992).
[CrossRef]

K.-E. Peiponen, E. M. Vartiainen, “Kramers–Kronig relations in optical data inversion,” Phys. Rev. B 44, 8301–8303 (1991).
[CrossRef]

Williams, M. Z.

K. F. Palmer, M. Z. Williams, “Determination of the optical constants of a thin film from transmittance measurements of a single film thickness,” Appl. Opt. 24, 1788–1798 (1985).
[CrossRef] [PubMed]

K. F. Palmer, M. Z. Williams, “Optical constant determination of thin films condensed on transmitting and reflecting surfaces,” Technical Report AEDC-TR-83-64 (AD-A140845) (Defense Technical Information Center, Fort Belvoir, Va., 1984).

K. F. Palmer, M. Z. Williams, B. A. Budde, W. T. Bertrand, “Optical analysis methods for material films condensed on cryogenic surfaces of spacecraft,” Technical Report AEDC-TR-94-3 (AD-A284014) (Defense Technical Information Center, Fort Belvoir, Va., 1994).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

Wood, B. E.

K. F. Palmer, J. A. Roux, B. E. Wood, “The infrared optical properties of mixtures of molecular species at 20 K,” Technical Report AEDC-TR-80-30 (AD-A094214) (Defense Technical Information Center, Fort Belvoir, Va., 1981).

Young, R. H.

Appl. Opt. (4)

Appl. Spectrosc. (1)

J. Chem. Phys. (2)

S. Maeda, G. Thyagarajan, P. N. Schatz, “Absolute infrared intensity measurements in thin films: II. Solids deposited on halide plates,” J. Chem. Phys. 38, 3474–3481 (1963).
[CrossRef]

J. S. Plaskett, P. N. Schatz, “On the Robinson and Price (Kramers–Kronig) method of interpreting reflection data taken through a transparent window,” J. Chem. Phys. 38, 612–617 (1963).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Commun. (1)

E. M. Vartiainen, K. E. Peiponen, T. Asakura, “Maximum entropy model in reflection spectra analysis,” Opt. Commun. 89, 37–40 (1992).
[CrossRef]

Opt. Spektrosk. (2)

E. A. Lupaskho, V. K. Miloslavskii, I. N. Shklyarevskii, “Use of the Kramers–Kronig dispersion relations in determining the phase shift occurring upon reflection of light from thin dielectric layers,” Opt. Spektrosk. 24, 257–262 (1968) [Opt. Spectrosc. 24, 132–134 (1968)].

E. A. Lupaskho, V. K. Miloslavskii, I. N. Shklyarevskii, “Use of the Kramers–Kronig dispersion relationships to calculate the phase of the wave reflected from thin dielectric layers,” Opt. Spektrosk. 29, 789–793 (1970) [Opt. Spectrosc. 29, 419–422 (1970)].

Phys. Rev. (1)

J. S. Toll, “Causality and the dispersion relation: logical foundations,” Phys. Rev. 104, 1760–1770 (1956).
[CrossRef]

Phys. Rev. B (1)

K.-E. Peiponen, E. M. Vartiainen, “Kramers–Kronig relations in optical data inversion,” Phys. Rev. B 44, 8301–8303 (1991).
[CrossRef]

Other (9)

K. F. Palmer, M. Z. Williams, “Optical constant determination of thin films condensed on transmitting and reflecting surfaces,” Technical Report AEDC-TR-83-64 (AD-A140845) (Defense Technical Information Center, Fort Belvoir, Va., 1984).

H. M. Nussenzveig, Causality and Dispersion Relations (Academic, New York, 1972).

P. J. Davis, Interpolation and Approximation (Dover, New York, 1975).

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1960).

K. F. Palmer, M. Z. Williams, B. A. Budde, W. T. Bertrand, “Optical analysis methods for material films condensed on cryogenic surfaces of spacecraft,” Technical Report AEDC-TR-94-3 (AD-A284014) (Defense Technical Information Center, Fort Belvoir, Va., 1994).

G. D. Guenther, Modern Optics (Wiley, New York, 1990).

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

K. F. Palmer, J. A. Roux, B. E. Wood, “The infrared optical properties of mixtures of molecular species at 20 K,” Technical Report AEDC-TR-80-30 (AD-A094214) (Defense Technical Information Center, Fort Belvoir, Va., 1981).

R. P. Boas, Invitation to Complex Analysis (Random House, New York, 1987).

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Figures (10)

Fig. 1
Fig. 1

Theoretical spectrum model with bands at ν = 1500, 2500, and 3500 cm-1.

Fig. 2
Fig. 2

Plots of theta differences (Δ N ) of theoretical spectrum for N = 10 reference points.

Fig. 3
Fig. 3

Plots of Δ N of theoretical spectrum for N Chebychev reference points. (a) N = 0, 1, 2, 3, and 5. (b) N = 10, 15, and 20. Note the change in scale in parts (a) and (b).

Fig. 4
Fig. 4

Transmittance and optical constants of hypothetical film material.

Fig. 5
Fig. 5

Differences between the determined and the original optical constants of hypothetical film with three reference points. (a) n-n orig and (b) k-k orig.

Fig. 6
Fig. 6

Differences between the determined and the original optical constants of hypothetical film with N = 1, 2, 4, and 9 Chebychev reference points. (a) n-n orig and (b) k-k orig.

Fig. 7
Fig. 7

Optical constants of CO2 film on 20-K germanium using nine reference points. (a) n and (b) k.

Fig. 8
Fig. 8

Differences between the determined and the original optical constants of CO2 film with N = 1, 2, 4, and 9 Chebychev reference points. (a) n-n orig and (b) k-k orig.

Fig. 9
Fig. 9

Contour path C for Hermite remainder calculation in Eq. (B3) of Appendix B.

Fig. 10
Fig. 10

Contour path for Hermite error term calculation in Appendix D.

Equations (45)

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Φ ν ν = - 1 π   P   0 ln   F ν d ν ν 2 - ν 2 + h ν ,
Φ ν ν = - 1 π   P   0 ln   R ν d ν ν 2 - ν 2 .
T ν = | n ˆ 3 ν / n ˆ 0 ν | t ˆ * ν t ˆ ν .
Φ ν ν = - 1 π   P   0 ln   T ν d ν ν 2 - ν 2 + 2 π d 1 .
Φ ν ν = - 1 π   P   0 ln   R ν d ν ν 2 - ν 2 + 1 ν j arctan u j ν .
Θ ν = Θ ν 1 - ν 2 - ν 1 2 π   P   0 ln   F ν d ν ν 2 - ν 2 ν 2 - ν 1 2 .
Θ ν Φ ν ν - h ν .
Θ ν = ν 2 - ν 2 2 ν 2 - ν N 2 ν 1 2 - ν 2 2 ν 1 2 - ν N 2   Θ ν 1 + + ν 2 - ν 1 2 ν 2 - ν j - 1 2 ν 2 - ν j + 1 2 ν 2 - ν N 2 ν j 2 - ν 1 2 ν j 2 - ν j - 1 2 ν j 2 - ν j + 1 2 ν j 2 - ν N 2   Θ ν j + + ν 2 - ν 1 2 ν 2 - ν N - 1 2 ν N 2 - ν 1 2 ν N 2 - ν N - 1 2   Θ ν N - ν 2 - ν 1 2 ν 2 - ν N 2 π   P   0 ln   F ν d ν ν 2 - ν 2 ν 2 - ν 1 2 ν 2 - ν N 2 .
Φ ν ν = ν 2 - ν 2 2 ν 2 - ν N 2 ν 1 2 - ν 2 2 ν 1 2 - ν N 2 Φ ν 1 ν 1 + + ν 2 - ν 1 2 ν 2 - ν j - 1 2 ν 2 - ν j + 1 2 ν 2 - ν N 2 ν j 2 - ν 1 2 ν j 2 - ν j - 1 2 ν j 2 - ν j + 1 2 ν j 2 - ν N 2 Φ ν j ν j + + ν 2 - ν 1 2 ν 2 - ν N - 1 2 ν N 2 - ν 1 2 ν N 2 - ν N - 1 2 Φ ν N ν N - ν 2 - ν 1 2 ν 2 - ν N 2 π P 0 ln F ν d ν ν 2 - ν 2 ν 2 - ν 1 2 ν 2 - ν N 2
Δ N Θ approx ν - Θ true ν = - ν 2 - ν 1 2 ν 2 - ν N 2 π × P   0 ν ln F approx ν - ln F true ν d ν ν 2 - ν 2 ν 2 - ν 1 2 ν 2 - ν N 2 - ν 2 - ν 1 2 ν 2 - ν N 2 π × P   ν u ln F approx ν - ln F true ν d ν ν 2 - ν 2 ν 2 - ν 1 2 ν 2 - ν N 2 .
w ν = ν 2 - ν 1 2 ν 2 - ν N 2 ,
ν zero   q = ν u 2 - ν 2 cos 2 q + 1 π 2 N + ν u 2 + ν 2 2 1 / 2 , q = 0 ,   1 , ,   N - 1
Δ N Aw ν π w ν p ν p 2 - ν 2 ,     N > 0 , Δ 0 A π ν p 2 - ν 2 .
Δ N Δ 0 ν 2 - ν 1 2 ν 2 - ν N 2 ν p 2 - ν 1 2 ν p 2 - ν N 2 < 1 .
n ˆ ν = 1 + 2 S ˆ ν 1 - S ˆ ν 1 / 2 ,
S ˆ ν = α F α ν α 2 - ν 2 - iq α ν ,
t ˆ ν = | n ˆ 0 ν T ν / n ˆ 3 ν | 1 / 2 exp i Φ ν .
Θ ν 1 - Θ ν ν 2 - ν 1 2 = 1 π   P   0 ln   F ν d ν ν 2 - ν 2 ν 2 - ν 1 2 .
Θ ν = ν 2 - ν 2 2 ν 1 2 - ν 2 2   Θ ν 1 + ν 2 - ν 1 2 ν 2 2 - ν 1 2   Θ ν 2 - ν 2 - ν 1 2 ν 2 - ν 2 2 π × P   0 ln   F ν d ν ν 2 - ν 2 ν 2 - ν 1 2 ν 2 - ν 2 2 ,
Θ ν 1 ν 2 - ν 1 2 ν 1 2 - ν 2 2 + Θ ν 2 ν 2 - ν 2 2 ν 2 2 - ν 1 2 - Θ ν ν 2 - ν 1 2 ν 2 - ν 2 2 = 1 π   P   0 ln   F ν d ν ν 2 - ν 2 ν 2 - ν 1 2 ν 2 - ν 2 2 .
Θ ν = ν 2 - ν 2 2 ν 2 - ν 3 2 ν 1 2 - ν 2 2 ν 1 2 - ν 3 2   Θ ν 1 + ν 2 - ν 1 2 ν 2 - ν 3 2 ν 2 2 - ν 1 2 ν 2 2 - ν 3 2   Θ ν 2 + ν 2 - ν 1 2 ν 2 - ν 2 2 ν 3 2 - ν 1 2 ν 3 2 - ν 2 2   Θ ν 3 - ν 2 - ν 1 2 ν 2 - ν 2 2 ν 2 - ν 3 2 π × P   0 ln   F ν d ν ν 2 - ν 2 ν 2 - ν 1 2 ν 2 - ν 2 2 ν 2 - ν 3 2 .
- 1 π   P   0 ln   F ν d ν ν 2 - ν 2 ν 2 - ν 1 2 ν 2 - ν N 2 = Θ ν - Θ ν 1 ν 2 - ν 1 2 ν 2 - ν N 2 + Θ ν 2 - Θ ν 1 ν 2 2 - ν 1 2 ν 2 2 - ν 2 ν 2 2 - ν N 2 + + Θ ν j - Θ ν 1 ν j 2 - ν 1 2 ν j 2 - ν j - 1 2 ν j 2 - ν 2 ν j 2 - ν j + 1 2 ν j 2 - ν N 2 + + Θ ν N - Θ ν 1 ν N 2 - ν 1 2 ν N 2 - ν 2 .
R n f ;   z f z - P n f ;   z = 1 2 π i   C z - z 0 z - z 1 z - z n f t d t t - z 0 t - z 1 t - z n t - z .
P n f ;   z = k = 0 n z - z 0 z - z 1 z - z k - 1 z - z k + 1 z - z n z k - z 0 z k - z 1 z k - z k - 1 z k - z k + 1 z k - z n × f z k .
H N ln   ĝ ;   ν = ln   ĝ ν ν - j = 1 N   l j ν ln   ĝ ν j ν j = 1 π i   C w ν ln   ĝ ν ˆ d ν ˆ w ν ˆ ν ˆ 2 - ν 2 ,
l j ν ν 2 - ν 1 2 ν 2 - ν j - 1 2 ν 2 - ν j + 1 2 ν 2 - ν N 2 ν j 2 - ν 1 2 ν j 2 - ν j - 1 2 ν j 2 - ν j + 1 2 ν j 2 - ν N 2 .
1 π i C 1 w ν ln   ĝ ν ˆ d ν ˆ w ν ˆ ν ˆ 2 - ν 2 = 1 π i   P   0 w ν ln   ĝ ν d ν w ν ν 2 - ν 2 + ln   ĝ ν 2 ν - j = 1 N   j ν ln   ĝ ν j 2 ν j .
lim L 1 π i C 2 w ν ln   ĝ L   exp i θ d L   exp i θ w L   exp i θ L   exp i θ 2 - ν 2 ] ,
1 π i C 3 w ν ln   ĝ ν ˆ d ν ˆ w ν ˆ ν ˆ 2 - ν 2 = - 1 π   P   0 w ν ln   ĝ i ν I d ν I w i ν I ν I 2 + ν 2 ,
ln   ĝ ν ν = j = 1 N   j ν ln   ĝ ν j ν j + 2 π i   P   0 w ν ln   ĝ ν d ν w ν ν 2 - ν 2 - 2 π   P   0 w ν ln   ĝ i ν I d ν I w i ν I ν I 2 - ν 2 .
Φ ν ν = j = 1 N   j ν Φ ν j ν j - 2 π   P   0 w ν ln | ĝ ν | d ν w ν ν 2 - ν 2 .
T n x = cos n   arccos   x ,     n = 0 ,   1 ,   2 , .
u = u u - u 2   x + u u + u 2 ,
x = cos 2 q + 1 π 2 n ,     q = 0 ,   1 ,   2 , ,   n - 1 .
f ν = - c 1 p ν + 1 p - ν ,
x = ν , x k = ν k ,     k = 1 ,   2 , ,   N , a = ν l ,   b = ν u , z k = a k + b k   i ,     k = 1 ,   2 ,   3 , z = x + y   i .
p z = z - z 1 z - z 1 * z - z 2 z - z 2 * × z - z 3 z - z 3 * .
Θ x = H 0 x = c   e 1 b 1 z 1 - z 2 z 1 - z 2 * z 1 - z 3 z 1 - z 3 * z 1 2 - x 2 + 1 b 2 z 2 - z 1 z 2 - z 1 * z 2 - z 3 z 2 - z 3 * z 2 2 - x 2 + 1 b 3 z 3 - z 1 z 3 - z 1 * z 3 - z 2 z 3 - z 2 * z 3 2 - x 2
Θ x - k = 1 N   k x Θ x k = H N x = c   e w x b 1 z 1 - z 2 z 1 - z 2 * z 1 - z 3 z 1 - z 3 * z 1 2 - x 2 w z 1 + w x b 2 z 2 - z 1 z 2 - z 1 * z 2 - z 3 z 2 - z 3 * z 2 2 - x 2 w z 2 + w x b 3 z 3 - z 1 z 3 - z 1 * z 3 - z 2 z 3 - z 2 * z 3 2 - x 2 w z 3 ,
w x = x 2 - x 1 2 x 2 - x 2 2 x 2 - x N 2 .
L f x = f x - f a 2 x ln a - x a + x + f b - f x 2 x ln b - x b + x ,     x a ,   b , L f a = f b - f a 2 a ln b - a b + a , L f b = f b - f a 2 b ln b - a b + a = a b L f a .
c x = 1 p z z 2 - x 2
H ˜ 0 x = 1 π L f x - 2 c   e = 1 3   c x × ln a - z b + z a + z b - z ,
L f x - k = 1 N   L f x k k x - 2 c   e = 1 3   c x w x w z ln a - z b + z a + z b - z .
Δ 0 = H ˜ 0 - H 0 = L f x π - 2 c × e = 1 3   c x 1 π ln a - z b + z a + z b - z - 1 , Δ N = H ˜ N - H N = 1 π L f x - k = 1 N   L f x k k x - 2 c   e = 1 3   c x w x w z × 1 π ln a - z b + z a + z b - z - 1 .

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