Abstract

A method is presented to deal with the numerical evaluation of Kramers–Kronig transforms (the Hilbert transforms of even and odd functions on the positive real axis). The general Hilbert transform is also treated. The functions involved must be continuous on the integration interval with suitable asymptotic behavior for large values of the argument and must have an appropriate functional form in the vicinity of the singularity of the integrand of the transform. The approach is based on a specialized Gaussian quadrature technique that uses the weight function log x-1. This choice allows the region in the vicinity of the singularity to be swept into the quadrature weights and abscissa values. Application to the Lorentzian and Gaussian line profiles is discussed.

© 2002 Optical Society of America

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  34. K. Diethelm, “The order of convergence of modified interpolatory quadratures for singular integrals of Cauchy type,” Z. Angew. Math. Mech. 75, S621–S622 (1995).
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    [CrossRef]
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    [CrossRef]
  39. K. Diethelm, “A method for the practical evaluation of the Hilbert transform on the real line,” J. Comput. Appl. Math. 112, 45–53 (1999).
    [CrossRef]
  40. K. Diethelm and P. Köhler, “Asymptotic behaviour of fixed-order error constants of modified quadrature formulae for Cauchy principal value integrals,” J. Inequal. Appl. 5, 167–190 (2000).
  41. F. W. King, G. J. Smethells, G. T. Helleloid, and P. J. Pelzl, “Numerical evaluation of Hilbert transforms for oscillatory functions: a convergence accelerator approach,” Comput. Phys. Commun. 145, 256–260 (2002).
    [CrossRef]
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    [CrossRef]
  44. D. M. Roessler, “Kramers–Kronig analysis of reflection data III. Approximations, with reference to sodium iodide,” Br. J. Appl. Phys. 17, 1313–1317 (1966).
    [CrossRef]
  45. K. Kozima, W. Suëtaka, and P. N. Schatz, “Optical constants of thin films by a Kramers–Kronig method,” J. Opt. Soc. Am. 56, 181–184 (1966).
    [CrossRef]
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  51. D. Parris and S. J. Van Der Walt, “A new numerical method for evaluating the Kramers–Kronig transformation,” Anal. Biochem. 68, 321–327 (1975).
    [CrossRef] [PubMed]
  52. A. Balzarotti, E. Colavita, S. Gentile and R. Rosei, “Kramers–Krönig analysis of modulated reflectance data investigation of errors,” Appl. Opt. 14, 2412–2417 (1975).
    [CrossRef] [PubMed]
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  54. K.-E. Peiponen and E. M. Vartiainen, “Kramers–Kronig relations in optical data inversion,” Phys. Rev. B 44, 8301–8303 (1991).
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  55. J. H. Bertie and S. L. Zhang, “Infrared intensities of liquids. IX. The Kramers–Kronig transform, and its approximation by the finite Hilbert transform via fast Fourier transforms,” Can. J. Chem. 70, 520–531 (1992).
    [CrossRef]
  56. P. P. Kircheva and G. B. Hadjichristov, “Kramers–Kronig relations in FWM spectroscopy,” J. Phys. B 27, 3781–3793 (1994).
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  57. J. H. Bertie and Z. Lan, “An accurate modified Kramers–Kronig transformation from reflectance to phase shift on attenuated total reflection,” J. Chem. Phys. 105, 8502–8514 (1996).
    [CrossRef]
  58. K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, “Phase retrieval in optical spectroscopy: resolving optical constants from power spectra,” Appl. Spectrosc. 50, 1283–1289 (1996).
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2002

F. W. King, G. J. Smethells, G. T. Helleloid, and P. J. Pelzl, “Numerical evaluation of Hilbert transforms for oscillatory functions: a convergence accelerator approach,” Comput. Phys. Commun. 145, 256–260 (2002).
[CrossRef]

2000

K. Diethelm and P. Köhler, “Asymptotic behaviour of fixed-order error constants of modified quadrature formulae for Cauchy principal value integrals,” J. Inequal. Appl. 5, 167–190 (2000).

1999

K. Diethelm, “A method for the practical evaluation of the Hilbert transform on the real line,” J. Comput. Appl. Math. 112, 45–53 (1999).
[CrossRef]

1997

K. Diethelm, “New error bounds for modified quadrature formulas for Cauchy principal value integrals,” J. Comput. Appl. Math. 82, 93–104 (1997).
[CrossRef]

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, “Complex analysis in dispersion theory,” Opt. Rev. 4, 433–441 (1997).
[CrossRef]

1996

J. H. Bertie and Z. Lan, “An accurate modified Kramers–Kronig transformation from reflectance to phase shift on attenuated total reflection,” J. Chem. Phys. 105, 8502–8514 (1996).
[CrossRef]

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, “Phase retrieval in optical spectroscopy: resolving optical constants from power spectra,” Appl. Spectrosc. 50, 1283–1289 (1996).
[CrossRef]

K. Diethelm, “A definiteness criterion for linear functionals and its application to Cauchy principal value quadrature,” J. Comput. Appl. Math. 66, 167–176 (1996).
[CrossRef]

K. Diethelm, “Definite quadrature formulae for Cauchy principal value integrals,” Bolyai Soc. Math. Stud. 5, 175–186 (1996).

K. Diethelm, “Peano kernels and bounds for the error constants of Gaussian and related quadrature rules for Cauchy principal value integrals,” Numer. Math. 73, 53–63 (1996).
[CrossRef]

1995

K. Diethelm, “Asymptotically sharp error bounds for a quadrature rule for Cauchy principal value integrals based on piecewise linear interpolation,” Approx. Theory Appl. (N.S.) 11, 78–89 (1995).

K. Diethelm, “Gaussian quadrature formulae of the third kind for Cauchy principal value integrals: basic properties and error estimates,” J. Comput. Appl. Math. 65, 97–114 (1995).
[CrossRef]

K. Diethelm, “The order of convergence of modified interpolatory quadratures for singular integrals of Cauchy type,” Z. Angew. Math. Mech. 75, S621–S622 (1995).

1994

B. D. Vecchia, “Two new formulas for the numerical evaluation of the Hilbert transform,” BIT Numer. Math. 34, 346–360 (1994).
[CrossRef]

K. Diethelm, “Error estimates for a quadrature rule for Cauchy principal value integrals,” Proc. Symp. Appl. Math. 48, 287–291 (1994).
[CrossRef]

K. Diethelm, “Nonoptimality of certain quadrature rules for Cauchy principal value integrals,” Z. Angew. Math. Mech. 74, T689–T690 (1994).

K. Diethelm, “Modified compound quadrature rules for strongly singular integrals,” Computing 52, 337–354 (1994).
[CrossRef]

K. Diethelm, “Uniform convergence of optimal order quadrature rules for Cauchy principal value integrals,” J. Comput. Appl. Math. 56, 321–329 (1994).
[CrossRef]

P. P. Kircheva and G. B. Hadjichristov, “Kramers–Kronig relations in FWM spectroscopy,” J. Phys. B 27, 3781–3793 (1994).
[CrossRef]

1992

J. H. Bertie and S. L. Zhang, “Infrared intensities of liquids. IX. The Kramers–Kronig transform, and its approximation by the finite Hilbert transform via fast Fourier transforms,” Can. J. Chem. 70, 520–531 (1992).
[CrossRef]

F. W. King, K. J. Dykema, and A. D. Lund, “Calculation of some integrals for the atomic three-electron problem,” Phys. Rev. A 46, 5406–5416 (1992).
[CrossRef] [PubMed]

1991

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

1989

P. Rabinowitz and D. S. Lubinsky, “Noninterpolatory integration rules for Cauchy principal value integrals,” Math. Comput. 53, 279–295 (1989).
[CrossRef]

1988

1986

G. A. Gazonas, “The numerical evaluation of Cauchy principal value integrals via the fast Fourier transform,” Int. J. Comput. Math. 18, 277–288 (1986).
[CrossRef]

P. Rabinowitz, “Some practical aspects in the numerical evaluation of Cauchy principal value integrals,” Int. J. Comput. Math. 20, 283–298 (1986).
[CrossRef]

1983

P. R. Rabinowitz, “Gauss–Kronrod integration rules for Cauchy principal value integrals,” Math. Comput. 41, 63–78 (1983).
[CrossRef]

1982

G. Monegato, “The numerical evaluation of one-dimensional Cauchy principal value integrals,” Computing 29, 337–354 (1982).
[CrossRef]

1981

T. Andersson, J. Johansson, and H. Eklund, “Numerical solution of the Hilbert transform for phase calculation from an amplitude spectrum,” Math. Computers Simul. 23, 262–266 (1981).
[CrossRef]

H.-P. Liu and D. D. Kosloff, “Numerical evaluation of the Hilbert transform by the fast Fourier transform (FFT) technique,” Geophys. J. R. Astron. Soc. 67, 791–799 (1981).
[CrossRef]

1980

O. E. Taurian, “A method and a program for the numerical evaluation of the Hilbert transform of a real function,” Comput. Phys. Commun. 20, 291–307 (1980).
[CrossRef]

1979

S. J. Collocott and G. J. Troup, “Adaptation: numerical solution of the Kramers–Kronig transforms by trapezoidal summation as compared to a Fourier method,” Comput. Phys. Commun. 17, 393–395 (1979).
[CrossRef]

1978

1977

F. W. King, “A Fourier series algorithm for the analysis of reflectance data,” J. Phys. C 10, 3199–3204 (1977).
[CrossRef]

M. Rasigni and G. Rasigni, “Optical constants of lithium deposits as detected from the Kramers–Kronig analysis,” J. Opt. Soc. Am. 67, 54–59 (1977).
[CrossRef]

S. J. Collocott, “Numerical solution of Kramers–Kronig transforms by a Fourier method,” Comp. Phys. Commun. 13, 203–206 (1977).
[CrossRef]

1975

D. Parris and S. J. Van Der Walt, “A new numerical method for evaluating the Kramers–Kronig transformation,” Anal. Biochem. 68, 321–327 (1975).
[CrossRef] [PubMed]

A. Balzarotti, E. Colavita, S. Gentile and R. Rosei, “Kramers–Krönig analysis of modulated reflectance data investigation of errors,” Appl. Opt. 14, 2412–2417 (1975).
[CrossRef] [PubMed]

D. W. Johnson, “A Fourier series method for numerical Kramers–Kronig analysis,” J. Phys. A. Math. Gen. 8, 490–495 (1975).
[CrossRef]

1973

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]

B. Danloy, “Numerical construction of Gaussian quadrature formulas for ∫01(−log x)xα f(x)dx and ∫01 Em(x)f(x)dx,” Math. Comput. 27, 861–869 (1973).

D. B. Hunter, “The numerical evaluation of Cauchy principal values of integrals by Romberg integration,” Numer. Math. 21, 185–192 (1973).
[CrossRef]

1972

K. Atkinson, “The numerical evaluation of the Cauchy transform on simple closed curves,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 9, 284–299 (1972).
[CrossRef]

D. F. Paget and D. Elliott, “An algorithm for the numerical evaluation of certain Cauchy principal value integrals,” Numer. Math. 19, 373–385 (1972).
[CrossRef]

R. A. Sack and A. F. Donovan, “An algorithm for Gaussian quadrature given modified moments,” Numer. Math. 18, 465–478 (1972).
[CrossRef]

R. K. Ahrenkiel, “Modified Kramers–Kronig analysis of optical spectra: erratum,” J. Opt. Soc. Am. 62, 1009 (1972).
[CrossRef]

J. D. Neufeld and G. Andermann, “Kramers–Kronig dispersion-analysis method for treating infrared transmittance data,” J. Opt. Soc. Am. 62, 1156–1162 (1972).
[CrossRef]

1971

1970

H. Morawitz, “A numerical approach to principal value integrals in dispersion relations,” J. Comput. Phys. 6, 120–123 (1970).
[CrossRef]

R. Piessens, “Numerical evaluation of Cauchy principal values of integrals,” BIT 10, 476–480 (1970).
[CrossRef]

1968

1967

C. A. Emeis, L. J. Oosterhoff, and G. de Vries, “Numerical evaluation of Kramers–Kronig relations,” Proc. R. Soc. London, Ser. A 297, 54–65 (1967).
[CrossRef]

L. M. Delves, “The numerical evaluation of principal value integrals,” Comput. J. (Cambridge) 10, 389–391 (1967).

1966

D. M. Roessler, “Kramers–Kronig analysis of reflection data III. Approximations, with reference to sodium iodide,” Br. J. Appl. Phys. 17, 1313–1317 (1966).
[CrossRef]

K. Kozima, W. Suëtaka, and P. N. Schatz, “Optical constants of thin films by a Kramers–Kronig method,” J. Opt. Soc. Am. 56, 181–184 (1966).
[CrossRef]

1965

G. Andermann, A. Caron, and D. A. Dows, “Kramers–Kronig dispersion analysis of infrared reflectance bands,” J. Opt. Soc. Am. 55, 1210–1216 (1965).
[CrossRef]

D. M. Roessler, “Kramers–Kronig analysis of non-normal incidence reflection,” Br. J. Appl. Phys. 16, 1359–1366 (1965).
[CrossRef]

D. G. Anderson, “Gaussian quadrature formula for ∫01−ln(x)f(x)dx,” Math. Comput. 19, 477–481 (1965).

1958

I. M. Longman, “On the numerical evaluation of Cauchy principal values of integrals,” Math. Tables Aids Comp. 12, 205–207 (1958).
[CrossRef]

1929

H. A. Kramers, “Die dispersion und absorption von Röntgenstrahlen,” Phys. Z. 30, 522–523 (1929).

1927

H. A. Kramers, “La diffusion de la lumière par les atomes,” Atti. Congr. Int. Fis. 2, 545–557 (1927).

1926

Ahrenkiel, R. K.

Andermann, G.

Anderson, D. G.

D. G. Anderson, “Gaussian quadrature formula for ∫01−ln(x)f(x)dx,” Math. Comput. 19, 477–481 (1965).

Andersson, T.

T. Andersson, J. Johansson, and H. Eklund, “Numerical solution of the Hilbert transform for phase calculation from an amplitude spectrum,” Math. Computers Simul. 23, 262–266 (1981).
[CrossRef]

Asakura, T.

Atkinson, K.

K. Atkinson, “The numerical evaluation of the Cauchy transform on simple closed curves,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 9, 284–299 (1972).
[CrossRef]

Balzarotti, A.

Bertie, J. H.

J. H. Bertie and Z. Lan, “An accurate modified Kramers–Kronig transformation from reflectance to phase shift on attenuated total reflection,” J. Chem. Phys. 105, 8502–8514 (1996).
[CrossRef]

J. H. Bertie and S. L. Zhang, “Infrared intensities of liquids. IX. The Kramers–Kronig transform, and its approximation by the finite Hilbert transform via fast Fourier transforms,” Can. J. Chem. 70, 520–531 (1992).
[CrossRef]

Caron, A.

Colavita, E.

Collocott, S. J.

S. J. Collocott and G. J. Troup, “Adaptation: numerical solution of the Kramers–Kronig transforms by trapezoidal summation as compared to a Fourier method,” Comput. Phys. Commun. 17, 393–395 (1979).
[CrossRef]

S. J. Collocott, “Numerical solution of Kramers–Kronig transforms by a Fourier method,” Comp. Phys. Commun. 13, 203–206 (1977).
[CrossRef]

Danloy, B.

B. Danloy, “Numerical construction of Gaussian quadrature formulas for ∫01(−log x)xα f(x)dx and ∫01 Em(x)f(x)dx,” Math. Comput. 27, 861–869 (1973).

de Vries, G.

C. A. Emeis, L. J. Oosterhoff, and G. de Vries, “Numerical evaluation of Kramers–Kronig relations,” Proc. R. Soc. London, Ser. A 297, 54–65 (1967).
[CrossRef]

Delves, L. M.

L. M. Delves, “The numerical evaluation of principal value integrals,” Comput. J. (Cambridge) 10, 389–391 (1967).

Diethelm, K.

K. Diethelm and P. Köhler, “Asymptotic behaviour of fixed-order error constants of modified quadrature formulae for Cauchy principal value integrals,” J. Inequal. Appl. 5, 167–190 (2000).

K. Diethelm, “A method for the practical evaluation of the Hilbert transform on the real line,” J. Comput. Appl. Math. 112, 45–53 (1999).
[CrossRef]

K. Diethelm, “New error bounds for modified quadrature formulas for Cauchy principal value integrals,” J. Comput. Appl. Math. 82, 93–104 (1997).
[CrossRef]

K. Diethelm, “A definiteness criterion for linear functionals and its application to Cauchy principal value quadrature,” J. Comput. Appl. Math. 66, 167–176 (1996).
[CrossRef]

K. Diethelm, “Definite quadrature formulae for Cauchy principal value integrals,” Bolyai Soc. Math. Stud. 5, 175–186 (1996).

K. Diethelm, “Peano kernels and bounds for the error constants of Gaussian and related quadrature rules for Cauchy principal value integrals,” Numer. Math. 73, 53–63 (1996).
[CrossRef]

K. Diethelm, “Asymptotically sharp error bounds for a quadrature rule for Cauchy principal value integrals based on piecewise linear interpolation,” Approx. Theory Appl. (N.S.) 11, 78–89 (1995).

K. Diethelm, “Gaussian quadrature formulae of the third kind for Cauchy principal value integrals: basic properties and error estimates,” J. Comput. Appl. Math. 65, 97–114 (1995).
[CrossRef]

K. Diethelm, “The order of convergence of modified interpolatory quadratures for singular integrals of Cauchy type,” Z. Angew. Math. Mech. 75, S621–S622 (1995).

K. Diethelm, “Error estimates for a quadrature rule for Cauchy principal value integrals,” Proc. Symp. Appl. Math. 48, 287–291 (1994).
[CrossRef]

K. Diethelm, “Nonoptimality of certain quadrature rules for Cauchy principal value integrals,” Z. Angew. Math. Mech. 74, T689–T690 (1994).

K. Diethelm, “Modified compound quadrature rules for strongly singular integrals,” Computing 52, 337–354 (1994).
[CrossRef]

K. Diethelm, “Uniform convergence of optimal order quadrature rules for Cauchy principal value integrals,” J. Comput. Appl. Math. 56, 321–329 (1994).
[CrossRef]

Donovan, A. F.

R. A. Sack and A. F. Donovan, “An algorithm for Gaussian quadrature given modified moments,” Numer. Math. 18, 465–478 (1972).
[CrossRef]

Dows, D. A.

Duesler, E.

Dykema, K. J.

F. W. King, K. J. Dykema, and A. D. Lund, “Calculation of some integrals for the atomic three-electron problem,” Phys. Rev. A 46, 5406–5416 (1992).
[CrossRef] [PubMed]

Eklund, H.

T. Andersson, J. Johansson, and H. Eklund, “Numerical solution of the Hilbert transform for phase calculation from an amplitude spectrum,” Math. Computers Simul. 23, 262–266 (1981).
[CrossRef]

Elliott, D.

D. F. Paget and D. Elliott, “An algorithm for the numerical evaluation of certain Cauchy principal value integrals,” Numer. Math. 19, 373–385 (1972).
[CrossRef]

Emeis, C. A.

C. A. Emeis, L. J. Oosterhoff, and G. de Vries, “Numerical evaluation of Kramers–Kronig relations,” Proc. R. Soc. London, Ser. A 297, 54–65 (1967).
[CrossRef]

Gazonas, G. A.

G. A. Gazonas, “The numerical evaluation of Cauchy principal value integrals via the fast Fourier transform,” Int. J. Comput. Math. 18, 277–288 (1986).
[CrossRef]

Gentile, S.

Hadjichristov, G. B.

P. P. Kircheva and G. B. Hadjichristov, “Kramers–Kronig relations in FWM spectroscopy,” J. Phys. B 27, 3781–3793 (1994).
[CrossRef]

Helleloid, G. T.

F. W. King, G. J. Smethells, G. T. Helleloid, and P. J. Pelzl, “Numerical evaluation of Hilbert transforms for oscillatory functions: a convergence accelerator approach,” Comput. Phys. Commun. 145, 256–260 (2002).
[CrossRef]

Hunter, D. B.

D. B. Hunter, “The numerical evaluation of Cauchy principal values of integrals by Romberg integration,” Numer. Math. 21, 185–192 (1973).
[CrossRef]

Ishida, H.

Johansson, J.

T. Andersson, J. Johansson, and H. Eklund, “Numerical solution of the Hilbert transform for phase calculation from an amplitude spectrum,” Math. Computers Simul. 23, 262–266 (1981).
[CrossRef]

Johnson, D. W.

D. W. Johnson, “A Fourier series method for numerical Kramers–Kronig analysis,” J. Phys. A. Math. Gen. 8, 490–495 (1975).
[CrossRef]

King, F. W.

F. W. King, G. J. Smethells, G. T. Helleloid, and P. J. Pelzl, “Numerical evaluation of Hilbert transforms for oscillatory functions: a convergence accelerator approach,” Comput. Phys. Commun. 145, 256–260 (2002).
[CrossRef]

F. W. King, K. J. Dykema, and A. D. Lund, “Calculation of some integrals for the atomic three-electron problem,” Phys. Rev. A 46, 5406–5416 (1992).
[CrossRef] [PubMed]

F. W. King, “Analysis of optical data by the conjugate Fourier-series approach,” J. Opt. Soc. Am. 68, 994–997 (1978).
[CrossRef]

F. W. King, “A Fourier series algorithm for the analysis of reflectance data,” J. Phys. C 10, 3199–3204 (1977).
[CrossRef]

Kircheva, P. P.

P. P. Kircheva and G. B. Hadjichristov, “Kramers–Kronig relations in FWM spectroscopy,” J. Phys. B 27, 3781–3793 (1994).
[CrossRef]

Knight, B. W.

Köhler, P.

K. Diethelm and P. Köhler, “Asymptotic behaviour of fixed-order error constants of modified quadrature formulae for Cauchy principal value integrals,” J. Inequal. Appl. 5, 167–190 (2000).

Kosloff, D. D.

H.-P. Liu and D. D. Kosloff, “Numerical evaluation of the Hilbert transform by the fast Fourier transform (FFT) technique,” Geophys. J. R. Astron. Soc. 67, 791–799 (1981).
[CrossRef]

Kozima, K.

Kramers, H. A.

H. A. Kramers, “Die dispersion und absorption von Röntgenstrahlen,” Phys. Z. 30, 522–523 (1929).

H. A. Kramers, “La diffusion de la lumière par les atomes,” Atti. Congr. Int. Fis. 2, 545–557 (1927).

Kronig, R. de L.

Lan, Z.

J. H. Bertie and Z. Lan, “An accurate modified Kramers–Kronig transformation from reflectance to phase shift on attenuated total reflection,” J. Chem. Phys. 105, 8502–8514 (1996).
[CrossRef]

Liu, H.-P.

H.-P. Liu and D. D. Kosloff, “Numerical evaluation of the Hilbert transform by the fast Fourier transform (FFT) technique,” Geophys. J. R. Astron. Soc. 67, 791–799 (1981).
[CrossRef]

Longman, I. M.

I. M. Longman, “On the numerical evaluation of Cauchy principal values of integrals,” Math. Tables Aids Comp. 12, 205–207 (1958).
[CrossRef]

Lubinsky, D. S.

P. Rabinowitz and D. S. Lubinsky, “Noninterpolatory integration rules for Cauchy principal value integrals,” Math. Comput. 53, 279–295 (1989).
[CrossRef]

Lund, A. D.

F. W. King, K. J. Dykema, and A. D. Lund, “Calculation of some integrals for the atomic three-electron problem,” Phys. Rev. A 46, 5406–5416 (1992).
[CrossRef] [PubMed]

Monegato, G.

G. Monegato, “The numerical evaluation of one-dimensional Cauchy principal value integrals,” Computing 29, 337–354 (1982).
[CrossRef]

Morawitz, H.

H. Morawitz, “A numerical approach to principal value integrals in dispersion relations,” J. Comput. Phys. 6, 120–123 (1970).
[CrossRef]

Neufeld, J. D.

Ohta, K.

Oosterhoff, L. J.

C. A. Emeis, L. J. Oosterhoff, and G. de Vries, “Numerical evaluation of Kramers–Kronig relations,” Proc. R. Soc. London, Ser. A 297, 54–65 (1967).
[CrossRef]

Paget, D. F.

D. F. Paget and D. Elliott, “An algorithm for the numerical evaluation of certain Cauchy principal value integrals,” Numer. Math. 19, 373–385 (1972).
[CrossRef]

Parris, D.

D. Parris and S. J. Van Der Walt, “A new numerical method for evaluating the Kramers–Kronig transformation,” Anal. Biochem. 68, 321–327 (1975).
[CrossRef] [PubMed]

Peiponen, K.-E.

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, “Complex analysis in dispersion theory,” Opt. Rev. 4, 433–441 (1997).
[CrossRef]

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, “Phase retrieval in optical spectroscopy: resolving optical constants from power spectra,” Appl. Spectrosc. 50, 1283–1289 (1996).
[CrossRef]

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

Pelzl, P. J.

F. W. King, G. J. Smethells, G. T. Helleloid, and P. J. Pelzl, “Numerical evaluation of Hilbert transforms for oscillatory functions: a convergence accelerator approach,” Comput. Phys. Commun. 145, 256–260 (2002).
[CrossRef]

Peterson, C. W.

Piessens, R.

R. Piessens, “Numerical evaluation of Cauchy principal values of integrals,” BIT 10, 476–480 (1970).
[CrossRef]

Rabinowitz, P.

P. Rabinowitz and D. S. Lubinsky, “Noninterpolatory integration rules for Cauchy principal value integrals,” Math. Comput. 53, 279–295 (1989).
[CrossRef]

P. Rabinowitz, “Some practical aspects in the numerical evaluation of Cauchy principal value integrals,” Int. J. Comput. Math. 20, 283–298 (1986).
[CrossRef]

Rabinowitz, P. R.

P. R. Rabinowitz, “Gauss–Kronrod integration rules for Cauchy principal value integrals,” Math. Comput. 41, 63–78 (1983).
[CrossRef]

Rasigni, G.

Rasigni, M.

Roessler, D. M.

D. M. Roessler, “Kramers–Kronig analysis of reflection data III. Approximations, with reference to sodium iodide,” Br. J. Appl. Phys. 17, 1313–1317 (1966).
[CrossRef]

D. M. Roessler, “Kramers–Kronig analysis of non-normal incidence reflection,” Br. J. Appl. Phys. 16, 1359–1366 (1965).
[CrossRef]

Rosei, R.

Sack, R. A.

R. A. Sack and A. F. Donovan, “An algorithm for Gaussian quadrature given modified moments,” Numer. Math. 18, 465–478 (1972).
[CrossRef]

Schatz, P. N.

Sloan, I. H.

I. H. Sloan, “The numerical evaluation of principal-value integrals,” J. Comput. Phys. 3, 332–333 (1968).
[CrossRef]

Smethells, G. J.

F. W. King, G. J. Smethells, G. T. Helleloid, and P. J. Pelzl, “Numerical evaluation of Hilbert transforms for oscillatory functions: a convergence accelerator approach,” Comput. Phys. Commun. 145, 256–260 (2002).
[CrossRef]

Suëtaka, W.

Taurian, O. E.

O. E. Taurian, “A method and a program for the numerical evaluation of the Hilbert transform of a real function,” Comput. Phys. Commun. 20, 291–307 (1980).
[CrossRef]

Troup, G. J.

S. J. Collocott and G. J. Troup, “Adaptation: numerical solution of the Kramers–Kronig transforms by trapezoidal summation as compared to a Fourier method,” Comput. Phys. Commun. 17, 393–395 (1979).
[CrossRef]

Van Der Walt, S. J.

D. Parris and S. J. Van Der Walt, “A new numerical method for evaluating the Kramers–Kronig transformation,” Anal. Biochem. 68, 321–327 (1975).
[CrossRef] [PubMed]

Vartiainen, E. M.

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, “Complex analysis in dispersion theory,” Opt. Rev. 4, 433–441 (1997).
[CrossRef]

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, “Phase retrieval in optical spectroscopy: resolving optical constants from power spectra,” Appl. Spectrosc. 50, 1283–1289 (1996).
[CrossRef]

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

Vecchia, B. D.

B. D. Vecchia, “Two new formulas for the numerical evaluation of the Hilbert transform,” BIT Numer. Math. 34, 346–360 (1994).
[CrossRef]

Wu, C. K.

Zhang, S. L.

J. H. Bertie and S. L. Zhang, “Infrared intensities of liquids. IX. The Kramers–Kronig transform, and its approximation by the finite Hilbert transform via fast Fourier transforms,” Can. J. Chem. 70, 520–531 (1992).
[CrossRef]

Anal. Biochem.

D. Parris and S. J. Van Der Walt, “A new numerical method for evaluating the Kramers–Kronig transformation,” Anal. Biochem. 68, 321–327 (1975).
[CrossRef] [PubMed]

Appl. Opt.

Appl. Spectrosc.

Approx. Theory Appl. (N.S.)

K. Diethelm, “Asymptotically sharp error bounds for a quadrature rule for Cauchy principal value integrals based on piecewise linear interpolation,” Approx. Theory Appl. (N.S.) 11, 78–89 (1995).

Atti. Congr. Int. Fis.

H. A. Kramers, “La diffusion de la lumière par les atomes,” Atti. Congr. Int. Fis. 2, 545–557 (1927).

BIT

R. Piessens, “Numerical evaluation of Cauchy principal values of integrals,” BIT 10, 476–480 (1970).
[CrossRef]

BIT Numer. Math.

B. D. Vecchia, “Two new formulas for the numerical evaluation of the Hilbert transform,” BIT Numer. Math. 34, 346–360 (1994).
[CrossRef]

Bolyai Soc. Math. Stud.

K. Diethelm, “Definite quadrature formulae for Cauchy principal value integrals,” Bolyai Soc. Math. Stud. 5, 175–186 (1996).

Br. J. Appl. Phys.

D. M. Roessler, “Kramers–Kronig analysis of non-normal incidence reflection,” Br. J. Appl. Phys. 16, 1359–1366 (1965).
[CrossRef]

D. M. Roessler, “Kramers–Kronig analysis of reflection data III. Approximations, with reference to sodium iodide,” Br. J. Appl. Phys. 17, 1313–1317 (1966).
[CrossRef]

Can. J. Chem.

J. H. Bertie and S. L. Zhang, “Infrared intensities of liquids. IX. The Kramers–Kronig transform, and its approximation by the finite Hilbert transform via fast Fourier transforms,” Can. J. Chem. 70, 520–531 (1992).
[CrossRef]

Comp. Phys. Commun.

S. J. Collocott, “Numerical solution of Kramers–Kronig transforms by a Fourier method,” Comp. Phys. Commun. 13, 203–206 (1977).
[CrossRef]

Comput. J. (Cambridge)

L. M. Delves, “The numerical evaluation of principal value integrals,” Comput. J. (Cambridge) 10, 389–391 (1967).

Comput. Phys. Commun.

S. J. Collocott and G. J. Troup, “Adaptation: numerical solution of the Kramers–Kronig transforms by trapezoidal summation as compared to a Fourier method,” Comput. Phys. Commun. 17, 393–395 (1979).
[CrossRef]

O. E. Taurian, “A method and a program for the numerical evaluation of the Hilbert transform of a real function,” Comput. Phys. Commun. 20, 291–307 (1980).
[CrossRef]

F. W. King, G. J. Smethells, G. T. Helleloid, and P. J. Pelzl, “Numerical evaluation of Hilbert transforms for oscillatory functions: a convergence accelerator approach,” Comput. Phys. Commun. 145, 256–260 (2002).
[CrossRef]

Computing

K. Diethelm, “Modified compound quadrature rules for strongly singular integrals,” Computing 52, 337–354 (1994).
[CrossRef]

G. Monegato, “The numerical evaluation of one-dimensional Cauchy principal value integrals,” Computing 29, 337–354 (1982).
[CrossRef]

Geophys. J. R. Astron. Soc.

H.-P. Liu and D. D. Kosloff, “Numerical evaluation of the Hilbert transform by the fast Fourier transform (FFT) technique,” Geophys. J. R. Astron. Soc. 67, 791–799 (1981).
[CrossRef]

Int. J. Comput. Math.

G. A. Gazonas, “The numerical evaluation of Cauchy principal value integrals via the fast Fourier transform,” Int. J. Comput. Math. 18, 277–288 (1986).
[CrossRef]

P. Rabinowitz, “Some practical aspects in the numerical evaluation of Cauchy principal value integrals,” Int. J. Comput. Math. 20, 283–298 (1986).
[CrossRef]

J. Chem. Phys.

J. H. Bertie and Z. Lan, “An accurate modified Kramers–Kronig transformation from reflectance to phase shift on attenuated total reflection,” J. Chem. Phys. 105, 8502–8514 (1996).
[CrossRef]

J. Comput. Appl. Math.

K. Diethelm, “A definiteness criterion for linear functionals and its application to Cauchy principal value quadrature,” J. Comput. Appl. Math. 66, 167–176 (1996).
[CrossRef]

K. Diethelm, “New error bounds for modified quadrature formulas for Cauchy principal value integrals,” J. Comput. Appl. Math. 82, 93–104 (1997).
[CrossRef]

K. Diethelm, “A method for the practical evaluation of the Hilbert transform on the real line,” J. Comput. Appl. Math. 112, 45–53 (1999).
[CrossRef]

K. Diethelm, “Uniform convergence of optimal order quadrature rules for Cauchy principal value integrals,” J. Comput. Appl. Math. 56, 321–329 (1994).
[CrossRef]

K. Diethelm, “Gaussian quadrature formulae of the third kind for Cauchy principal value integrals: basic properties and error estimates,” J. Comput. Appl. Math. 65, 97–114 (1995).
[CrossRef]

J. Comput. Phys.

I. H. Sloan, “The numerical evaluation of principal-value integrals,” J. Comput. Phys. 3, 332–333 (1968).
[CrossRef]

H. Morawitz, “A numerical approach to principal value integrals in dispersion relations,” J. Comput. Phys. 6, 120–123 (1970).
[CrossRef]

J. Inequal. Appl.

K. Diethelm and P. Köhler, “Asymptotic behaviour of fixed-order error constants of modified quadrature formulae for Cauchy principal value integrals,” J. Inequal. Appl. 5, 167–190 (2000).

J. Opt. Soc. Am.

G. Andermann, A. Caron, and D. A. Dows, “Kramers–Kronig dispersion analysis of infrared reflectance bands,” J. Opt. Soc. Am. 55, 1210–1216 (1965).
[CrossRef]

K. Kozima, W. Suëtaka, and P. N. Schatz, “Optical constants of thin films by a Kramers–Kronig method,” J. Opt. Soc. Am. 56, 181–184 (1966).
[CrossRef]

M. Rasigni and G. Rasigni, “Optical constants of lithium deposits as detected from the Kramers–Kronig analysis,” J. Opt. Soc. Am. 67, 54–59 (1977).
[CrossRef]

G. Andermann, C. K. Wu, and E. Duesler, “Kramers–Kronig phase-angle partitioning method for disclosing systematic errors in infrared reflectance data,” J. Opt. Soc. Am. 58, 1663–1664 (1968).
[CrossRef]

R. K. Ahrenkiel, “Modified Kramers–Kronig analysis of optical spectra,” J. Opt. Soc. Am. 61, 1651–1655 (1971).
[CrossRef]

R. K. Ahrenkiel, “Modified Kramers–Kronig analysis of optical spectra: erratum,” J. Opt. Soc. Am. 62, 1009 (1972).
[CrossRef]

J. D. Neufeld and G. Andermann, “Kramers–Kronig dispersion-analysis method for treating infrared transmittance data,” J. Opt. Soc. Am. 62, 1156–1162 (1972).
[CrossRef]

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]

F. W. King, “Analysis of optical data by the conjugate Fourier-series approach,” J. Opt. Soc. Am. 68, 994–997 (1978).
[CrossRef]

R. de L. Kronig, “On the theory of dispersion of x-rays,” J. Opt. Soc. Am. 12, 547–557 (1926).
[CrossRef]

J. Phys. A. Math. Gen.

D. W. Johnson, “A Fourier series method for numerical Kramers–Kronig analysis,” J. Phys. A. Math. Gen. 8, 490–495 (1975).
[CrossRef]

J. Phys. B

P. P. Kircheva and G. B. Hadjichristov, “Kramers–Kronig relations in FWM spectroscopy,” J. Phys. B 27, 3781–3793 (1994).
[CrossRef]

J. Phys. C

F. W. King, “A Fourier series algorithm for the analysis of reflectance data,” J. Phys. C 10, 3199–3204 (1977).
[CrossRef]

Math. Comput.

D. G. Anderson, “Gaussian quadrature formula for ∫01−ln(x)f(x)dx,” Math. Comput. 19, 477–481 (1965).

B. Danloy, “Numerical construction of Gaussian quadrature formulas for ∫01(−log x)xα f(x)dx and ∫01 Em(x)f(x)dx,” Math. Comput. 27, 861–869 (1973).

P. Rabinowitz and D. S. Lubinsky, “Noninterpolatory integration rules for Cauchy principal value integrals,” Math. Comput. 53, 279–295 (1989).
[CrossRef]

P. R. Rabinowitz, “Gauss–Kronrod integration rules for Cauchy principal value integrals,” Math. Comput. 41, 63–78 (1983).
[CrossRef]

Math. Computers Simul.

T. Andersson, J. Johansson, and H. Eklund, “Numerical solution of the Hilbert transform for phase calculation from an amplitude spectrum,” Math. Computers Simul. 23, 262–266 (1981).
[CrossRef]

Math. Tables Aids Comp.

I. M. Longman, “On the numerical evaluation of Cauchy principal values of integrals,” Math. Tables Aids Comp. 12, 205–207 (1958).
[CrossRef]

Numer. Math.

K. Diethelm, “Peano kernels and bounds for the error constants of Gaussian and related quadrature rules for Cauchy principal value integrals,” Numer. Math. 73, 53–63 (1996).
[CrossRef]

D. F. Paget and D. Elliott, “An algorithm for the numerical evaluation of certain Cauchy principal value integrals,” Numer. Math. 19, 373–385 (1972).
[CrossRef]

D. B. Hunter, “The numerical evaluation of Cauchy principal values of integrals by Romberg integration,” Numer. Math. 21, 185–192 (1973).
[CrossRef]

R. A. Sack and A. F. Donovan, “An algorithm for Gaussian quadrature given modified moments,” Numer. Math. 18, 465–478 (1972).
[CrossRef]

Opt. Rev.

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, “Complex analysis in dispersion theory,” Opt. Rev. 4, 433–441 (1997).
[CrossRef]

Phys. Rev. A

F. W. King, K. J. Dykema, and A. D. Lund, “Calculation of some integrals for the atomic three-electron problem,” Phys. Rev. A 46, 5406–5416 (1992).
[CrossRef] [PubMed]

Phys. Rev. B

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

Phys. Z.

H. A. Kramers, “Die dispersion und absorption von Röntgenstrahlen,” Phys. Z. 30, 522–523 (1929).

Proc. R. Soc. London, Ser. A

C. A. Emeis, L. J. Oosterhoff, and G. de Vries, “Numerical evaluation of Kramers–Kronig relations,” Proc. R. Soc. London, Ser. A 297, 54–65 (1967).
[CrossRef]

Proc. Symp. Appl. Math.

K. Diethelm, “Error estimates for a quadrature rule for Cauchy principal value integrals,” Proc. Symp. Appl. Math. 48, 287–291 (1994).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal.

K. Atkinson, “The numerical evaluation of the Cauchy transform on simple closed curves,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 9, 284–299 (1972).
[CrossRef]

Z. Angew. Math. Mech.

K. Diethelm, “Nonoptimality of certain quadrature rules for Cauchy principal value integrals,” Z. Angew. Math. Mech. 74, T689–T690 (1994).

K. Diethelm, “The order of convergence of modified interpolatory quadratures for singular integrals of Cauchy type,” Z. Angew. Math. Mech. 75, S621–S622 (1995).

Other

P. Rabinowitz, “The numerical evaluation of Cauchy principal value integrals,” in Symposium on Numerical Mathematics (University of Natal, Durban, South Africa, 1978), pp. 53–82.

P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation (Academic, New York, 1971), Vol. 1.

J. N. Pandey, The Hilbert Transform of Schwartz Distributions and Applications (Wiley-Interscience, New York, 1996).

E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, 2nd ed. (Oxford University, London, 1948).

S. Wolfram, The Mathematica Book, 4th ed. (Cambridge University, Cambridge, England, 1999).

D. H. Bailey, “Automatic translation of Fortran programs to multiprecision,” RNR Technical Rep. RNR-91–025, which is available from the author: dbailey@nas.nasa.gov.

D. H. Bailey, “MPFUN: a portable high performance multiprecision package,” RNR Technical Rep. RNR-90–022, which is available from the author: dbailey@nas.nasa.gov.

D. Y. Smith, “Dispersion theory, sum rules, and their application to the analysis of optical data,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 35–68.

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, Dispersion, Complex Analysis and Optical Spectroscopy (Springer-Verlag, Berlin, 1999).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77, 2nd ed. (Cambridge University, Cambridge, England, 1992).

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

V. I. Krylov and A. A. Pal’tsev, “On numerical integration of functions with logarithmic and power singularities,” Vesci Akad. Nauk BSSR Ser. Fiz. Techn. Nauk 14–23 (1963).

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Tables (6)

Tables Icon

Table 1 Some Common Functions and the Integration Ranges for which {wi, xi} Are Available as a Function of N

Tables Icon

Table 2 Abscissas and Weights for a Gaussian Quadrature with a Weight Function of log x-1

Tables Icon

Table 3 Abscissas and Weights for a Gaussian Quadrature with a Weight Function of log x-1

Tables Icon

Table 4 Hilbert Transform of the Lorentzian by Use of a Logarithmic Gaussian Quadraturea

Tables Icon

Table 5 Hilbert Transform of the Lorentzian by Use of a Logarithmic Gaussian Quadraturea

Tables Icon

Table 6 Hilbert Transform of a Gaussian by Use of a Logarithmic Gaussian Quadraturea

Equations (64)

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

(Hf)(x)=1πP-f(s)dsx-s.
(Hf)(x)=1πlim0+-x-f(s)dsx-s+x+f(s)dsx-s.
(Hf)(x)=1πP0f(y)x-y+f(-y)x+ydy.
(Hf)(x)=2xπP0f(y)x2-y2dy.
(Hf)(x)=2πP0yf(y)x2-y2dy.
f(x)=g(x)+ih(x),
h(x)=(Hg)(x),
g(x)=-(Hh)(x).
abf(x)dxi=1Nwif(xi),
f(x)=W(x)p(x),
W(x)0forx[a, b],
mj=abW(x)xjdx,
abf(x)dx=abW(x)p(x)dxi=1Nwip(xi).
f(x)=exp(-ax)b+x3,a,b>0.
0f(x)dx=0exp(-ax)b+x3dx=a0exp(-x)a3b+x3dxi=1Nwig(xi)
I=01log1+x1-xdx.
I=01log1+x1-xf(x)dx,
(f, g)=abW(x)f(x)g(x)dx.
p(x)=xj+a1xj-1++aj
(pj, pk)=0forjk.
p0(x)=1,
p1(x)=x-α0,
pi+1(x)=(x-αi)pi(x)-βipi-1(x)fori1,
αi=(xpi, pi)(pi, pi)fori0,
βi=(pi, pi)(pi-1, pi-1)fori1.
i=1Npj(x)wi=0forj=1,2 ,, N-1(p0, p0)forj=0
wi=(pN-1, pN-1)pN-1(xi)pN(xi),
abW(x)p(x)dx=i=1Nwip(xi),
I=01log(1/x)f(x)dx,
W(x)=log(1/x).
mi=abW(x)ρi(x)dx,
ρ0(x)=1,
ρ1(x)=x-c0,
ρi+1(x)=(x-ci)ρi(x)-diρi-1(x)fori1,
limx0{f[(1+x)c]-f[(1-x)c]}=xm withm>0.
(Hf)(x)=-1πP-f[x(s+1)]dss(x0)=-1πP--1f[x(s+1)]dss+-11f[x(s+1)]dss+1f[x(s+1)]dss=-1πP-11f[x(s+1)]dss+1f[x(s+1)]dss-1f[x(1-s)]dss.
P-11f[x(s+1)]dss
=lim01{f[x(s+1)]-f[x(1-s)]}dss
=01{f[x(s+1)]-[x(1-s)]}dss,
P-11f[x(s+1)]dss=01{f[x(1+s)]-f[x(1-s)]}d log sdsds={f[x(1+s)]-f[x(1-s)]}log s|01+01log s-1{f[x(1+s)]-f[x(1-s)]}ds=01log s-1{f[x(1+s)]-f[x(1-s)]}ds,
lims0{f[(1+s)x]-f[(1-s)x]}log s=lims0 sm log s=0,
P1f[x(s+1)]dss-1f[x(1-s)]dss=P01{f[x(1+t-1)]-f[x(1-t-1)]}dtt=01{f[x(1+t-1)]-f[x(1-t-1)]}d log tdtdt={f[x(1+t-1)]-f[x(1-t-1)]}log t|01+01log t-1{f[x(1+t-1)]-f[x(1-t-1)]}dt.
limt0{f[x(1+t-1)]-f[x(1-t-1)]}log t=limt0 tn log t=0,
(Hf)(x)=01log s-1K(s, x)ds,
K(s, x)=π-1{f[x(1-s)]-f[x(1+s)]+f[x(1-s-1)]-f[x(1+s-1)]}forx0.
(Hf)(0)=01log s-1K(s, 0)ds,
K(s, 0)=π-1[f(-s)-f(s)+f(-s-1)-f(s-1)].
K(s, 0)=0,
K(s, x)=π-1{f[x(1-s)]-f[x(1+s)]+f[x(s-1-1)]-f[x(s-1+1)]}forx0.
K(s, 0)=-2π-1{f(s)+f(s-1)},
K(s, x)=π-1{f[x(1-s)]-f[x(1+s)]-f[x(s-1-1)]-f[x(s-1+1)]}forx0.
(Hf)(x)=01log s-1K(s, x)dsforf(x)even,
(Hf)(x)=01log s-1K1(s, x)dsforf(x)odd,
K1(s, x)=1πx{g[x(1-s)]-g[x(1+s)]+g[x(s-1-1)]-g[x(s-1+1)]}forx0,
(Hf)(x)=i=1N wiK(xi, x),
01 sm log s-1ds=1(m+1)2.
I(x)=1πaa2+(x-x0)2,
(HI)(x)=1π(x-x0)a2+(x-x0)2.
I(x)=A exp[-a(x-x0)2],
H[exp(-ax2)]=-i exp(-ax2)erf(iax),
erf(z)=2π0zexp(-s2)ds.
H[exp(-ax2)]=2aπx exp(-ax2) 1F112;32;ax2=2aπx 1F11;32;-ax2.
 1F1(α;β;z)=k=0(α)k zk(β)kk!,
(α)k=α(α+1)(α+2)(α+k-1)=Γ(α+k)Γ(α).

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