Abstract

Abel inverse integral to obtain local field distributions from path-integrated measurements in an axisymmetric medium is an ill-posed problem with the integrant diverging at the lower integration limit. Existing methods to evaluate this integral can be broadly categorized as numerical integration techniques, semianalytical techniques, and least-squares whole-curve-fit techniques. In this study, Simpson’s 1/3rd rule (a numerical integration technique), one-point and two-point formulas (semianalytical techniques), and the Guass–Hermite product polynomial method (a least-squares whole-curve-fit technique) are compared for accuracy and error propagation in Abel inversion of deflectometric data. For data acquired at equally spaced radial intervals, the deconvolved field can be expressed as a linear combination (weighted sum) of measured data. This approach permits use of the uncertainty analysis principle to compute error propagation by the integration algorithm. Least-squares curve-fit techniques should be avoided because of poor inversion accuracy with large propagation of measurement error. The two-point formula is recommended to achieve high inversion accuracy with minimum error propagation.

© 2009 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  38. A. K. Shenoy, A. K. Agrawal, and S. R. Gollahalli, “Quantitative evaluation of flow computations by rainbow schlieren deflectometry,” AIAA J. 36, 1953-1960 (1998).
    [CrossRef]
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2007 (1)

J. R. Camacho, F. N. Beg, and P. Lee1, “Comparison of sensitivities of Moire deflectometry and interferometry to measure electron densities in z-pinch plasmas,” J. Phys. D. 40, 2026-2032 (2007).
[CrossRef]

2006 (1)

T. Wong and A. K. Agrawal, “Quantitative measurements in an unsteady flame using high-speed rainbow schlieren deflectometry,” Meas. Sci. Technol. 17, 1503-1510 (2006).
[CrossRef]

2005 (3)

B. S. Yildirim and A. K. Agrawal, “Full-field concentration measurements of self-excited oscillations in momentum-dominated helium jets,” Exp. Fluids 38, 161-173(2005).
[CrossRef]

R. K. Paul, J. T. Andrews, K. Bose, and P. K. Barhai, “Reconstruction errors in Abel inversion,” Plasma Devices Oper. 13, 281-290 (2005).
[CrossRef]

E. Ampem-Lassen, S. T. Huntington, N. M. Dragomir, K. A. Nugent, and A. Roberts, “Refractive index profiling of axially symmetric optical fibers: a new technique,” Opt. Express 13, 3277-3282 (2005).
[CrossRef] [PubMed]

2003 (1)

K. S. Pasumarthi and A. K. Agrawal, “Schlieren measurements and analysis of concentration field in self-excited helium jets,” Phys. Fluids 15, 3683-3692 (2003).
[CrossRef]

2002 (2)

A. K. Agrawal, K. N. Alammar, and S. R. Gollahalli, “Application of rainbow schlieren deflectometry to measure temperature and oxygen concentration in a laminar jet diffusion flame,” Exp. Fluids 32, 689-691 (2002).

V. Dribinski, A. Ossadtchi, V. A. Mandeleshtam, and H. Reisler, “Reconstruction of Abel-transformable images: the Gaussian basis-set expansion Abel transform method,” Rev. Sci. Instrum. 73, 2634-2642 (2002).
[CrossRef]

2001 (1)

K. M. Green, M. C. Borras, P. P. Woskov, G. J. Flores III, K. Hadidi, and P. Thomas, “Electronic excitation temperature profiles in an air microwave plasma torch,” IEEE Trans. Plasma Sci. 29, 399-406 (2001).
[CrossRef]

1999 (1)

A. T. Ramsey and M. Diesso, “Abel inversions: error propagation and inversion reliability,” Rev. Sci. Instrum. 70, 380-383 (1999).
[CrossRef]

1998 (2)

A. K. Shenoy, A. K. Agrawal, and S. R. Gollahalli, “Quantitative evaluation of flow computations by rainbow schlieren deflectometry,” AIAA J. 36, 1953-1960 (1998).
[CrossRef]

K. Al-Ammar, A. K. Agrawal, S. R. Gollahalli, and D. Griffin, “Concentration measurements in an axisymmetric helium jet using rainbow schlieren deflectometry,” Exp. Fluids 25, 89-95 (1998).
[CrossRef]

1995 (1)

1994 (2)

1992 (1)

1988 (2)

M. Kalal and K. A. Nugent, “Abel inversion using fast Fourier transforms,” Appl. Opt. 27, 1956-1959 (1988).
[CrossRef] [PubMed]

L. M. Smith, “Nonstationary noise effects in the Abel inversion,” IEEE Trans. Inf. Theory 34, 158-161 (1988).
[CrossRef]

1987 (1)

1984 (1)

1983 (2)

K. Tatekura, “Determination of the index profile of optical fibers from transverse interferograms using Fourier theory,” Appl. Opt. 22, 460 (1983).
[CrossRef] [PubMed]

M. Deutsch and I. Beniaminy, “Inversion of Abel's integral equation for experimental data,” J. Appl. Phys. 54, 137-143(1983).
[CrossRef]

1981 (2)

1978 (1)

P. Andanson, B. Cheminat, and A. M. Halbique, “Numerical solution of the Abel integral equation: application to plasma spectroscopy,” J. Phys. D Appl. Phys. 11, 209-215 (1978).
[CrossRef]

1974 (2)

U. Buck, “Inversion of molecular scattering data,” Rev. Mod. Phys. 46, 369-389 (1974).
[CrossRef]

K. E. Atkinson, “The numerical solution of an Abel integral equation by a product trapezoidal method,” SIAM J. Numer. Anal. 11, 97-101 (1974).
[CrossRef]

1969 (1)

G. N. Minerbo and M. E. Levy, “Inversion of Abel's integral equation by means of orthogonal polynomials,” SIAM J. Numer. Anal. 6, 598-616 (1969).
[CrossRef]

1966 (2)

R. Gorenflo and Y. Kovetz, “Solution of an Abel-type integral equation in the presence of noise by quadratic programming,” Numer. Math. 8, 392-406 (1966).
[CrossRef]

C. J. Cremers and R. C. Birkebak, “Application of the Abel integral equation to spectrographic data,” Appl. Opt. 5, 1057-1064 (1966).
[CrossRef] [PubMed]

1963 (1)

W. Frie, “Zur auswertung der Abelschen integralgleichung,” Ann. Phys. 465, 332-339 (1963).
[CrossRef]

1961 (1)

1960 (2)

1956 (1)

R. N. Bracewell, “Strip integration in radio astronomy,” Aust. J. Phys. 9, 198-217 (1956).
[CrossRef]

1826 (1)

N. H. Abel, “Auflosung einer mechanischen Aufgabe,” J. Reine Angew. Math. 1, 153-157 (1826).
[CrossRef]

Abel, N. H.

N. H. Abel, “Auflosung einer mechanischen Aufgabe,” J. Reine Angew. Math. 1, 153-157 (1826).
[CrossRef]

Agrawal, A. K.

T. Wong and A. K. Agrawal, “Quantitative measurements in an unsteady flame using high-speed rainbow schlieren deflectometry,” Meas. Sci. Technol. 17, 1503-1510 (2006).
[CrossRef]

B. S. Yildirim and A. K. Agrawal, “Full-field concentration measurements of self-excited oscillations in momentum-dominated helium jets,” Exp. Fluids 38, 161-173(2005).
[CrossRef]

K. S. Pasumarthi and A. K. Agrawal, “Schlieren measurements and analysis of concentration field in self-excited helium jets,” Phys. Fluids 15, 3683-3692 (2003).
[CrossRef]

A. K. Agrawal, K. N. Alammar, and S. R. Gollahalli, “Application of rainbow schlieren deflectometry to measure temperature and oxygen concentration in a laminar jet diffusion flame,” Exp. Fluids 32, 689-691 (2002).

K. Al-Ammar, A. K. Agrawal, S. R. Gollahalli, and D. Griffin, “Concentration measurements in an axisymmetric helium jet using rainbow schlieren deflectometry,” Exp. Fluids 25, 89-95 (1998).
[CrossRef]

A. K. Shenoy, A. K. Agrawal, and S. R. Gollahalli, “Quantitative evaluation of flow computations by rainbow schlieren deflectometry,” AIAA J. 36, 1953-1960 (1998).
[CrossRef]

Alammar, K. N.

A. K. Agrawal, K. N. Alammar, and S. R. Gollahalli, “Application of rainbow schlieren deflectometry to measure temperature and oxygen concentration in a laminar jet diffusion flame,” Exp. Fluids 32, 689-691 (2002).

Al-Ammar, K.

K. Al-Ammar, A. K. Agrawal, S. R. Gollahalli, and D. Griffin, “Concentration measurements in an axisymmetric helium jet using rainbow schlieren deflectometry,” Exp. Fluids 25, 89-95 (1998).
[CrossRef]

Ampem-Lassen, E.

Andanson, P.

P. Andanson, B. Cheminat, and A. M. Halbique, “Numerical solution of the Abel integral equation: application to plasma spectroscopy,” J. Phys. D Appl. Phys. 11, 209-215 (1978).
[CrossRef]

Andrews, J. T.

R. K. Paul, J. T. Andrews, K. Bose, and P. K. Barhai, “Reconstruction errors in Abel inversion,” Plasma Devices Oper. 13, 281-290 (2005).
[CrossRef]

Atkinson, K. E.

K. E. Atkinson, “The numerical solution of an Abel integral equation by a product trapezoidal method,” SIAM J. Numer. Anal. 11, 97-101 (1974).
[CrossRef]

Barhai, P. K.

R. K. Paul, J. T. Andrews, K. Bose, and P. K. Barhai, “Reconstruction errors in Abel inversion,” Plasma Devices Oper. 13, 281-290 (2005).
[CrossRef]

Bar-Ziv, E.

Beg, F. N.

J. R. Camacho, F. N. Beg, and P. Lee1, “Comparison of sensitivities of Moire deflectometry and interferometry to measure electron densities in z-pinch plasmas,” J. Phys. D. 40, 2026-2032 (2007).
[CrossRef]

Beniaminy, I.

M. Deutsch and I. Beniaminy, “Inversion of Abel's integral equation for experimental data,” J. Appl. Phys. 54, 137-143(1983).
[CrossRef]

Birkebak, R. C.

Bockasten, K.

Borras, M. C.

K. M. Green, M. C. Borras, P. P. Woskov, G. J. Flores III, K. Hadidi, and P. Thomas, “Electronic excitation temperature profiles in an air microwave plasma torch,” IEEE Trans. Plasma Sci. 29, 399-406 (2001).
[CrossRef]

Bose, K.

R. K. Paul, J. T. Andrews, K. Bose, and P. K. Barhai, “Reconstruction errors in Abel inversion,” Plasma Devices Oper. 13, 281-290 (2005).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, “Strip integration in radio astronomy,” Aust. J. Phys. 9, 198-217 (1956).
[CrossRef]

Buchele, D. R.

Buck, U.

U. Buck, “Inversion of molecular scattering data,” Rev. Mod. Phys. 46, 369-389 (1974).
[CrossRef]

Camacho, J. R.

J. R. Camacho, F. N. Beg, and P. Lee1, “Comparison of sensitivities of Moire deflectometry and interferometry to measure electron densities in z-pinch plasmas,” J. Phys. D. 40, 2026-2032 (2007).
[CrossRef]

Cheminat, B.

P. Andanson, B. Cheminat, and A. M. Halbique, “Numerical solution of the Abel integral equation: application to plasma spectroscopy,” J. Phys. D Appl. Phys. 11, 209-215 (1978).
[CrossRef]

Coleman, H. W.

H. W. Coleman and W. G. Steele, Jr., Experimentation and Uncertainty Analysis for Engineers, 2nd ed. (Wiley, 1999).

Cremers, C. J.

Dasch, C. J.

Deutsch, M.

S. Guerona and M. Deutsch, “A fast Abel inversion algorithm,” J. Appl. Phys. 75, 4313-4318 (1994).
[CrossRef]

M. Deutsch and I. Beniaminy, “Inversion of Abel's integral equation for experimental data,” J. Appl. Phys. 54, 137-143(1983).
[CrossRef]

Diesso, M.

A. T. Ramsey and M. Diesso, “Abel inversions: error propagation and inversion reliability,” Rev. Sci. Instrum. 70, 380-383 (1999).
[CrossRef]

Dragomir, N. M.

Dribinski, V.

V. Dribinski, A. Ossadtchi, V. A. Mandeleshtam, and H. Reisler, “Reconstruction of Abel-transformable images: the Gaussian basis-set expansion Abel transform method,” Rev. Sci. Instrum. 73, 2634-2642 (2002).
[CrossRef]

Farrell, P. V.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992).

Flores, G. J.

K. M. Green, M. C. Borras, P. P. Woskov, G. J. Flores III, K. Hadidi, and P. Thomas, “Electronic excitation temperature profiles in an air microwave plasma torch,” IEEE Trans. Plasma Sci. 29, 399-406 (2001).
[CrossRef]

Freeman, M. P.

Frie, W.

W. Frie, “Zur auswertung der Abelschen integralgleichung,” Ann. Phys. 465, 332-339 (1963).
[CrossRef]

Glantschnig, W. J.

Glatt, I.

Gollahalli, S. R.

A. K. Agrawal, K. N. Alammar, and S. R. Gollahalli, “Application of rainbow schlieren deflectometry to measure temperature and oxygen concentration in a laminar jet diffusion flame,” Exp. Fluids 32, 689-691 (2002).

K. Al-Ammar, A. K. Agrawal, S. R. Gollahalli, and D. Griffin, “Concentration measurements in an axisymmetric helium jet using rainbow schlieren deflectometry,” Exp. Fluids 25, 89-95 (1998).
[CrossRef]

A. K. Shenoy, A. K. Agrawal, and S. R. Gollahalli, “Quantitative evaluation of flow computations by rainbow schlieren deflectometry,” AIAA J. 36, 1953-1960 (1998).
[CrossRef]

Gorenflo, R.

R. Gorenflo and Y. Kovetz, “Solution of an Abel-type integral equation in the presence of noise by quadratic programming,” Numer. Math. 8, 392-406 (1966).
[CrossRef]

R. Gorenflo and S. Vessella, “Abel integral equations: analysis and applications,” in Lecture Notes in Mathematics, A. Dold, B. Eckmann, and F. Takens, eds. (Springer-Verlag, 1980).

Green, K. M.

K. M. Green, M. C. Borras, P. P. Woskov, G. J. Flores III, K. Hadidi, and P. Thomas, “Electronic excitation temperature profiles in an air microwave plasma torch,” IEEE Trans. Plasma Sci. 29, 399-406 (2001).
[CrossRef]

Greenberg, P. S.

Griffin, D.

K. Al-Ammar, A. K. Agrawal, S. R. Gollahalli, and D. Griffin, “Concentration measurements in an axisymmetric helium jet using rainbow schlieren deflectometry,” Exp. Fluids 25, 89-95 (1998).
[CrossRef]

Guerona, S.

S. Guerona and M. Deutsch, “A fast Abel inversion algorithm,” J. Appl. Phys. 75, 4313-4318 (1994).
[CrossRef]

Hadidi, K.

K. M. Green, M. C. Borras, P. P. Woskov, G. J. Flores III, K. Hadidi, and P. Thomas, “Electronic excitation temperature profiles in an air microwave plasma torch,” IEEE Trans. Plasma Sci. 29, 399-406 (2001).
[CrossRef]

Halbique, A. M.

P. Andanson, B. Cheminat, and A. M. Halbique, “Numerical solution of the Abel integral equation: application to plasma spectroscopy,” J. Phys. D Appl. Phys. 11, 209-215 (1978).
[CrossRef]

Hammond, D. C.

Hofeldt, D. L.

Holliday, A.

Huntington, S. T.

Kafri, O.

Kalal, M.

Katz, S.

Keren, E.

Klimek, R. B.

Kovetz, Y.

R. Gorenflo and Y. Kovetz, “Solution of an Abel-type integral equation in the presence of noise by quadratic programming,” Numer. Math. 8, 392-406 (1966).
[CrossRef]

Lee1, P.

J. R. Camacho, F. N. Beg, and P. Lee1, “Comparison of sensitivities of Moire deflectometry and interferometry to measure electron densities in z-pinch plasmas,” J. Phys. D. 40, 2026-2032 (2007).
[CrossRef]

Levy, M. E.

G. N. Minerbo and M. E. Levy, “Inversion of Abel's integral equation by means of orthogonal polynomials,” SIAM J. Numer. Anal. 6, 598-616 (1969).
[CrossRef]

Mandeleshtam, V. A.

V. Dribinski, A. Ossadtchi, V. A. Mandeleshtam, and H. Reisler, “Reconstruction of Abel-transformable images: the Gaussian basis-set expansion Abel transform method,” Rev. Sci. Instrum. 73, 2634-2642 (2002).
[CrossRef]

Minerbo, G. N.

G. N. Minerbo and M. E. Levy, “Inversion of Abel's integral equation by means of orthogonal polynomials,” SIAM J. Numer. Anal. 6, 598-616 (1969).
[CrossRef]

Nestor, O. H.

O. H. Nestor and H. N. Olson, “Numerical methods for reducing line and surface probe data,” SIAM Rev. 2, 200-207(1960).
[CrossRef]

Nugent, K. A.

Olson, H. N.

O. H. Nestor and H. N. Olson, “Numerical methods for reducing line and surface probe data,” SIAM Rev. 2, 200-207(1960).
[CrossRef]

Ossadtchi, A.

V. Dribinski, A. Ossadtchi, V. A. Mandeleshtam, and H. Reisler, “Reconstruction of Abel-transformable images: the Gaussian basis-set expansion Abel transform method,” Rev. Sci. Instrum. 73, 2634-2642 (2002).
[CrossRef]

Pasumarthi, K. S.

K. S. Pasumarthi and A. K. Agrawal, “Schlieren measurements and analysis of concentration field in self-excited helium jets,” Phys. Fluids 15, 3683-3692 (2003).
[CrossRef]

Paul, R. K.

R. K. Paul, J. T. Andrews, K. Bose, and P. K. Barhai, “Reconstruction errors in Abel inversion,” Plasma Devices Oper. 13, 281-290 (2005).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992).

Ramsey, A. T.

A. T. Ramsey and M. Diesso, “Abel inversions: error propagation and inversion reliability,” Rev. Sci. Instrum. 70, 380-383 (1999).
[CrossRef]

Reisler, H.

V. Dribinski, A. Ossadtchi, V. A. Mandeleshtam, and H. Reisler, “Reconstruction of Abel-transformable images: the Gaussian basis-set expansion Abel transform method,” Rev. Sci. Instrum. 73, 2634-2642 (2002).
[CrossRef]

Roberts, A.

Rubinstein, R.

Shenoy, A. K.

A. K. Shenoy, A. K. Agrawal, and S. R. Gollahalli, “Quantitative evaluation of flow computations by rainbow schlieren deflectometry,” AIAA J. 36, 1953-1960 (1998).
[CrossRef]

Smith, L. M.

L. M. Smith, “Nonstationary noise effects in the Abel inversion,” IEEE Trans. Inf. Theory 34, 158-161 (1988).
[CrossRef]

Steele, W. G.

H. W. Coleman and W. G. Steele, Jr., Experimentation and Uncertainty Analysis for Engineers, 2nd ed. (Wiley, 1999).

Tatekura, K.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992).

Thomas, P.

K. M. Green, M. C. Borras, P. P. Woskov, G. J. Flores III, K. Hadidi, and P. Thomas, “Electronic excitation temperature profiles in an air microwave plasma torch,” IEEE Trans. Plasma Sci. 29, 399-406 (2001).
[CrossRef]

Vasil'ev, L. A.

L. A. Vasil'ev, Schlieren Methods (Israel Program for Scientific Translations, 1971).

Vessella, S.

R. Gorenflo and S. Vessella, “Abel integral equations: analysis and applications,” in Lecture Notes in Mathematics, A. Dold, B. Eckmann, and F. Takens, eds. (Springer-Verlag, 1980).

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992).

Wong, T.

T. Wong and A. K. Agrawal, “Quantitative measurements in an unsteady flame using high-speed rainbow schlieren deflectometry,” Meas. Sci. Technol. 17, 1503-1510 (2006).
[CrossRef]

Woskov, P. P.

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Figures (8)

Fig. 1
Fig. 1

Comparison of inversion coefficient ( D 10 j ) for different algorithms ( N = 25 ).

Fig. 2
Fig. 2

Test profiles of (a) refractive-index difference (to be reconstructed) and (b) transverse deflection angle (input data).

Fig. 3
Fig. 3

Inversion accuracy of different algorithms: (a) inverted profile and (b) error in inversion.

Fig. 4
Fig. 4

Effect of buffer zone on curve-fit error using GHPP.

Fig. 5
Fig. 5

Inversion error with and without a buffer zone in the GHPP method.

Fig. 6
Fig. 6

Error propagation for different algorithms using a uniform error in input data.

Fig. 7
Fig. 7

Error propagation for different algorithms: (a) random error in input data and (b) propagated error profiles.

Fig. 8
Fig. 8

Error propagation for different algorithms: (a) Gaussian distribution error in input data and (b) propagated error profiles.

Equations (20)

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P ( x ) = 2 x R f ( r ) r r 2 x 2 d r ,
f ( r ) = 1 π r R d P ( x ) d x d x x 2 r 2 .
Θ ( x ) = 2 x x R δ r d r r 2 x 2 .
δ ( r ) = 1 π r R Θ ( x ) d x x 2 r 2 .
f ( r i ) = 1 Δ r · j = i D i j · P j
δ ( r i ) = j = i N + 1 D i j · Θ j .
δ ( r i ) = 1 π r i R Θ ( r ) d r r 2 r i 2 = 1 π j = i , incr = 2 N 1 r j r j + 2 Θ ( r j ) d r j r j 2 r i 2 = 1 π j = i , incr = 2 N 1 [ Δ r 3 · ( Θ j r j 2 r i 2 + 4 Θ j + 1 r j + 1 2 r i 2 + Θ j + 2 r j + 2 2 r i 2 ) ] .
D ij = 1 3 π · [ 2 + ( 1 + ( 1 ) ( j i + 1 ) ) ] ( j 1 ) 2 ( i 1 ) 2 if     j > i and j N + 1 , = 1 3 π · [ 2 + ( 1 + ( 1 ) ( j i + 1 ) ) ] 2 · ( j 1 ) 2 ( i 1 ) 2 if     j = N + 1 , = 0 if     j < i .
D i i = D i ( i + 1 ) for     i N , = 0 for     i = N + 1.
δ ( r ) = 1 π · r Θ ( x ) · d x x 2 r 2 = 1 π · j = 1 N ( 0 Δ r ( Θ j + Θ j + 1 2 ) · d l ( r j + l ) 2 r i 2 ) ,
δ ( r i ) = 1 π · j = 1 N ( 0 1 ( Θ j + Θ j + 1 2 ) · d β ( ( j 1 ) + β ) 2 ( i 1 ) 2 ) ,
D i j = 1 2 π · ln ( j + j 2 ( i 1 ) 2 ( j 2 ) + ( j 2 ) 2 ( i 1 ) 2 ) if       j > i and j 2 , = 1 2 π · ( 2 + ln ( j + j 2 ( i 1 ) 2 ( j 1 ) + ( j 1 ) 2 ( i 1 ) 2 ) ) if     j > i and j = 2 , = 1 2 π · ln ( j + j 2 ( i 1 ) 2 ( j 1 ) + ( j 1 ) 2 ( i 1 ) 2 ) if     j = i and i 1 , = 0 if     j < i or j = i = 1.
δ ( r ) = 1 π · r Θ ( x ) · d x x 2 r 2 = 1 π · j = 1 N ( 0 Δ r ( Θ j + l · ( Θ j + 1 Θ j ) Δ r ) · d l ( r j + l ) 2 r i 2 ) .
D i j = 1 π · ( A i , j A i , ( j 1 ) j · B i , j + ( j 2 ) · B i , ( j 1 ) ) if       j > i and j 2 , = 1 π · ( A i , j j · B i , j 1 ) if       j > i and j = 2 , = 1 π · ( A i , j j · B i , j ) if     j = i and i 1 , = 0 if     j = i = 1   or   j < i
A i , j = j 2 ( i 1 ) 2 ( j 1 ) 2 ( i 1 ) 2 , B i , j = ln ( j + j 2 ( i 1 ) 2 ( j 1 ) + ( j 1 ) 2 ( i 1 ) 2 ) .
Θ ( x ) = ( p = 0 6 a ( 2 p + 1 ) · q = 0 6 H ( 2 p + 1 ) , ( 2 q + 1 ) · y ( 2 q + 1 ) ) · e x 2 .
δ ( r i ) = 1 π · r i Θ ( x ) · d x x 2 r i 2 = 1 π · r i ( p = 0 6 a ( 2 p + 1 ) · q = 0 6 H ( 2 p + 1 ) , ( 2 q + 1 ) · x ( 2 q + 1 ) ) · e x 2 · d x x 2 r i 2 δ ( r i ) = 1 π · ( p = 0 6 a ( 2 p + 1 ) · q = 0 6 H ( 2 p + 1 ) , ( 2 q + 1 ) · I q , i ) ,
D i j = 1 π 3 / 2 · p = 0 6 ( 1 ( 2 p + 1 ) ! · 2 ( 2 p + 1 ) · q = 0 6 ( H ( 2 p + 1 ) , ( 2 q + 1 ) · ( j 1 ) ( 2 q + 1 ) · ( Δ r ) ( 2 q + 2 ) ) · q = 0 6 H ( 2 p + 1 ) , ( 2 q + 1 ) · I q , i ) if     j = 1   or N + 1 , = 1 π 3 / 2 · p = 0 6 ( 1 ( 2 p + 1 ) ! · 2 ( 2 p + 1 ) · q = 0 6 ( H ( 2 p + 1 ) , ( 2 q + 1 ) · ( 2 · ( j 1 ) ( 2 q + 1 ) ) · ( Δ r ) ( 2 q + 2 ) ) · q = 0 6 H ( 2 p + 1 ) , ( 2 q + 1 ) · I q , i ) if     1 < j < N + 1.
δ ( r ) = ( e ( r 1 ) 2 + e ( r + 1 ) 2 ) .
U δ ( i ) = j = i N + 1 ( ( Θ j ( j = i N + 1 D i j · Θ j ) ) 2 · U Θ j 2 ) = j = i N + 1 ( D i j · U Θ j ) 2 ,

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