Abstract

An algorithm is presented for the calculation of the Kramers-Kronig transform of a spectrum via a piecewise Laurent polynomial method. This algorithm is demonstrated to be highly accurate, while also being computationally efficient. The algorithm places no requirements on data point spacing and is capable of integrating across the full spectrum (i.e. from zero to infinity). Further, we present a computer application designed to aid in calculating the Kramers-Kronig transform on near-edge experimental X-ray absorption spectra (extended with atomic scattering factor data) in order to produce the dispersive part of the X-ray refractive index, including near-edge features.

© 2014 Optical Society of America

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References

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  4. M. Mezger, B. Jérôme, J. B. Kortright, M. Valvidares, E. M. Gullikson, A. Giglia, N. Mahne, and S. Nannarone, “Molecular orientation in soft matter thin films studied by resonant soft x-ray reflectivity,” Phys. Rev. B 83, 155406 (2011).
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    [Crossref]
  20. J. J. Hoyt, D. de Fontaine, and W. K. Warburton, “Determination of the anomalous scattering factors for Cu, Ni and Ti using the dispersion relation,” J. Appl. Crystallogr. 17, 344–351 (1984).
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    [Crossref]

2013 (1)

H. Yan, C. Wang, A. R. McCarn, and H. Ade, “Accurate and facile determination of the index of refraction of organic thin films near the carbon 1s absorption edge,” Phys. Rev. Lett. 110, 177401 (2013).
[Crossref] [PubMed]

2012 (1)

B. A. Collins, J. E. Cochran, H. Yan, E. Gann, C. Hub, R. Fink, C. Wang, T. Schuettfort, C. R. McNeill, M. L. Chabinyc, and H. Ade, “Polarized x-ray scattering reveals non-crystalline orientational ordering in organic films,” Nature Materials 11, 536–543 (2012).
[Crossref] [PubMed]

2011 (2)

M. Mezger, B. Jérôme, J. B. Kortright, M. Valvidares, E. M. Gullikson, A. Giglia, N. Mahne, and S. Nannarone, “Molecular orientation in soft matter thin films studied by resonant soft x-ray reflectivity,” Phys. Rev. B 83, 155406 (2011).
[Crossref]

B. Watts, S. Swaraj, D. Nordlund, J. Lüning, and H. Ade, “Calibrated NEXAFS spectra of common conjugated polymers,” J. Chem. Phys. 134, 024707 (2011).
[Crossref]

2010 (1)

S. Swaraj, C. Wang, H. Yan, B. Watts, J. Lüning, C. R. McNeill, and H. Ade, “Nanomorphology of bulk hetero-junction photovoltaic thin films probed with resonant soft x-ray scattering,” Nano Lett. 10, 2863–2869 (2010).
[Crossref] [PubMed]

2007 (1)

M. Nandagopal and N. Arunajadai, “On the evaluation of finite hilbert transforms,” Comp. Sci. Eng. 9, 90–95 (2007).
[Crossref]

2006 (1)

B. Watts, L. Thomsen, and P. C. Dastoor, “Methods in carbon K-edge NEXAFS: experiment and analysis,” J. Electron Spectrosc. Relat. Phenom. 151, 105–120 (2006).
[Crossref]

2005 (1)

C. Wang, T. Araki, and H. Ade, “Soft x-ray resonant reflectivity of low-Z material thin films,” Appl. Phys. Lett. 87, 214109 (2005).
[Crossref]

2002 (2)

P. Bruzzoni, R. M. Carranza, J. R. C. Lacoste, and E. A. Crespo, “Kramers-kronig transforms calculation with a fast convolution algorithm,” Electrochimica Acta 48, 341–347 (2002).
[Crossref]

F. W. King, “Efficient numerical approach to the evaluation of kramers-kronig transforms,” J. Opt. Soc. Am. B 19, 2427–2436 (2002).
[Crossref]

2001 (2)

G. Evans and R. F. Pettifer, “COOCH: a program for deriving anomalous-scattering factors from X-ray fluorescence spectra,” J. Appl. Crystallogr. 34, 82–86 (2001).
[Crossref]

J.-L. Hodeau, V. Favre-Nicolin, S. Bos, H. Renevier, E. Lorenzo, and J.-F. Berar, “Resonant diffraction,” Chem. Rev. 101, 1843–1867 (2001).
[Crossref] [PubMed]

1998 (1)

J. O. Cross, M. Newville, J. J. Rehr, L. B. Sorensen, C. E. Bouldin, G. Watson, T. Gouder, G. H. Lander, and M. I. Bell, “Inclusion of local structure effects in theoretical x-ray resonant scattering amplitudes using ab initio x-ray-absorption spectra calculations,” Phys. Rev. B 58, 11215 (1998).
[Crossref]

1997 (1)

D. C. Bazin, D. A. Sayers, and J. J. Rehr, “Comparison between x-ray absorption spectroscopy, anomalous wide angle x-ray scattering, anomalous small angle x-ray scattering, and diffraction anomalous fine structure techniques applied to nanometer-scale metallic clusters,” J. Phys. Chem. B 101, 11040–11050 (1997).
[Crossref]

1995 (1)

A. Natarajan and N. Mohankumar, “A comparison of some quadrature methods for approximating cauchy principal value integrals,” J. Comp. Phys. 116, 265–368 (1995).
[Crossref]

1993 (1)

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: Photoabsorption, scattering, transmission, and reflection at e = 50–30,000 ev, z = 1–92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[Crossref]

1992 (1)

H. Stragier, J. O. Cross, J. J. Rehr, L. B. Sorensen, C. E. Bouldin, and J. C. Woicik, “Diffraction anomalous fine structure: A new x-ray structural technique,” Phys. Rev. Lett. 69, 3064–3067 (1992).
[Crossref] [PubMed]

1991 (2)

W. A. Hendrickson, “Determination of macromolecular structures from anomalous diffraction of synchrotron radiation,” Science 254, 51–58 (1991).
[Crossref] [PubMed]

T. Hasagawa and T. Torii, “An automatic quadrature for cauchy principal value integrals,” Mathematics of Computation 56, 741–754 (1991).
[Crossref]

1990 (1)

W. A. Hendrickson, J. R. Horton, and D. M. LeMaster, “Selenomethionyl proteins produced for analysis by multiwavelength anomalous diffraction (mad): a vehicle for direct determination of three-dimensional structure,” EMBO J. 9, 1665–1672 (1990).
[PubMed]

1988 (1)

1987 (2)

G. Tsamasphyros and P. Androulidakis, “The tanh transformation for the solution of singular integral equations,” Intl. J. Num. Meth. Eng. 4, 543–556 (1987).
[Crossref]

M. Iri, S. Moriguti, and Y. Takasawa, “On certain quadrature formula,” J. Comp. Appl. Mathem. 17, 3–20 (1987).
[Crossref]

1984 (1)

J. J. Hoyt, D. de Fontaine, and W. K. Warburton, “Determination of the anomalous scattering factors for Cu, Ni and Ti using the dispersion relation,” J. Appl. Crystallogr. 17, 344–351 (1984).
[Crossref]

1980 (1)

J. Karle, “Some developments in anomalous dispersion for the structural investigation of macromolecular systems in biology,” Int. J. Quantum Chem. 18, 357–367 (1980).
[Crossref]

1974 (1)

H. Takhashi and M. Mori, “Double exponential formulas for numerical integration,” Publ. RIMS, Kyoto Univ. 9, 721–741 (1974).
[Crossref]

1973 (1)

1970 (1)

D. T. Cromer and D. Liberman, “Relativistic calculation of anomalous scattering factors of X rays,” J. Chem. Phys. 53, 1891–1898 (1970).
[Crossref]

1926 (1)

Ade, H.

H. Yan, C. Wang, A. R. McCarn, and H. Ade, “Accurate and facile determination of the index of refraction of organic thin films near the carbon 1s absorption edge,” Phys. Rev. Lett. 110, 177401 (2013).
[Crossref] [PubMed]

B. A. Collins, J. E. Cochran, H. Yan, E. Gann, C. Hub, R. Fink, C. Wang, T. Schuettfort, C. R. McNeill, M. L. Chabinyc, and H. Ade, “Polarized x-ray scattering reveals non-crystalline orientational ordering in organic films,” Nature Materials 11, 536–543 (2012).
[Crossref] [PubMed]

B. Watts, S. Swaraj, D. Nordlund, J. Lüning, and H. Ade, “Calibrated NEXAFS spectra of common conjugated polymers,” J. Chem. Phys. 134, 024707 (2011).
[Crossref]

S. Swaraj, C. Wang, H. Yan, B. Watts, J. Lüning, C. R. McNeill, and H. Ade, “Nanomorphology of bulk hetero-junction photovoltaic thin films probed with resonant soft x-ray scattering,” Nano Lett. 10, 2863–2869 (2010).
[Crossref] [PubMed]

C. Wang, T. Araki, and H. Ade, “Soft x-ray resonant reflectivity of low-Z material thin films,” Appl. Phys. Lett. 87, 214109 (2005).
[Crossref]

Androulidakis, P.

G. Tsamasphyros and P. Androulidakis, “The tanh transformation for the solution of singular integral equations,” Intl. J. Num. Meth. Eng. 4, 543–556 (1987).
[Crossref]

Araki, T.

C. Wang, T. Araki, and H. Ade, “Soft x-ray resonant reflectivity of low-Z material thin films,” Appl. Phys. Lett. 87, 214109 (2005).
[Crossref]

Arunajadai, N.

M. Nandagopal and N. Arunajadai, “On the evaluation of finite hilbert transforms,” Comp. Sci. Eng. 9, 90–95 (2007).
[Crossref]

Bazin, D. C.

D. C. Bazin, D. A. Sayers, and J. J. Rehr, “Comparison between x-ray absorption spectroscopy, anomalous wide angle x-ray scattering, anomalous small angle x-ray scattering, and diffraction anomalous fine structure techniques applied to nanometer-scale metallic clusters,” J. Phys. Chem. B 101, 11040–11050 (1997).
[Crossref]

Bell, M. I.

J. O. Cross, M. Newville, J. J. Rehr, L. B. Sorensen, C. E. Bouldin, G. Watson, T. Gouder, G. H. Lander, and M. I. Bell, “Inclusion of local structure effects in theoretical x-ray resonant scattering amplitudes using ab initio x-ray-absorption spectra calculations,” Phys. Rev. B 58, 11215 (1998).
[Crossref]

Berar, J.-F.

J.-L. Hodeau, V. Favre-Nicolin, S. Bos, H. Renevier, E. Lorenzo, and J.-F. Berar, “Resonant diffraction,” Chem. Rev. 101, 1843–1867 (2001).
[Crossref] [PubMed]

Biggs, F.

F. Biggs and R. Lighthill, “Sand87-0070: Analytical approximations for X-ray cross sections III,” (1988).

Bos, S.

J.-L. Hodeau, V. Favre-Nicolin, S. Bos, H. Renevier, E. Lorenzo, and J.-F. Berar, “Resonant diffraction,” Chem. Rev. 101, 1843–1867 (2001).
[Crossref] [PubMed]

Bouldin, C. E.

J. O. Cross, M. Newville, J. J. Rehr, L. B. Sorensen, C. E. Bouldin, G. Watson, T. Gouder, G. H. Lander, and M. I. Bell, “Inclusion of local structure effects in theoretical x-ray resonant scattering amplitudes using ab initio x-ray-absorption spectra calculations,” Phys. Rev. B 58, 11215 (1998).
[Crossref]

H. Stragier, J. O. Cross, J. J. Rehr, L. B. Sorensen, C. E. Bouldin, and J. C. Woicik, “Diffraction anomalous fine structure: A new x-ray structural technique,” Phys. Rev. Lett. 69, 3064–3067 (1992).
[Crossref] [PubMed]

Bruzzoni, P.

P. Bruzzoni, R. M. Carranza, J. R. C. Lacoste, and E. A. Crespo, “Kramers-kronig transforms calculation with a fast convolution algorithm,” Electrochimica Acta 48, 341–347 (2002).
[Crossref]

Carranza, R. M.

P. Bruzzoni, R. M. Carranza, J. R. C. Lacoste, and E. A. Crespo, “Kramers-kronig transforms calculation with a fast convolution algorithm,” Electrochimica Acta 48, 341–347 (2002).
[Crossref]

Chabinyc, M. L.

B. A. Collins, J. E. Cochran, H. Yan, E. Gann, C. Hub, R. Fink, C. Wang, T. Schuettfort, C. R. McNeill, M. L. Chabinyc, and H. Ade, “Polarized x-ray scattering reveals non-crystalline orientational ordering in organic films,” Nature Materials 11, 536–543 (2012).
[Crossref] [PubMed]

Cochran, J. E.

B. A. Collins, J. E. Cochran, H. Yan, E. Gann, C. Hub, R. Fink, C. Wang, T. Schuettfort, C. R. McNeill, M. L. Chabinyc, and H. Ade, “Polarized x-ray scattering reveals non-crystalline orientational ordering in organic films,” Nature Materials 11, 536–543 (2012).
[Crossref] [PubMed]

Collins, B. A.

B. A. Collins, J. E. Cochran, H. Yan, E. Gann, C. Hub, R. Fink, C. Wang, T. Schuettfort, C. R. McNeill, M. L. Chabinyc, and H. Ade, “Polarized x-ray scattering reveals non-crystalline orientational ordering in organic films,” Nature Materials 11, 536–543 (2012).
[Crossref] [PubMed]

Crespo, E. A.

P. Bruzzoni, R. M. Carranza, J. R. C. Lacoste, and E. A. Crespo, “Kramers-kronig transforms calculation with a fast convolution algorithm,” Electrochimica Acta 48, 341–347 (2002).
[Crossref]

Cromer, D. T.

D. T. Cromer and D. Liberman, “Relativistic calculation of anomalous scattering factors of X rays,” J. Chem. Phys. 53, 1891–1898 (1970).
[Crossref]

Cross, J. O.

J. O. Cross, M. Newville, J. J. Rehr, L. B. Sorensen, C. E. Bouldin, G. Watson, T. Gouder, G. H. Lander, and M. I. Bell, “Inclusion of local structure effects in theoretical x-ray resonant scattering amplitudes using ab initio x-ray-absorption spectra calculations,” Phys. Rev. B 58, 11215 (1998).
[Crossref]

H. Stragier, J. O. Cross, J. J. Rehr, L. B. Sorensen, C. E. Bouldin, and J. C. Woicik, “Diffraction anomalous fine structure: A new x-ray structural technique,” Phys. Rev. Lett. 69, 3064–3067 (1992).
[Crossref] [PubMed]

Dastoor, P. C.

B. Watts, L. Thomsen, and P. C. Dastoor, “Methods in carbon K-edge NEXAFS: experiment and analysis,” J. Electron Spectrosc. Relat. Phenom. 151, 105–120 (2006).
[Crossref]

Davis, J. C.

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: Photoabsorption, scattering, transmission, and reflection at e = 50–30,000 ev, z = 1–92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[Crossref]

de Fontaine, D.

J. J. Hoyt, D. de Fontaine, and W. K. Warburton, “Determination of the anomalous scattering factors for Cu, Ni and Ti using the dispersion relation,” J. Appl. Crystallogr. 17, 344–351 (1984).
[Crossref]

Evans, G.

G. Evans and R. F. Pettifer, “COOCH: a program for deriving anomalous-scattering factors from X-ray fluorescence spectra,” J. Appl. Crystallogr. 34, 82–86 (2001).
[Crossref]

Favre-Nicolin, V.

J.-L. Hodeau, V. Favre-Nicolin, S. Bos, H. Renevier, E. Lorenzo, and J.-F. Berar, “Resonant diffraction,” Chem. Rev. 101, 1843–1867 (2001).
[Crossref] [PubMed]

Fink, R.

B. A. Collins, J. E. Cochran, H. Yan, E. Gann, C. Hub, R. Fink, C. Wang, T. Schuettfort, C. R. McNeill, M. L. Chabinyc, and H. Ade, “Polarized x-ray scattering reveals non-crystalline orientational ordering in organic films,” Nature Materials 11, 536–543 (2012).
[Crossref] [PubMed]

Gann, E.

B. A. Collins, J. E. Cochran, H. Yan, E. Gann, C. Hub, R. Fink, C. Wang, T. Schuettfort, C. R. McNeill, M. L. Chabinyc, and H. Ade, “Polarized x-ray scattering reveals non-crystalline orientational ordering in organic films,” Nature Materials 11, 536–543 (2012).
[Crossref] [PubMed]

Giglia, A.

M. Mezger, B. Jérôme, J. B. Kortright, M. Valvidares, E. M. Gullikson, A. Giglia, N. Mahne, and S. Nannarone, “Molecular orientation in soft matter thin films studied by resonant soft x-ray reflectivity,” Phys. Rev. B 83, 155406 (2011).
[Crossref]

Gouder, T.

J. O. Cross, M. Newville, J. J. Rehr, L. B. Sorensen, C. E. Bouldin, G. Watson, T. Gouder, G. H. Lander, and M. I. Bell, “Inclusion of local structure effects in theoretical x-ray resonant scattering amplitudes using ab initio x-ray-absorption spectra calculations,” Phys. Rev. B 58, 11215 (1998).
[Crossref]

Gullikson, E. M.

M. Mezger, B. Jérôme, J. B. Kortright, M. Valvidares, E. M. Gullikson, A. Giglia, N. Mahne, and S. Nannarone, “Molecular orientation in soft matter thin films studied by resonant soft x-ray reflectivity,” Phys. Rev. B 83, 155406 (2011).
[Crossref]

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: Photoabsorption, scattering, transmission, and reflection at e = 50–30,000 ev, z = 1–92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[Crossref]

Hasagawa, T.

T. Hasagawa and T. Torii, “An automatic quadrature for cauchy principal value integrals,” Mathematics of Computation 56, 741–754 (1991).
[Crossref]

Hendrickson, W. A.

W. A. Hendrickson, “Determination of macromolecular structures from anomalous diffraction of synchrotron radiation,” Science 254, 51–58 (1991).
[Crossref] [PubMed]

W. A. Hendrickson, J. R. Horton, and D. M. LeMaster, “Selenomethionyl proteins produced for analysis by multiwavelength anomalous diffraction (mad): a vehicle for direct determination of three-dimensional structure,” EMBO J. 9, 1665–1672 (1990).
[PubMed]

Henke, B. L.

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: Photoabsorption, scattering, transmission, and reflection at e = 50–30,000 ev, z = 1–92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[Crossref]

Hodeau, J.-L.

J.-L. Hodeau, V. Favre-Nicolin, S. Bos, H. Renevier, E. Lorenzo, and J.-F. Berar, “Resonant diffraction,” Chem. Rev. 101, 1843–1867 (2001).
[Crossref] [PubMed]

Horton, J. R.

W. A. Hendrickson, J. R. Horton, and D. M. LeMaster, “Selenomethionyl proteins produced for analysis by multiwavelength anomalous diffraction (mad): a vehicle for direct determination of three-dimensional structure,” EMBO J. 9, 1665–1672 (1990).
[PubMed]

Hoyt, J. J.

J. J. Hoyt, D. de Fontaine, and W. K. Warburton, “Determination of the anomalous scattering factors for Cu, Ni and Ti using the dispersion relation,” J. Appl. Crystallogr. 17, 344–351 (1984).
[Crossref]

Hub, C.

B. A. Collins, J. E. Cochran, H. Yan, E. Gann, C. Hub, R. Fink, C. Wang, T. Schuettfort, C. R. McNeill, M. L. Chabinyc, and H. Ade, “Polarized x-ray scattering reveals non-crystalline orientational ordering in organic films,” Nature Materials 11, 536–543 (2012).
[Crossref] [PubMed]

Iri, M.

M. Iri, S. Moriguti, and Y. Takasawa, “On certain quadrature formula,” J. Comp. Appl. Mathem. 17, 3–20 (1987).
[Crossref]

Ishida, H.

Jérôme, B.

M. Mezger, B. Jérôme, J. B. Kortright, M. Valvidares, E. M. Gullikson, A. Giglia, N. Mahne, and S. Nannarone, “Molecular orientation in soft matter thin films studied by resonant soft x-ray reflectivity,” Phys. Rev. B 83, 155406 (2011).
[Crossref]

Karle, J.

J. Karle, “Some developments in anomalous dispersion for the structural investigation of macromolecular systems in biology,” Int. J. Quantum Chem. 18, 357–367 (1980).
[Crossref]

King, F. W.

Knight, B. W.

Kortright, J. B.

M. Mezger, B. Jérôme, J. B. Kortright, M. Valvidares, E. M. Gullikson, A. Giglia, N. Mahne, and S. Nannarone, “Molecular orientation in soft matter thin films studied by resonant soft x-ray reflectivity,” Phys. Rev. B 83, 155406 (2011).
[Crossref]

Kronig, R. D. L.

Lacoste, J. R. C.

P. Bruzzoni, R. M. Carranza, J. R. C. Lacoste, and E. A. Crespo, “Kramers-kronig transforms calculation with a fast convolution algorithm,” Electrochimica Acta 48, 341–347 (2002).
[Crossref]

Lander, G. H.

J. O. Cross, M. Newville, J. J. Rehr, L. B. Sorensen, C. E. Bouldin, G. Watson, T. Gouder, G. H. Lander, and M. I. Bell, “Inclusion of local structure effects in theoretical x-ray resonant scattering amplitudes using ab initio x-ray-absorption spectra calculations,” Phys. Rev. B 58, 11215 (1998).
[Crossref]

LeMaster, D. M.

W. A. Hendrickson, J. R. Horton, and D. M. LeMaster, “Selenomethionyl proteins produced for analysis by multiwavelength anomalous diffraction (mad): a vehicle for direct determination of three-dimensional structure,” EMBO J. 9, 1665–1672 (1990).
[PubMed]

Liberman, D.

D. T. Cromer and D. Liberman, “Relativistic calculation of anomalous scattering factors of X rays,” J. Chem. Phys. 53, 1891–1898 (1970).
[Crossref]

Lighthill, R.

F. Biggs and R. Lighthill, “Sand87-0070: Analytical approximations for X-ray cross sections III,” (1988).

Lorenzo, E.

J.-L. Hodeau, V. Favre-Nicolin, S. Bos, H. Renevier, E. Lorenzo, and J.-F. Berar, “Resonant diffraction,” Chem. Rev. 101, 1843–1867 (2001).
[Crossref] [PubMed]

Lüning, J.

B. Watts, S. Swaraj, D. Nordlund, J. Lüning, and H. Ade, “Calibrated NEXAFS spectra of common conjugated polymers,” J. Chem. Phys. 134, 024707 (2011).
[Crossref]

S. Swaraj, C. Wang, H. Yan, B. Watts, J. Lüning, C. R. McNeill, and H. Ade, “Nanomorphology of bulk hetero-junction photovoltaic thin films probed with resonant soft x-ray scattering,” Nano Lett. 10, 2863–2869 (2010).
[Crossref] [PubMed]

Mahne, N.

M. Mezger, B. Jérôme, J. B. Kortright, M. Valvidares, E. M. Gullikson, A. Giglia, N. Mahne, and S. Nannarone, “Molecular orientation in soft matter thin films studied by resonant soft x-ray reflectivity,” Phys. Rev. B 83, 155406 (2011).
[Crossref]

McCarn, A. R.

H. Yan, C. Wang, A. R. McCarn, and H. Ade, “Accurate and facile determination of the index of refraction of organic thin films near the carbon 1s absorption edge,” Phys. Rev. Lett. 110, 177401 (2013).
[Crossref] [PubMed]

McNeill, C. R.

B. A. Collins, J. E. Cochran, H. Yan, E. Gann, C. Hub, R. Fink, C. Wang, T. Schuettfort, C. R. McNeill, M. L. Chabinyc, and H. Ade, “Polarized x-ray scattering reveals non-crystalline orientational ordering in organic films,” Nature Materials 11, 536–543 (2012).
[Crossref] [PubMed]

S. Swaraj, C. Wang, H. Yan, B. Watts, J. Lüning, C. R. McNeill, and H. Ade, “Nanomorphology of bulk hetero-junction photovoltaic thin films probed with resonant soft x-ray scattering,” Nano Lett. 10, 2863–2869 (2010).
[Crossref] [PubMed]

Mezger, M.

M. Mezger, B. Jérôme, J. B. Kortright, M. Valvidares, E. M. Gullikson, A. Giglia, N. Mahne, and S. Nannarone, “Molecular orientation in soft matter thin films studied by resonant soft x-ray reflectivity,” Phys. Rev. B 83, 155406 (2011).
[Crossref]

Mohankumar, N.

A. Natarajan and N. Mohankumar, “A comparison of some quadrature methods for approximating cauchy principal value integrals,” J. Comp. Phys. 116, 265–368 (1995).
[Crossref]

Mori, M.

H. Takhashi and M. Mori, “Double exponential formulas for numerical integration,” Publ. RIMS, Kyoto Univ. 9, 721–741 (1974).
[Crossref]

Moriguti, S.

M. Iri, S. Moriguti, and Y. Takasawa, “On certain quadrature formula,” J. Comp. Appl. Mathem. 17, 3–20 (1987).
[Crossref]

Nandagopal, M.

M. Nandagopal and N. Arunajadai, “On the evaluation of finite hilbert transforms,” Comp. Sci. Eng. 9, 90–95 (2007).
[Crossref]

Nannarone, S.

M. Mezger, B. Jérôme, J. B. Kortright, M. Valvidares, E. M. Gullikson, A. Giglia, N. Mahne, and S. Nannarone, “Molecular orientation in soft matter thin films studied by resonant soft x-ray reflectivity,” Phys. Rev. B 83, 155406 (2011).
[Crossref]

Natarajan, A.

A. Natarajan and N. Mohankumar, “A comparison of some quadrature methods for approximating cauchy principal value integrals,” J. Comp. Phys. 116, 265–368 (1995).
[Crossref]

Newville, M.

J. O. Cross, M. Newville, J. J. Rehr, L. B. Sorensen, C. E. Bouldin, G. Watson, T. Gouder, G. H. Lander, and M. I. Bell, “Inclusion of local structure effects in theoretical x-ray resonant scattering amplitudes using ab initio x-ray-absorption spectra calculations,” Phys. Rev. B 58, 11215 (1998).
[Crossref]

Nordlund, D.

B. Watts, S. Swaraj, D. Nordlund, J. Lüning, and H. Ade, “Calibrated NEXAFS spectra of common conjugated polymers,” J. Chem. Phys. 134, 024707 (2011).
[Crossref]

Ohta, K.

Peterson, C. W.

Pettifer, R. F.

G. Evans and R. F. Pettifer, “COOCH: a program for deriving anomalous-scattering factors from X-ray fluorescence spectra,” J. Appl. Crystallogr. 34, 82–86 (2001).
[Crossref]

Rehr, J. J.

J. O. Cross, M. Newville, J. J. Rehr, L. B. Sorensen, C. E. Bouldin, G. Watson, T. Gouder, G. H. Lander, and M. I. Bell, “Inclusion of local structure effects in theoretical x-ray resonant scattering amplitudes using ab initio x-ray-absorption spectra calculations,” Phys. Rev. B 58, 11215 (1998).
[Crossref]

D. C. Bazin, D. A. Sayers, and J. J. Rehr, “Comparison between x-ray absorption spectroscopy, anomalous wide angle x-ray scattering, anomalous small angle x-ray scattering, and diffraction anomalous fine structure techniques applied to nanometer-scale metallic clusters,” J. Phys. Chem. B 101, 11040–11050 (1997).
[Crossref]

H. Stragier, J. O. Cross, J. J. Rehr, L. B. Sorensen, C. E. Bouldin, and J. C. Woicik, “Diffraction anomalous fine structure: A new x-ray structural technique,” Phys. Rev. Lett. 69, 3064–3067 (1992).
[Crossref] [PubMed]

Renevier, H.

J.-L. Hodeau, V. Favre-Nicolin, S. Bos, H. Renevier, E. Lorenzo, and J.-F. Berar, “Resonant diffraction,” Chem. Rev. 101, 1843–1867 (2001).
[Crossref] [PubMed]

Sayers, D. A.

D. C. Bazin, D. A. Sayers, and J. J. Rehr, “Comparison between x-ray absorption spectroscopy, anomalous wide angle x-ray scattering, anomalous small angle x-ray scattering, and diffraction anomalous fine structure techniques applied to nanometer-scale metallic clusters,” J. Phys. Chem. B 101, 11040–11050 (1997).
[Crossref]

Schuettfort, T.

B. A. Collins, J. E. Cochran, H. Yan, E. Gann, C. Hub, R. Fink, C. Wang, T. Schuettfort, C. R. McNeill, M. L. Chabinyc, and H. Ade, “Polarized x-ray scattering reveals non-crystalline orientational ordering in organic films,” Nature Materials 11, 536–543 (2012).
[Crossref] [PubMed]

Sorensen, L. B.

J. O. Cross, M. Newville, J. J. Rehr, L. B. Sorensen, C. E. Bouldin, G. Watson, T. Gouder, G. H. Lander, and M. I. Bell, “Inclusion of local structure effects in theoretical x-ray resonant scattering amplitudes using ab initio x-ray-absorption spectra calculations,” Phys. Rev. B 58, 11215 (1998).
[Crossref]

H. Stragier, J. O. Cross, J. J. Rehr, L. B. Sorensen, C. E. Bouldin, and J. C. Woicik, “Diffraction anomalous fine structure: A new x-ray structural technique,” Phys. Rev. Lett. 69, 3064–3067 (1992).
[Crossref] [PubMed]

Stragier, H.

H. Stragier, J. O. Cross, J. J. Rehr, L. B. Sorensen, C. E. Bouldin, and J. C. Woicik, “Diffraction anomalous fine structure: A new x-ray structural technique,” Phys. Rev. Lett. 69, 3064–3067 (1992).
[Crossref] [PubMed]

Swaraj, S.

B. Watts, S. Swaraj, D. Nordlund, J. Lüning, and H. Ade, “Calibrated NEXAFS spectra of common conjugated polymers,” J. Chem. Phys. 134, 024707 (2011).
[Crossref]

S. Swaraj, C. Wang, H. Yan, B. Watts, J. Lüning, C. R. McNeill, and H. Ade, “Nanomorphology of bulk hetero-junction photovoltaic thin films probed with resonant soft x-ray scattering,” Nano Lett. 10, 2863–2869 (2010).
[Crossref] [PubMed]

Takasawa, Y.

M. Iri, S. Moriguti, and Y. Takasawa, “On certain quadrature formula,” J. Comp. Appl. Mathem. 17, 3–20 (1987).
[Crossref]

Takhashi, H.

H. Takhashi and M. Mori, “Double exponential formulas for numerical integration,” Publ. RIMS, Kyoto Univ. 9, 721–741 (1974).
[Crossref]

Thomsen, L.

B. Watts, L. Thomsen, and P. C. Dastoor, “Methods in carbon K-edge NEXAFS: experiment and analysis,” J. Electron Spectrosc. Relat. Phenom. 151, 105–120 (2006).
[Crossref]

Torii, T.

T. Hasagawa and T. Torii, “An automatic quadrature for cauchy principal value integrals,” Mathematics of Computation 56, 741–754 (1991).
[Crossref]

Tsamasphyros, G.

G. Tsamasphyros and P. Androulidakis, “The tanh transformation for the solution of singular integral equations,” Intl. J. Num. Meth. Eng. 4, 543–556 (1987).
[Crossref]

Valvidares, M.

M. Mezger, B. Jérôme, J. B. Kortright, M. Valvidares, E. M. Gullikson, A. Giglia, N. Mahne, and S. Nannarone, “Molecular orientation in soft matter thin films studied by resonant soft x-ray reflectivity,” Phys. Rev. B 83, 155406 (2011).
[Crossref]

Wang, C.

H. Yan, C. Wang, A. R. McCarn, and H. Ade, “Accurate and facile determination of the index of refraction of organic thin films near the carbon 1s absorption edge,” Phys. Rev. Lett. 110, 177401 (2013).
[Crossref] [PubMed]

B. A. Collins, J. E. Cochran, H. Yan, E. Gann, C. Hub, R. Fink, C. Wang, T. Schuettfort, C. R. McNeill, M. L. Chabinyc, and H. Ade, “Polarized x-ray scattering reveals non-crystalline orientational ordering in organic films,” Nature Materials 11, 536–543 (2012).
[Crossref] [PubMed]

S. Swaraj, C. Wang, H. Yan, B. Watts, J. Lüning, C. R. McNeill, and H. Ade, “Nanomorphology of bulk hetero-junction photovoltaic thin films probed with resonant soft x-ray scattering,” Nano Lett. 10, 2863–2869 (2010).
[Crossref] [PubMed]

C. Wang, T. Araki, and H. Ade, “Soft x-ray resonant reflectivity of low-Z material thin films,” Appl. Phys. Lett. 87, 214109 (2005).
[Crossref]

Warburton, W. K.

J. J. Hoyt, D. de Fontaine, and W. K. Warburton, “Determination of the anomalous scattering factors for Cu, Ni and Ti using the dispersion relation,” J. Appl. Crystallogr. 17, 344–351 (1984).
[Crossref]

Watson, G.

J. O. Cross, M. Newville, J. J. Rehr, L. B. Sorensen, C. E. Bouldin, G. Watson, T. Gouder, G. H. Lander, and M. I. Bell, “Inclusion of local structure effects in theoretical x-ray resonant scattering amplitudes using ab initio x-ray-absorption spectra calculations,” Phys. Rev. B 58, 11215 (1998).
[Crossref]

Watts, B.

B. Watts, S. Swaraj, D. Nordlund, J. Lüning, and H. Ade, “Calibrated NEXAFS spectra of common conjugated polymers,” J. Chem. Phys. 134, 024707 (2011).
[Crossref]

S. Swaraj, C. Wang, H. Yan, B. Watts, J. Lüning, C. R. McNeill, and H. Ade, “Nanomorphology of bulk hetero-junction photovoltaic thin films probed with resonant soft x-ray scattering,” Nano Lett. 10, 2863–2869 (2010).
[Crossref] [PubMed]

B. Watts, L. Thomsen, and P. C. Dastoor, “Methods in carbon K-edge NEXAFS: experiment and analysis,” J. Electron Spectrosc. Relat. Phenom. 151, 105–120 (2006).
[Crossref]

Woicik, J. C.

H. Stragier, J. O. Cross, J. J. Rehr, L. B. Sorensen, C. E. Bouldin, and J. C. Woicik, “Diffraction anomalous fine structure: A new x-ray structural technique,” Phys. Rev. Lett. 69, 3064–3067 (1992).
[Crossref] [PubMed]

Yan, H.

H. Yan, C. Wang, A. R. McCarn, and H. Ade, “Accurate and facile determination of the index of refraction of organic thin films near the carbon 1s absorption edge,” Phys. Rev. Lett. 110, 177401 (2013).
[Crossref] [PubMed]

B. A. Collins, J. E. Cochran, H. Yan, E. Gann, C. Hub, R. Fink, C. Wang, T. Schuettfort, C. R. McNeill, M. L. Chabinyc, and H. Ade, “Polarized x-ray scattering reveals non-crystalline orientational ordering in organic films,” Nature Materials 11, 536–543 (2012).
[Crossref] [PubMed]

S. Swaraj, C. Wang, H. Yan, B. Watts, J. Lüning, C. R. McNeill, and H. Ade, “Nanomorphology of bulk hetero-junction photovoltaic thin films probed with resonant soft x-ray scattering,” Nano Lett. 10, 2863–2869 (2010).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

C. Wang, T. Araki, and H. Ade, “Soft x-ray resonant reflectivity of low-Z material thin films,” Appl. Phys. Lett. 87, 214109 (2005).
[Crossref]

Appl. Spectrosc. (1)

At. Data Nucl. Data Tables (1)

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: Photoabsorption, scattering, transmission, and reflection at e = 50–30,000 ev, z = 1–92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[Crossref]

Chem. Rev. (1)

J.-L. Hodeau, V. Favre-Nicolin, S. Bos, H. Renevier, E. Lorenzo, and J.-F. Berar, “Resonant diffraction,” Chem. Rev. 101, 1843–1867 (2001).
[Crossref] [PubMed]

Comp. Sci. Eng. (1)

M. Nandagopal and N. Arunajadai, “On the evaluation of finite hilbert transforms,” Comp. Sci. Eng. 9, 90–95 (2007).
[Crossref]

Electrochimica Acta (1)

P. Bruzzoni, R. M. Carranza, J. R. C. Lacoste, and E. A. Crespo, “Kramers-kronig transforms calculation with a fast convolution algorithm,” Electrochimica Acta 48, 341–347 (2002).
[Crossref]

EMBO J. (1)

W. A. Hendrickson, J. R. Horton, and D. M. LeMaster, “Selenomethionyl proteins produced for analysis by multiwavelength anomalous diffraction (mad): a vehicle for direct determination of three-dimensional structure,” EMBO J. 9, 1665–1672 (1990).
[PubMed]

Int. J. Quantum Chem. (1)

J. Karle, “Some developments in anomalous dispersion for the structural investigation of macromolecular systems in biology,” Int. J. Quantum Chem. 18, 357–367 (1980).
[Crossref]

Intl. J. Num. Meth. Eng. (1)

G. Tsamasphyros and P. Androulidakis, “The tanh transformation for the solution of singular integral equations,” Intl. J. Num. Meth. Eng. 4, 543–556 (1987).
[Crossref]

J. Appl. Crystallogr. (2)

J. J. Hoyt, D. de Fontaine, and W. K. Warburton, “Determination of the anomalous scattering factors for Cu, Ni and Ti using the dispersion relation,” J. Appl. Crystallogr. 17, 344–351 (1984).
[Crossref]

G. Evans and R. F. Pettifer, “COOCH: a program for deriving anomalous-scattering factors from X-ray fluorescence spectra,” J. Appl. Crystallogr. 34, 82–86 (2001).
[Crossref]

J. Chem. Phys. (2)

D. T. Cromer and D. Liberman, “Relativistic calculation of anomalous scattering factors of X rays,” J. Chem. Phys. 53, 1891–1898 (1970).
[Crossref]

B. Watts, S. Swaraj, D. Nordlund, J. Lüning, and H. Ade, “Calibrated NEXAFS spectra of common conjugated polymers,” J. Chem. Phys. 134, 024707 (2011).
[Crossref]

J. Comp. Appl. Mathem. (1)

M. Iri, S. Moriguti, and Y. Takasawa, “On certain quadrature formula,” J. Comp. Appl. Mathem. 17, 3–20 (1987).
[Crossref]

J. Comp. Phys. (1)

A. Natarajan and N. Mohankumar, “A comparison of some quadrature methods for approximating cauchy principal value integrals,” J. Comp. Phys. 116, 265–368 (1995).
[Crossref]

J. Electron Spectrosc. Relat. Phenom. (1)

B. Watts, L. Thomsen, and P. C. Dastoor, “Methods in carbon K-edge NEXAFS: experiment and analysis,” J. Electron Spectrosc. Relat. Phenom. 151, 105–120 (2006).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. B (1)

J. Phys. Chem. B (1)

D. C. Bazin, D. A. Sayers, and J. J. Rehr, “Comparison between x-ray absorption spectroscopy, anomalous wide angle x-ray scattering, anomalous small angle x-ray scattering, and diffraction anomalous fine structure techniques applied to nanometer-scale metallic clusters,” J. Phys. Chem. B 101, 11040–11050 (1997).
[Crossref]

Mathematics of Computation (1)

T. Hasagawa and T. Torii, “An automatic quadrature for cauchy principal value integrals,” Mathematics of Computation 56, 741–754 (1991).
[Crossref]

Nano Lett. (1)

S. Swaraj, C. Wang, H. Yan, B. Watts, J. Lüning, C. R. McNeill, and H. Ade, “Nanomorphology of bulk hetero-junction photovoltaic thin films probed with resonant soft x-ray scattering,” Nano Lett. 10, 2863–2869 (2010).
[Crossref] [PubMed]

Nature Materials (1)

B. A. Collins, J. E. Cochran, H. Yan, E. Gann, C. Hub, R. Fink, C. Wang, T. Schuettfort, C. R. McNeill, M. L. Chabinyc, and H. Ade, “Polarized x-ray scattering reveals non-crystalline orientational ordering in organic films,” Nature Materials 11, 536–543 (2012).
[Crossref] [PubMed]

Phys. Rev. B (2)

M. Mezger, B. Jérôme, J. B. Kortright, M. Valvidares, E. M. Gullikson, A. Giglia, N. Mahne, and S. Nannarone, “Molecular orientation in soft matter thin films studied by resonant soft x-ray reflectivity,” Phys. Rev. B 83, 155406 (2011).
[Crossref]

J. O. Cross, M. Newville, J. J. Rehr, L. B. Sorensen, C. E. Bouldin, G. Watson, T. Gouder, G. H. Lander, and M. I. Bell, “Inclusion of local structure effects in theoretical x-ray resonant scattering amplitudes using ab initio x-ray-absorption spectra calculations,” Phys. Rev. B 58, 11215 (1998).
[Crossref]

Phys. Rev. Lett. (2)

H. Yan, C. Wang, A. R. McCarn, and H. Ade, “Accurate and facile determination of the index of refraction of organic thin films near the carbon 1s absorption edge,” Phys. Rev. Lett. 110, 177401 (2013).
[Crossref] [PubMed]

H. Stragier, J. O. Cross, J. J. Rehr, L. B. Sorensen, C. E. Bouldin, and J. C. Woicik, “Diffraction anomalous fine structure: A new x-ray structural technique,” Phys. Rev. Lett. 69, 3064–3067 (1992).
[Crossref] [PubMed]

Publ. RIMS, Kyoto Univ. (1)

H. Takhashi and M. Mori, “Double exponential formulas for numerical integration,” Publ. RIMS, Kyoto Univ. 9, 721–741 (1974).
[Crossref]

Science (1)

W. A. Hendrickson, “Determination of macromolecular structures from anomalous diffraction of synchrotron radiation,” Science 254, 51–58 (1991).
[Crossref] [PubMed]

Other (3)

M. Newville, “Homepage of the diffkk computer program,” (2014). http://cars9.uchicago.edu/dafs/diffkk/ .

Center for X-ray Optics, “CXRO: X-ray interactions with matter,” (2014). http://henke.lbl.gov/optical_constants/ .

F. Biggs and R. Lighthill, “Sand87-0070: Analytical approximations for X-ray cross sections III,” (1988).

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Figures (6)

Fig. 1
Fig. 1 The graphical user interface for KKcalc, demonstrating calculation of the detailed cobalt K-edge scattering factors. The imaginary scattering factors are displayed as a combination of a black line (representing isolated atom data from Henke et al.) and blue points (experimental data taken from a user-defined file) and the calculated real scattering factors are shown by a green line.
Fig. 2
Fig. 2 The model extinction coefficient spectrum, k (ν), and the analytic solution to its Kramers-Kronig transform, Δn (ν)), plotted as described by Ohta and Ishida [14]. Their differences from a piecewise polynomial approximation of the input function, kPP (ν), and its transform using the algorithm presented here, ΔnPP (ν), are also plotted to demonstrate a close fit to the analytical functions.
Fig. 3
Fig. 3 Scattering factors for elemental carbon. The imaginary part is shown in blue, while the real part is plotted in black (as published by Henke et al.), green (Eq. (10)) and yellow (Eq. (10), but restricted to integrating the range 10 eV to 30 keV). The inset shows detail around the K absorption edge.
Fig. 4
Fig. 4 Scattering factors for elemental iron. The imaginary part is shown in blue, while the real part is plotted in black (as published by Henke et al.), green (Eq. (10)) and yellow (Eq. (10), but restricted to integrating the range 10 eV to 30 keV). Insets show detail around the M (top) and L (bottom) absorption edges.
Fig. 5
Fig. 5 Scattering factors for elemental lead. The imaginary part is shown in blue, while the real part is plotted in black (as published by Henke et al.), green (Eq. (10)) and yellow (Eq. (10), but restricted to integrating the range 10 eV to 30 keV). The inset shows detail around the M absorption edge.
Fig. 6
Fig. 6 Fitting results (circle) of optical constants, dispersion part δ (left), and absorption part β (right), for PMMA and calculation (solid lines) using Kramers-Kronig relation via the doubly-subtractive (red) and piecewise polynomial (orange) algorithms. The corresponding values derived from Henke’s database are shown for comparison (dashed line).

Tables (1)

Tables Icon

Table 1 Listing of coefficients and terms produced by an −3 ≤ n ≤ 1 implementation of Eq. (7), as given by Eq. (10). Zeros indicate cancelling of the sum terms, while blank table elements indicate combinations of terms and coefficients outside the range of the sum limits.

Equations (13)

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

n ( E ) = 1 δ ( E ) i β ( E ) = 1 r 0 2 π λ 2 q N q ( f 1 q ( E ) + i f 2 q ( E ) )
f 1 ( E ) = Z * 2 π P 0 x f 2 ( x ) x 2 E 2 d x
f 2 ( x ) F ( x ) = { p 1 ( x ) x 1 x x 2 p 2 ( x ) x 2 x x 3 p i 1 ( x ) x i 1 x x i
f 1 ( E ) = Z * 2 π P x 1 x i x F ( x ) x 2 E 2 d x = Z * + 2 π j = 1 i 1 P x 1 x i x p j ( x ) x 2 E 2 = Z * + 1 π E j = 1 i 1 P x j x j + 1 x p j ( x ) x + E 1 π E j = 1 i 1 P x j x j + 1 x p j ( x ) x E d x
x j x j + 1 x p j ( x ) x + E d x = x j x j + 1 n = M N x a j , n x n x + E d x = n = M N a j , n x j x j + 1 x n + 1 x + E d x = n = 0 N a j , n ( k = 1 n + 1 ( E ) n k + 1 k ( x j + 1 k x j k ) ) + n = M N a j , n ( E ) n + 1 ln x j + 1 + E x j + E n = M 2 a j , n ( E ) n + 1 ln x j + 1 x j + n = M 3 a j , n ( k = n + 2 1 ( E ) n k + 1 k ( x j + 1 k x j k ) )
x j x j + 1 x p j ( x ) x E d x = n = 0 N a j , n ( k = 1 n + 1 E n k + 1 k ( x j + 1 k x j k ) ) + n = M N a j , n E n + 1 ln x j + 1 E x j E n = M 2 a j , n E n + 1 ln x j + 1 x j + n = M 3 a j , n ( k = n + 2 1 E n k + 1 k ( x j + 1 k x j k ) )
x j 1 x j + 1 x F ( x ) x E d x = n = 0 N ( k = 1 n + 1 E n k + 1 k ( a j 1 , n ( E k x j 1 k ) + a j , n ( x j + 1 k E k ) ) ) + E p j ( E ) ln x j + 1 E x j 1 E n = M 2 ( a j 1 , n E n + 1 ln E x j 1 + a j , n E n + 1 ln x j + 1 E ) + n = M 3 ( k = n + 2 1 E n k + 1 k ( a j 1 , n ( E k x j 1 k ) + a j , n ( x j + 1 k E k ) ) )
j = 1 i 1 x j x j + 1 x p j ( x ) x E d x = j = 1 i 1 { n = 0 N a j , n ( k = 1 n + 1 E n k + 1 k ( x j + 1 k x j k ) ) + ( 1 δ E , x j + 1 ) n = M N a j , n E n + 1 ln x j + 1 E + δ E , x j + 1 x j δ E , x j E n = M 2 a j , n E n + 1 ln x j + 1 x j + n = M 3 a j , n ( k = n + 2 1 E n k + 1 k ( x j + 1 k x j k ) ) }
δ f , g = { 0 f g 1 f = g
f 1 ( E ) = Z * + 1 π j = 1 i 1 { n = 0 N a j , n ( k = 1 n + 1 ( E ) n k + E n k k ( x j + 1 k x j k ) ) n = M N a j , n ( E ) n ln x j + 1 + E x j + E ( 1 δ E , x j + 1 ) n = M N a j , n E n ln x j + 1 E + δ E , x j + 1 x j δ E , x j E + n = M 2 a j , n ( ( E ) n + E n ) ln x j + 1 x j + n = M 3 a j , n ( k = n + 2 1 ( E ) n k + E n k k ( x j + 1 k x j k ) ) }
f 2 ( x ) = a 1 x + a 0 + a 1 x 1 + a 2 x 2 + a 3 x 3
Z * = q n q ( Z q ( Z q 82.5 ) 2.37 )
f 1 ( E ) = Z * + 1 π E j = 1 i { ( E 2 a j 1 E a j , 0 + a j , 1 a j , 2 E + a j , 3 E 2 ) ln x j + 1 + E x j + E ( 1 δ E , x j + 1 ) ( E 2 a j , 1 + E a j , 0 + a j , 1 + a j , 2 E + a j , 3 E 2 ) ln x j + 1 E + δ E , x j + 1 x j δ E , x j E 2 E a j , 1 ( x j + 1 x j ) + 2 a j , 2 E ln x j + 1 x j + 2 a j , 3 E ( x j + 1 1 x j 1 ) }

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