Abstract

When the real (imaginary) part of the transfer function of a causal linear system is known for all frequencies, the imaginary (real) part may be calculated for all frequencies from the Kramers–Kronig relations. For physical systems, the real part could be the refractive index or the amplitude, and the imaginary part could be the extinction coefficient or the phase. Experimentally these quantities are known only for limited-frequency intervals. This paper presents generalized Kramers–Kronig relations, from which the real and imaginary parts may be calculated for all frequencies from knowledge of these parts for at least partly overlapping frequency intervals. When the procedure is applied to experimental data, errors are introduced. Certain types of errors of the known real and imaginary parts completely destroy the possibility of calculating the unknown parts, whereas others give negligible errors. The existence of a filter with the property of allowing the restoration of a truncated spectrum is established. The transfer function and the impulse response function of this filter are given.

© 1982 Optical Society of America

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  1. R. de L. Kronig, “On the theory of dispersion of x-rays,” J. Opt. Soc. Am. 12, 547–557 (1926).
    [Crossref]
  2. H. A. M. Kramers, “La diffusion de la lumiere par les atomes,” in Atti del Congresso Internazionale dei Fisici, Settembre 1927 (N. Zanichelli, Bologna, Italy, 1928), pp. 545–557.
  3. H. A. Kramers, “Die Dispersion und Absorption von Röntgenstralen,” Phys. Z. 30, 522–523 (1929).
  4. J. Tauc, “Optical properties of semiconductors” in Proceedings of the International School of Physics “Enrico Fermi,” The Optical Properties of Solids, J. Tauc, ed. (Academic, New York, London, 1966), pp. 63–89.
  5. T. S. Moss, G. J. Burrell, and B. Ellis, Semiconductor Opto-Electronics (Butterworth, London, 1973), Chap. 2.
  6. H. W. Bode, Network Analysis and Feedback Amplifier Design (Van Nostrand, New York, 1945), Chap. XIV.
  7. R. V. Churchill, Complex Variables and Applications (McGraw-Hill, Tokyo, 1960), Chap. 11.
  8. D. J. Patil, “Representation of Hp-functions,” Bull. Am. Math. Soc. 78, 617–620 (1972).
    [Crossref]
  9. D. J. Patil, “Recapturing H2-functions on a polydisc,” Trans. Am. Math. Soc. 188, 97–103 (1974).
  10. R. N. Mukherjee, “Representations of H2-functions on the real line,” Boll. U.M.I. 10, 666–671 (1974).
  11. R. Z. Bachrach and F. C. Brown, “Exciton-optical properties of TlBr and TlCl,” Phys. Rev. B 1, 818–831 (1970).
    [Crossref]
  12. R. K. Ahrenkiel, “Modified Kramers–Kronig analysis of optical spectra,” J. Opt. Soc. Am. 61, 1651–1655 (1971).
    [Crossref]
  13. E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, 2nd ed. (Oxford U. Press, London, 1959), Chap. V (especially Theorem 95).
  14. G. Dahlquist and Å. Björck, Numerical Methods (Prentice Hall, Englewood Cliffs, N.J., 1974), Chaps. 4 and 5.
  15. A. H. Stroud and D. Secrest, Gaussian Quadrature Formulas (Prentice Hall, Englewood Cliffs, N.J., 1966).
  16. H. Riesel, Department of Numerical and Computing Science, Royal Institute of Technology, Stockholm, Sweden (personal communication).
  17. R. W. Schafer, R. M. Mersereau, and M. A. Richards, “Constrained iterative restoration algorithms,” Proc. IEEE 69, 432–450 (1981).
    [Crossref]
  18. I. N. Sneddon, Special Functions of Mathematical Physics and Chemistry (Oliver and Boyd, Edinburgh, Scotland, 1956), Chap. IV.
  19. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), Entry 3.911.1.
  20. C. W. Peterson and B. W. Knight, “Causality calculations in the time domain: an efficient alternative to the Kramers–Kronig method,” J. Opt. Soc. Am. 63, 1238–1242 (1973).
    [Crossref]
  21. See Ref. 13, especially Chap. V, Sec. 1.

1981 (1)

R. W. Schafer, R. M. Mersereau, and M. A. Richards, “Constrained iterative restoration algorithms,” Proc. IEEE 69, 432–450 (1981).
[Crossref]

1974 (2)

D. J. Patil, “Recapturing H2-functions on a polydisc,” Trans. Am. Math. Soc. 188, 97–103 (1974).

R. N. Mukherjee, “Representations of H2-functions on the real line,” Boll. U.M.I. 10, 666–671 (1974).

1973 (1)

1972 (1)

D. J. Patil, “Representation of Hp-functions,” Bull. Am. Math. Soc. 78, 617–620 (1972).
[Crossref]

1971 (1)

1970 (1)

R. Z. Bachrach and F. C. Brown, “Exciton-optical properties of TlBr and TlCl,” Phys. Rev. B 1, 818–831 (1970).
[Crossref]

1929 (1)

H. A. Kramers, “Die Dispersion und Absorption von Röntgenstralen,” Phys. Z. 30, 522–523 (1929).

1926 (1)

Ahrenkiel, R. K.

Bachrach, R. Z.

R. Z. Bachrach and F. C. Brown, “Exciton-optical properties of TlBr and TlCl,” Phys. Rev. B 1, 818–831 (1970).
[Crossref]

Björck, Å.

G. Dahlquist and Å. Björck, Numerical Methods (Prentice Hall, Englewood Cliffs, N.J., 1974), Chaps. 4 and 5.

Bode, H. W.

H. W. Bode, Network Analysis and Feedback Amplifier Design (Van Nostrand, New York, 1945), Chap. XIV.

Brown, F. C.

R. Z. Bachrach and F. C. Brown, “Exciton-optical properties of TlBr and TlCl,” Phys. Rev. B 1, 818–831 (1970).
[Crossref]

Burrell, G. J.

T. S. Moss, G. J. Burrell, and B. Ellis, Semiconductor Opto-Electronics (Butterworth, London, 1973), Chap. 2.

Churchill, R. V.

R. V. Churchill, Complex Variables and Applications (McGraw-Hill, Tokyo, 1960), Chap. 11.

Dahlquist, G.

G. Dahlquist and Å. Björck, Numerical Methods (Prentice Hall, Englewood Cliffs, N.J., 1974), Chaps. 4 and 5.

Ellis, B.

T. S. Moss, G. J. Burrell, and B. Ellis, Semiconductor Opto-Electronics (Butterworth, London, 1973), Chap. 2.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), Entry 3.911.1.

Knight, B. W.

Kramers, H. A.

H. A. Kramers, “Die Dispersion und Absorption von Röntgenstralen,” Phys. Z. 30, 522–523 (1929).

Kramers, H. A. M.

H. A. M. Kramers, “La diffusion de la lumiere par les atomes,” in Atti del Congresso Internazionale dei Fisici, Settembre 1927 (N. Zanichelli, Bologna, Italy, 1928), pp. 545–557.

Kronig, R. de L.

Mersereau, R. M.

R. W. Schafer, R. M. Mersereau, and M. A. Richards, “Constrained iterative restoration algorithms,” Proc. IEEE 69, 432–450 (1981).
[Crossref]

Moss, T. S.

T. S. Moss, G. J. Burrell, and B. Ellis, Semiconductor Opto-Electronics (Butterworth, London, 1973), Chap. 2.

Mukherjee, R. N.

R. N. Mukherjee, “Representations of H2-functions on the real line,” Boll. U.M.I. 10, 666–671 (1974).

Patil, D. J.

D. J. Patil, “Recapturing H2-functions on a polydisc,” Trans. Am. Math. Soc. 188, 97–103 (1974).

D. J. Patil, “Representation of Hp-functions,” Bull. Am. Math. Soc. 78, 617–620 (1972).
[Crossref]

Peterson, C. W.

Richards, M. A.

R. W. Schafer, R. M. Mersereau, and M. A. Richards, “Constrained iterative restoration algorithms,” Proc. IEEE 69, 432–450 (1981).
[Crossref]

Riesel, H.

H. Riesel, Department of Numerical and Computing Science, Royal Institute of Technology, Stockholm, Sweden (personal communication).

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), Entry 3.911.1.

Schafer, R. W.

R. W. Schafer, R. M. Mersereau, and M. A. Richards, “Constrained iterative restoration algorithms,” Proc. IEEE 69, 432–450 (1981).
[Crossref]

Secrest, D.

A. H. Stroud and D. Secrest, Gaussian Quadrature Formulas (Prentice Hall, Englewood Cliffs, N.J., 1966).

Sneddon, I. N.

I. N. Sneddon, Special Functions of Mathematical Physics and Chemistry (Oliver and Boyd, Edinburgh, Scotland, 1956), Chap. IV.

Stroud, A. H.

A. H. Stroud and D. Secrest, Gaussian Quadrature Formulas (Prentice Hall, Englewood Cliffs, N.J., 1966).

Tauc, J.

J. Tauc, “Optical properties of semiconductors” in Proceedings of the International School of Physics “Enrico Fermi,” The Optical Properties of Solids, J. Tauc, ed. (Academic, New York, London, 1966), pp. 63–89.

Titchmarsh, E. C.

E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, 2nd ed. (Oxford U. Press, London, 1959), Chap. V (especially Theorem 95).

Boll. U.M.I. (1)

R. N. Mukherjee, “Representations of H2-functions on the real line,” Boll. U.M.I. 10, 666–671 (1974).

Bull. Am. Math. Soc. (1)

D. J. Patil, “Representation of Hp-functions,” Bull. Am. Math. Soc. 78, 617–620 (1972).
[Crossref]

J. Opt. Soc. Am. (3)

Phys. Rev. B (1)

R. Z. Bachrach and F. C. Brown, “Exciton-optical properties of TlBr and TlCl,” Phys. Rev. B 1, 818–831 (1970).
[Crossref]

Phys. Z. (1)

H. A. Kramers, “Die Dispersion und Absorption von Röntgenstralen,” Phys. Z. 30, 522–523 (1929).

Proc. IEEE (1)

R. W. Schafer, R. M. Mersereau, and M. A. Richards, “Constrained iterative restoration algorithms,” Proc. IEEE 69, 432–450 (1981).
[Crossref]

Trans. Am. Math. Soc. (1)

D. J. Patil, “Recapturing H2-functions on a polydisc,” Trans. Am. Math. Soc. 188, 97–103 (1974).

Other (12)

H. A. M. Kramers, “La diffusion de la lumiere par les atomes,” in Atti del Congresso Internazionale dei Fisici, Settembre 1927 (N. Zanichelli, Bologna, Italy, 1928), pp. 545–557.

J. Tauc, “Optical properties of semiconductors” in Proceedings of the International School of Physics “Enrico Fermi,” The Optical Properties of Solids, J. Tauc, ed. (Academic, New York, London, 1966), pp. 63–89.

T. S. Moss, G. J. Burrell, and B. Ellis, Semiconductor Opto-Electronics (Butterworth, London, 1973), Chap. 2.

H. W. Bode, Network Analysis and Feedback Amplifier Design (Van Nostrand, New York, 1945), Chap. XIV.

R. V. Churchill, Complex Variables and Applications (McGraw-Hill, Tokyo, 1960), Chap. 11.

I. N. Sneddon, Special Functions of Mathematical Physics and Chemistry (Oliver and Boyd, Edinburgh, Scotland, 1956), Chap. IV.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), Entry 3.911.1.

E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, 2nd ed. (Oxford U. Press, London, 1959), Chap. V (especially Theorem 95).

G. Dahlquist and Å. Björck, Numerical Methods (Prentice Hall, Englewood Cliffs, N.J., 1974), Chaps. 4 and 5.

A. H. Stroud and D. Secrest, Gaussian Quadrature Formulas (Prentice Hall, Englewood Cliffs, N.J., 1966).

H. Riesel, Department of Numerical and Computing Science, Royal Institute of Technology, Stockholm, Sweden (personal communication).

See Ref. 13, especially Chap. V, Sec. 1.

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Figures (5)

Fig. 1
Fig. 1

Known spectrum [Eqs. (14)] truncated at ω = ω1. By using the real (R) and the imaginary (I) parts for ωω1 (dashed lines), the spectrum has been restored for ω > ω1 (□). The solid lines show the true continuation of the spectrum.

Fig. 2
Fig. 2

Real and imaginary parts of the truncated spectrum are such that the refractive index and the extinction coefficient curves of the Drude–Lorentz absorption model4,5 are obtained for ωω1 [Eqs. (15)]. Graphical symbols are the same as for Fig. 1.

Fig. 3
Fig. 3

Quotient FRcalc(ω)/FRtrue(ω) for the spectrum in Fig. 1 for various values of parameters used in the numerical integration. The accuracy is simultaneously increased in the intervals ω = (0; 1) (where the spectrum is known) and ω = (1; 5) (where the spectrum is unknown) when one or more of the integration parameters are changed. This offers a method to check the numerical integration. H is the integration step length and U is the upper integration boundary (approximating infinity).

Fig. 4
Fig. 4

A set of possible errors ΔFR(ω) and ΔFI(ω) in the real and imaginary parts FR(ω) and FI(ω) of the known part of the spectrum (dashed lines). The error for ωω1, will be interpreted as the low-frequency parts of an analytic function that may (but does not always have to) have considerable real and imaginary parts at ω > ω1 (solid lines). In such a case the restored spectrum is destroyed.

Fig. 5
Fig. 5

The restored spectrum may be obtained from the output of a causal filter. The figure shows the real part HR(ω) of the transfer function H(ω) of the spectrum-restoring filter for λ = 104 and λ = 108. The accuracy of the restored spectrum is increased when the parameter λ increases.

Tables (1)

Tables Icon

Table 1 Coefficients b2n (Calculated for λ = 1028) of the Impulse Response Function h 0 ( t ) = 4 ω 1 / π n = 0 b 2 n J 2 n ( ω 1 t ) of the Spectrum-Restoring Filter

Equations (94)

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F ( ω ) = F R ( ω ) + i F I ( ω ) ,
F R ( ω ) = 1 π - F I ( ω ) ω - ω d ω ,
F I ( ω ) = - 1 π - F R ( ω ) ω - ω d ω .
y ( t ) = - f ( t - u ) x ( u ) d u = 0 t f ( t - u ) x ( u ) d u
F ( Ω ) = - f ( t ) e i Ω t d t = F R ( Ω ) + i F I ( Ω )
F R λ ( ω ) = λ 1 / 4 cos G ( ω ) 1 2 π × ω ω 1 - F R ( ω ) sin G ( ω ) + F I ( ω ) cos G ( ω ) ω - ω d ω + λ 1 / 4 sin G ( ω ) 1 2 π × ω ω 1 F R ( ω ) cos G ( ω ) + F I ( ω ) sin G ( ω ) ω - ω d ω ,
F I λ ( ω ) = λ 1 / 4 sin G ( ω ) 1 2 π × ω ω 1 - F R ( ω ) sin G ( ω ) + F I ( ω ) cos G ( ω ) ω - ω d ω - λ 1 / 4 cos G ( ω ) 1 2 π × ω ω 1 F R ( ω ) cos G ( ω ) + F I ( ω ) sin G ( ω ) ω - ω d ω ,
G ( ω ) = 1 4 π ln ( 1 + λ ) ln | ω - ω 1 ω + ω 1 | .
F R λ ( ω ) = 1 π ω ω 1 F I ( ω ) ω - ω C ( λ , ω , ω ) d ω + 1 π ω ω 1 F R ( ω ) ω - ω S ( λ , ω , ω ) d ω ,
F I λ ( ω ) = - 1 π ω ω 1 F R ( ω ) ω - ω C ( λ , ω , ω ) d ω + 1 π ω ω 1 F I ( ω ) ω - ω S ( λ , ω , ω ) d ω ,
S ( λ , ω , ω ) = λ 1 / 4 2 sin [ G ( ω ) - G ( ω ) ] , C ( λ , ω , ω ) = λ 1 / 4 2 cos [ G ( ω ) - G ( ω ) ] .
T R ( ω ) = F R ( ω ) H R ( ω ) - F I ( ω ) H I ( ω ) = 1 π - T I ( ω ) ω - ω d ω ,
T I ( ω ) = F R ( ω ) H I ( ω ) + F I ( ω ) H R ( ω ) = - 1 π - T R ( ω ) ω - ω d ω .
F R ( ω ) = H R ( ω ) H 2 1 π - T I ( ω ) ω - ω d ω - H I ( ω ) H 2 1 π - T R ( ω ) ω - ω d ω ,
F I ( ω ) = - H R ( ω ) H 2 1 π - T R ( ω ) ω - ω d ω - H I ( ω ) H 2 1 π - T I ( ω ) ω - ω d ω .
H 2 = H R 2 ( ω ) + H I 2 ( ω ) .
F R ( ω ) = H R ( ω ) H 2 1 π ω ω 1 T I ( ω ) ω - ω d ω - H I ( ω ) H 2 1 π ω ω 1 T R ( ω ) ω - ω d ω + H R ( ω ) H 2 1 π ω > ω 1 T I ( ω ) ω - ω d ω - H I ( ω ) H 2 1 π ω > ω 1 T R ( ω ) ω - ω d ω ,
F I ( ω ) = - H R ( ω ) H 2 1 π ω ω 1 T R ( ω ) ω - ω d ω - H I ( ω ) H 2 1 π ω ω 1 T I ( ω ) ω - ω d ω - H R ( ω ) H 2 1 π ω > ω 1 T R ( ω ) ω - ω d ω - H I ( ω ) H 2 1 π ω > ω 1 T I ( ω ) ω - ω d ω .
H p ( λ , ω ) = { λ p e - i G ( ω ) , ω ω 1 2 λ p - 1 / 4 e - i G ( ω ) , ω > ω 1 , G ( ω ) = 1 4 π ln ( 1 + λ ) ln | ω - ω 1 ω + ω 1 | .
H 0 ( λ , ω ) H ( ω ) = { e - i G ( ω ) , ω ω 1 2 λ - 1 / 4 e - i G ( ω ) , ω > ω 1 .
F R ( ω ) = cos G ( ω ) 1 π × ω ω 1 - F R ( ω ) sin G ( ω ) + F I ( ω ) cos G ( ω ) ω - ω d ω + sin G ( ω ) 1 π × ω ω 1 F R ( ω ) cos G ( ω ) + F I ( ω ) sin G ( ω ) ω - ω d ω ,
F I ( ω ) = sin G ( ω ) 1 π × ω ω 1 - F R ( ω ) sin G ( ω ) + F I ( ω ) cos G ( ω ) ω - ω d ω - cos G ( ω ) 1 π × ω ω 1 F R ( ω ) cos G ( ω ) + F I ( ω ) sin G ( ω ) ω - ω d ω .
F R ( 0 ) = 1 π ω ω 1 - F R ( ω ) sin G ( ω ) + F I ( ω ) cos G ( ω ) ω d ω ,
M ( ω ) = M ( λ , ω ) = { e i G ( ω ) , ω < ω 1 λ 1 / 4 2 e i G ( ω ) , ω ω 1 .
F R λ ( ω ) = λ 1 / 4 cos G ( ω ) 1 2 π × ω ω 1 F R ( ω ) sin G ( ω ) + F I ( ω ) cos G ( ω ) ω - ω d ω - λ 1 / 4 sin G ( ω ) 1 2 π × ω ω 1 F R ( ω ) cos G ( ω ) - F I ( ω ) sin G ( ω ) ω - ω d ω ,
F I λ ( ω ) = - λ 1 / 4 sin G ( ω ) 1 2 π × ω ω 1 F R ( ω ) sin G ( ω ) + F I ( ω ) cos G ( ω ) ω - ω d ω - λ 1 / 4 cos G ( ω ) 1 2 π × ω ω 1 F R ( ω ) cos G ( ω ) - F I ( ω ) sin G ( ω ) ω - ω d ω .
H I ( ω ) = δ ( ω - ω 2 ) - δ ( ω + ω 2 ) ,
H R ( ω ) = 2 π ω 2 ω 2 2 - ω 2 .
F I ( ω ) = F I SKK ( ω ) = ω ω 2 F I ( ω 2 ) - 2 π ω ( ω 2 - ω 2 2 ) 0 F R ( ω ) d ω ( ω 2 2 - ω 2 ) ( ω 2 - ω 2 ) , F R ( ω ) = F R SKK ( ω ) = F R ( ω 2 ) + 2 π ( ω 2 - ω 2 2 ) 0 ω F I ( ω ) d ω ( ω 2 2 - ω 2 ) ( ω 2 - ω 2 ) .
F R ( ω ) = i n i 2 - k i 2 - 1 = i N i e 2 m 0 ω i 2 - ω 2 ( ω i 2 - ω 2 ) 2 + g i 2 ω 2 , F I ( ω ) = i 2 n i k i = i N i e 2 m 0 g i ω ( ω i 2 - ω 2 ) 2 + g i 2 ω 2 .
F R ( ω ) = 1 1 + ω 2 ,
F I ( ω ) = ω 1 + ω 2             ( ω ω 1 = 1 ) .
F R ( ω ) = 4 - ω 2 ( 4 - ω 2 ) 2 + ω 2 + 3 16 - ω 2 ( 16 - ω 2 ) 2 + 4 ω 2 ,
F I ( ω ) = ω ( 4 - ω 2 ) 2 + ω 2 + 3 2 ω ( 16 - ω 2 ) 2 + 4 ω 2 ,
x = - ln ( ω 1 - ω ω 1 + ω ) ,
F R λ ( ω ) = K ( λ , ω ) [ ω sin G ( ω ) ( I 1 - I 2 ) + cos G ( ω ) ( I 3 + I 4 ) ] ,
F I λ ( ω ) = K ( λ , ω ) [ - ω cos G ( ω ) ( I 1 - I 2 ) + sin G ( ω ) ( I 3 + I 4 ) ] ,
K ( λ , ω ) = - λ 14 π ω 1 ω 1 2 + ω 2 , I 1 = 0 F R [ ω ( x ) ] cos ( γ x ) N ( ω , ω 1 , x ) d x 0 u ( ) d x , I 2 = 0 F I [ ω ( x ) ] sin ( γ x ) N ( ω , ω 1 , x ) d x , I 3 = 0 ω ( x ) F I [ ω ( x ) ] cos ( γ x ) N ( ω , ω 1 , x ) d x , I 4 = 0 ω ( x ) F R [ ω ( x ) ] sin ( γ x ) N ( ω , ω 1 , x ) d x , N ( ω , ω 1 , x ) = 1 + ( e - x + e x ) 2 ( ω 2 + ω 1 2 ) ω 2 - ω 1 2 , γ = 1 4 π ln ( 1 + λ ) .
I 1 = 0 F R [ ω ( x ) ] cos ( γ x ) N ( ω , ω 1 , x ) d x = 0 { F R [ ω ( ω ) ] N ( ω , ω 1 , x ) - F R ( ω 1 ) ω 2 - ω 1 2 2 ( ω 2 + ω 1 2 ) } cos ( γ x ) d x + F R ( ω 1 ) ω 2 - ω 1 2 2 ( ω 2 + ω 1 2 ) 0 e - x cos ( γ x ) d x ,
0 e - x cos ( γ x ) d x = 1 / ( 1 + γ 2 ) .
F R calc ( ω ) = A F R t ( ω ) + ( B - A ) λ 1 / 4 1 2 π ω ω 1 F I ( ω ) ω - ω cos [ G ( ω ) - G ( ω ) ] d ω .
h ( t ) = 1 2 π - H ( ω ) e - i ω t d ω = 1 2 π ω ω 1 e - i G ( ω ) e - i ω t d ω + λ - 1 / 4 π ω > ω 1 e - i G ( ω ) e - i ω t d ω h 0 ( t ) + h 1 ( t ) .
h 0 ( t ) = ω 1 2 π - 1 1 e - i G ( x ) e - i x ω 1 t d x ,
G ( x ) = 1 4 π ln ( 1 + λ ) ln ( 1 - x 1 + x ) .
e - i x ω 1 t = l = - J l ( ω 1 t ) e - i l arcsin x ,
h 0 ( t ) = h 0 A ( t ) + h 0 S ( t ) = 2 h 0 A ( t ) = 4 ω 1 π m = 0 a 2 m + 1 J 2 m + 1 ( ω 1 t ) = 2 h 0 S ( t ) = 4 ω 1 π n = 0 b 2 n J 2 n ( ω 1 t ) ,
a 2 m + 1 = - 0 1 sin G ( x ) sin [ ( 2 m + 1 ) arcsin x ] d x ,
b 2 n = 0 1 cos G ( x ) cos [ 2 n arcsin x ] d x .
g λ ( z ) = λ h λ ( z ) 1 2 π i E h λ * ( t ) g ( t ) t - z d t ,
h λ ( z ) = exp [ - 1 4 π i ln ( 1 + λ ) E 1 + t z t - z 1 1 + t 2 d t ] .
E A 1 + t z t - z 1 1 + t 2 d t = E A d t t - z - 1 2 E A 2 t 1 + t 2 d t ,
E A d t t - z = E A d ( τ + i σ ) ( τ - x ) - i ( y - σ ) = ln | x 0 + x x 0 - x x 1 - x x 1 + x | + i π R ( x , x 0 , x 1 )
R ( x , x 0 , x 1 ) = { 1 for             x 0 x x 1 0 elsewhere .
E A 2 t 1 + t 2 d t = I E A ln ( 1 + t 2 ) = - 2 π i
h λ ( x ) = exp ( - 1 4 π i ln ( 1 + λ ) × { i π [ 1 + R ( x , x 0 , x 1 ) ] + ln | x 0 + x x 0 - x x 1 - x x 1 + x | } ) = ( 1 + λ ) - [ 1 + R ( x , x 0 , x 1 ) ] / 4 × exp [ i 4 π ln ( 1 + λ ) ln | x 0 + x x 0 - x x 1 - x x 1 + x | ] ,
h λ ( x ) = ( 1 + λ ) - 1 / 2 e i G ( x )             for             x 0 x x 1 ,
h λ ( x ) = ( 1 + λ ) - 1 / 4 e i G ( x ) ,             elsewhere ,
G ( x ) = 1 4 π ln ( 1 + λ ) ln | x 0 + x x 0 - x x 1 - x x 1 + x | .
E A h λ * ( t ) g ( t ) t - z d t = E A h λ * ( t ) g ( t ) ( τ - x ) - i ( y - σ ) d t = E A h λ * ( t ) g ( t ) τ - x d t + i π 1 π E A ( y - σ ) h λ * ( t ) g ( t ) ( τ - x ) 2 + ( y - σ ) 2 d t x 0 τ x 1 h λ * ( τ ) g ( τ ) τ - x d τ + i π h λ * ( x ) g ( x ) R ( x , x 0 , x 1 ) ,
g λ ( x ) = e i G ( x ) 1 2 π i x 0 τ x 1 e - i G ( τ ) g ( τ ) τ - x d τ + 1 2 g ( x ) .
g λ ( x ) = λ 1 / 4 e i G ( x ) 1 2 π i x 0 τ x 1 e i G ( τ ) g ( τ ) τ - x d τ .
F R λ ( ω ) = λ 1 / 4 cos G ( ω ) 1 2 π × ω 0 ω ω 1 - F R ( ω ) sin G ( ω ) + F I ( ω ) cos G ( ω ) ω - ω d ω + λ 1 / 4 sin G ( ω ) 1 2 π × ω 0 ω ω 1 F R ( ω ) cos G ( ω ) + F I ( ω ) sin G ( ω ) ω - ω d ω ,
F I λ ( ω ) = λ 1 / 4 sin G ( ω ) 1 2 π × ω 0 ω ω 1 - F R ( ω ) sin G ( ω ) + F I ( ω ) cos G ( ω ) ω - ω d ω - λ 1 / 4 cos G ( ω ) 1 2 π × ω 0 ω ω 1 F R ( ω ) cos G ( ω ) + F I ( ω ) sin G ( ω ) ω - ω d ω ,
Φ R ( ω ) = cos G ( ω ) 1 π × ω ω 1 - F R ( ω ) sin G ( ω ) + F I ( ω ) cos G ( ω ) ω - ω d ω + sin G ( ω ) 1 π × ω ω 1 F R ( ω ) cos G ( ω ) + F I ( ω ) sin G ( ω ) ω - ω d ω = 1 π ω ω 1 F R ( ω ) sin [ G ( ω ) - G ( ω ) ] ω - ω d ω + 1 π ω ω 1 F I ( ω ) cos [ G ( ω ) - G ( ω ) ] ω - ω d ω .
G ( ω ) - G ( ω ) = G ( ω 1 x ) - G ( ω 1 y ) = γ [ ln ( 1 - x 1 + x ) - ln ( 1 - y 1 + y ) ] ,
γ = 1 4 π ln ( 1 + λ ) , x = ω / ω 1 ,             x 1 , y = ω / ω 1 ,             y 1. G ( ω 1 x ) - G ( ω 1 y ) = 2 γ [ ( y - x ) + ( y 3 - x 3 ) + ( y 2 - x 5 ) + ] , 2 γ [ ( y - x ) + u ( x , y ) ] ,
ln ( 1 - x 1 + x ) = - 2 ( x + 1 3 x 3 + 1 5 x 5 + )
Φ R ( ω ) = Φ R ( ω 1 x ) = 1 π - 1 1 F R ( ω 1 y ) sin [ 2 γ ( y - x ) + 2 γ u ( x , y ) ] y - x d y + 1 π - 1 1 F I ( ω 1 y ) cos [ 2 γ ( y - x ) + 2 γ u ( x , y ) ] y - x d y = 1 π - 1 1 F R ( ω 1 y ) cos [ 2 γ u ( x , y ) ] sin 2 γ ( y - x ) y - x d y + 1 π - 1 1 F R ( ω 1 y ) sin [ 2 γ u ( x , y ) ] cos 2 γ ( y - x ) y - x d y + 1 π - 1 1 F I ( ω 1 y ) cos [ 2 γ u ( x , y ) ] cos 2 γ ( y - x ) y - x d y - 1 π - 1 1 F I ( ω 1 y ) sin [ 2 γ u ( x , y ) ] sin 2 γ ( y - x ) y - x d y .
lim γ 1 π sin 2 γ ( y - x ) y - x = δ ( y - x )
1 π - 1 1 F R ( ω 1 y ) cos [ 2 γ u ( x , y ) ] sin 2 γ ( y - x ) y - x d y = F R ( ω 1 x ) cos [ 2 γ u ( x , x ) ] = F R ( ω 1 x ) = F R ( ω ) ,
1 π - 1 1 F I ( ω 1 y ) sin [ 2 γ u ( x , y ) ] sin 2 γ ( y - x ) y - x d y = F I ( ω 1 x ) sin [ 2 γ u ( x , x ) ] = 0.
- 1 1 F R ( ω 1 y ) sin [ 2 γ u ( x , y ) ] cos 2 γ ( y - x ) y - x d y - 1 x ( ) d y + x 1 ( ) d y
- 1 x F R ( ω 1 y ) sin [ 2 γ u ( x , y ) ] cos 2 γ ( y - x ) y - x d y = - - ln 2 γ ( 1 + x ) F R [ ω 1 ( x - 1 2 γ e - t ) ] × sin [ 2 γ u ( x , x - 1 2 γ e - t ) ] cos ( e - t ) d t - F R ( ω 1 x ) sin [ 2 γ u ( x , x ) ] - cos ( e - t ) d t = 0
x 1 F R ( ω 1 y ) sin [ 2 γ u ( x , y ) cos 2 γ ( y - x ) y - x d y 0
lim γ - 1 1 F R ( ω 1 y ) sin [ 2 γ u ( x , y ) ] cos 2 γ ( y - x ) y - x d y = 0.
- 1 1 F I ( ω 1 y ) cos [ 2 γ u ( x , y ) ] cos 2 γ ( y - x ) y - x d y - 1 x ( ) d y + x 1 ( ) d y ,
- 1 x ( ) d y = - 1 x F I ( ω 1 y ) cos [ 2 γ u ( x , y ) ] × cos 2 γ ( y - x ) y - x d y = - - ln 2 γ ( 1 + x ) F I [ ω 1 ( x - 1 2 γ e - t ) ] × cos [ 2 γ u ( x , x - 1 2 γ e - t ) ] cos ( e - t ) d t - F I ( ω 1 x ) - cos ( e - t ) d t
x 1 F I ( ω 1 y ) cos [ 2 γ u ( x , y ) ] cos 2 γ ( y - x ) y - x d y + F I ( ω 1 x ) - cos ( e - t ) d t when γ . lim γ - 1 1 F I ( ω 1 y ) cos [ 2 γ u ( x , y ) ] cos 2 γ ( y - x ) y - x d y = 0.
h 0 ( t ) = ω 1 2 π - 1 1 e - i G ( x ) e - i x ω 1 t d x ,
G ( x ) = 1 4 π ln ( 1 + λ ) ln ( 1 - x 1 + x ) .
e - i x ω 1 t = l = - J l ( ω 1 t ) e - i l arcsin x
h 0 ( t ) = ω 1 2 π l = - [ - 1 1 e - i G ( x ) e - i l arcsin x d x ] J l ( ω 1 t ) .
- 1 1 e - i G ( x ) e - i l arcsin x d x = - 2 0 1 sin G ( x ) sin ( l arcsin x ) d x + 2 0 1 cos G ( x ) cos ( l arcsin x ) d x , 2 ( a l + b l )
a - l = - a l , b - l = + b l , a 0 = 0 ,
b 0 = 0 1 cos G ( x ) d x = 2 0 e - t ( 1 + e - t ) 2 cos ( γ t ) d t = - 2 0 d ( 1 1 + e t ) cos ( γ t ) = γ π [ sinh ( γ π ) ] - 1 2 γ π e - γ π λ - 1 / 4 2 ln ( λ )
0 sin ( γ x ) 1 + e x d x = 1 2 γ - π 2 [ sin h ( γ π ) ] - 1 .
J - l ( z ) = ( - 1 ) l J l ( z ) ,
h 0 ( t ) = ω 1 π l = 1 ( a - l + b - l ) J - l ( ω 1 t ) + ( a l + b l ) J l ( ω 1 t ) = ω 1 π l = 1 [ 1 - ( - 1 ) l ] a l J l ( ω 1 t ) + [ 1 + ( - 1 ) l ] b l J l ( ω 1 t ) = 2 ω 1 π m = 0 a 2 m + 1 J 2 m + 1 ( ω 1 t ) + 2 ω 1 π n = 1 b 2 n J 2 n ( ω 1 t ) ,
a 2 m + 1 = - 0 1 sin G ( x ) sin [ ( 2 m + 1 ) arcsin x ] d x ,
b 2 n = 0 1 cos G ( x ) cos [ 2 n arcsin x ] d x
J l ( - z ) = ( - 1 ) l J l ( z ) ,
h 0 ( t ) = h 0 A ( t ) + h 0 S ( t ) ,
h 0 A ( t ) = 2 ω 1 π m = 0 a 2 m + 1 J 2 m + 1 ( ω 1 t ) , h 0 S ( t ) = 2 ω 1 π n = 1 b 2 n J 2 n ( ω 1 t ) .
h 0 ( t ) = 2 h 0 A ( t ) = 2 h 0 S ( t ) = 4 ω 1 π m = 0 a 2 m + 1 J 2 m + 1 ( ω 1 t ) = 4 ω 1 π n = 1 b 2 n J 2 n ( ω 1 t ) .