Abstract

An efficient algorithm for the accurate computation of the linear canonical transform with complex transform parameters and with complex output variable is presented. Sampling issues are discussed and the requirements for different cases given. Simulations are provided to validate the results.

© 2011 Optical Society of America

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References

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  1. K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, 1979).
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  9. P. A. Bélanger, A. Hardy, and A. E. Siegman, “Resonant modes of optical cavities with phase-conjugate mirrors,” Appl. Opt. 19, 602–609 (1980).
    [CrossRef] [PubMed]
  10. K. K. Sharma, “Fractional Laplace transform,” Signal, Image and Video Processing 4, 377–379 (2009).
    [CrossRef]
  11. B. Deng, R. Tao, and Y. Wang, “Convolution theorems for the linear canonical transform and their applications,” Science in China Ser. F 49, 592–603 (2006).
    [CrossRef]
  12. B.-Z. Li, R. Tao, and Y. Wang, “New sampling formulae related to linear canonical transform,” Signal Process. 87, 983–990 (2007).
    [CrossRef]
  13. J.-J. Ding, “Research of fractional Fourier transform and linear canonical transform,” Ph.D. dissertation (National Taiwan University, 2001).
  14. J. Zhao, R. Tao, and Y. Wang, “Sampling rate conversion for linear canonical transform,” Signal Process. 88, 2825–2832 (2008).
    [CrossRef]
  15. F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293–297 (1981).
    [CrossRef]
  16. J. J. Healy and J. T. Sheridan, “Sampling and discretization of the linear canonical transform,” Signal Process. 89, 641–648(2009).
    [CrossRef]
  17. J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “An additional sampling criterion for the linear canonical transform,” Opt. Lett. 33, 2599–2601 (2008).
    [CrossRef] [PubMed]
  18. F. Oktem and H. M. Ozaktas, “Exact relation between continuous and discrete linear canonical transforms,” IEEE Signal Process. Lett. 16, 727–730 (2009).
    [CrossRef]
  19. A. Stern, “Sampling of linear canonical transformed signals,” Signal Process. 86, 1421–1425 (2006).
    [CrossRef]
  20. J. J. Healy and J. T. Sheridan, “Fast linear canonical transforms,” J. Opt. Soc. Am. A 27, 21–30 (2010).
    [CrossRef]
  21. B. M. Hennelly and J. T. Sheridan, “Fast numerical algorithm for the linear canonical transform,” J. Opt. Soc. Am. A 22, 928–937(2005).
    [CrossRef]
  22. A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
    [CrossRef]
  23. H. M. Ozaktas, A. Koç, I. Sari, and M. A. Kutay, “Efficient computation of quadratic-phase integrals in optics,” Opt Lett. 31, 35–37 (2006).
    [CrossRef] [PubMed]
  24. J. J. Healy and J. T. Sheridan, “Reevaluation of the direct method of calculating Fresnel and other linear canonical transforms,” Opt. Lett. 35, 947–949 (2010).
    [CrossRef] [PubMed]
  25. A. Koç, H. M. Ozaktas, and L. Hesselink, “Fast and accurate algorithm for the computation of complex linear canonical transforms,” J. Opt. Soc. Am. A 27, 1896–1908 (2010).
    [CrossRef]
  26. J. J. Healy and J. T. Sheridan, “Space–bandwidth ratio as a means of choosing between Fresnel and other linear canonical transform algorithms,” J. Opt. Soc. Am. A 28, 786–790 (2011).
    [CrossRef]
  27. F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula,” Appl. Opt. 45, 1102–1110 (2006).
    [CrossRef] [PubMed]
  28. F. Jia, “Study on the principle and applications of digital holography,” Master’s dissertation (Northwest University, 2008).
  29. C. Liu, D. Wang, and Y. Zhang, “Comparison and verification of numerical reconstruction methods in digital holography,” Opt. Eng. 48, 105802 (2009).
    [CrossRef]
  30. H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
    [CrossRef]
  31. J. W. Brown and R. V. Churchill, Complex Variables and Applications (McGraw- Hill, 2004).

2011 (1)

2010 (3)

2009 (4)

C. Liu, D. Wang, and Y. Zhang, “Comparison and verification of numerical reconstruction methods in digital holography,” Opt. Eng. 48, 105802 (2009).
[CrossRef]

K. K. Sharma, “Fractional Laplace transform,” Signal, Image and Video Processing 4, 377–379 (2009).
[CrossRef]

J. J. Healy and J. T. Sheridan, “Sampling and discretization of the linear canonical transform,” Signal Process. 89, 641–648(2009).
[CrossRef]

F. Oktem and H. M. Ozaktas, “Exact relation between continuous and discrete linear canonical transforms,” IEEE Signal Process. Lett. 16, 727–730 (2009).
[CrossRef]

2008 (3)

J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “An additional sampling criterion for the linear canonical transform,” Opt. Lett. 33, 2599–2601 (2008).
[CrossRef] [PubMed]

J. Zhao, R. Tao, and Y. Wang, “Sampling rate conversion for linear canonical transform,” Signal Process. 88, 2825–2832 (2008).
[CrossRef]

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

2007 (1)

B.-Z. Li, R. Tao, and Y. Wang, “New sampling formulae related to linear canonical transform,” Signal Process. 87, 983–990 (2007).
[CrossRef]

2006 (4)

H. M. Ozaktas, A. Koç, I. Sari, and M. A. Kutay, “Efficient computation of quadratic-phase integrals in optics,” Opt Lett. 31, 35–37 (2006).
[CrossRef] [PubMed]

F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula,” Appl. Opt. 45, 1102–1110 (2006).
[CrossRef] [PubMed]

B. Deng, R. Tao, and Y. Wang, “Convolution theorems for the linear canonical transform and their applications,” Science in China Ser. F 49, 592–603 (2006).
[CrossRef]

A. Stern, “Sampling of linear canonical transformed signals,” Signal Process. 86, 1421–1425 (2006).
[CrossRef]

2005 (1)

1998 (1)

1996 (1)

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

1994 (1)

1993 (1)

1987 (1)

1981 (1)

F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293–297 (1981).
[CrossRef]

1980 (2)

P. A. Bélanger, A. Hardy, and A. E. Siegman, “Resonant modes of optical cavities with phase-conjugate mirrors,” Appl. Opt. 19, 602–609 (1980).
[CrossRef] [PubMed]

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

1979 (1)

1970 (1)

Abe, S.

Arikan, O.

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

Bartelt, H. O.

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Bastiaans, M. J.

Bélanger, P. A.

Bozdagi, G.

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

Brenner, K.-H.

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Brown, J. W.

J. W. Brown and R. V. Churchill, Complex Variables and Applications (McGraw- Hill, 2004).

Candan, C.

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

Churchill, R. V.

J. W. Brown and R. V. Churchill, Complex Variables and Applications (McGraw- Hill, 2004).

Collins, S. A.

Deng, B.

B. Deng, R. Tao, and Y. Wang, “Convolution theorems for the linear canonical transform and their applications,” Science in China Ser. F 49, 592–603 (2006).
[CrossRef]

Ding, J.-J.

J.-J. Ding, “Research of fractional Fourier transform and linear canonical transform,” Ph.D. dissertation (National Taiwan University, 2001).

Gori, F.

F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293–297 (1981).
[CrossRef]

Grum, T. P.

Hanson, S. G.

Hardy, A.

Healy, J. J.

Hennelly, B. M.

Hesselink, L.

Jia, F.

F. Jia, “Study on the principle and applications of digital holography,” Master’s dissertation (Northwest University, 2008).

Koç, A.

A. Koç, H. M. Ozaktas, and L. Hesselink, “Fast and accurate algorithm for the computation of complex linear canonical transforms,” J. Opt. Soc. Am. A 27, 1896–1908 (2010).
[CrossRef]

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

H. M. Ozaktas, A. Koç, I. Sari, and M. A. Kutay, “Efficient computation of quadratic-phase integrals in optics,” Opt Lett. 31, 35–37 (2006).
[CrossRef] [PubMed]

Kutay, M. A.

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

H. M. Ozaktas, A. Koç, I. Sari, and M. A. Kutay, “Efficient computation of quadratic-phase integrals in optics,” Opt Lett. 31, 35–37 (2006).
[CrossRef] [PubMed]

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

Li, B.-Z.

B.-Z. Li, R. Tao, and Y. Wang, “New sampling formulae related to linear canonical transform,” Signal Process. 87, 983–990 (2007).
[CrossRef]

Liu, C.

C. Liu, D. Wang, and Y. Zhang, “Comparison and verification of numerical reconstruction methods in digital holography,” Opt. Eng. 48, 105802 (2009).
[CrossRef]

Lohmann, A. W.

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Oktem, F.

F. Oktem and H. M. Ozaktas, “Exact relation between continuous and discrete linear canonical transforms,” IEEE Signal Process. Lett. 16, 727–730 (2009).
[CrossRef]

Ozaktas, H. M.

A. Koç, H. M. Ozaktas, and L. Hesselink, “Fast and accurate algorithm for the computation of complex linear canonical transforms,” J. Opt. Soc. Am. A 27, 1896–1908 (2010).
[CrossRef]

F. Oktem and H. M. Ozaktas, “Exact relation between continuous and discrete linear canonical transforms,” IEEE Signal Process. Lett. 16, 727–730 (2009).
[CrossRef]

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

H. M. Ozaktas, A. Koç, I. Sari, and M. A. Kutay, “Efficient computation of quadratic-phase integrals in optics,” Opt Lett. 31, 35–37 (2006).
[CrossRef] [PubMed]

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

Rose, B.

Sari, I.

H. M. Ozaktas, A. Koç, I. Sari, and M. A. Kutay, “Efficient computation of quadratic-phase integrals in optics,” Opt Lett. 31, 35–37 (2006).
[CrossRef] [PubMed]

Sharma, K. K.

K. K. Sharma, “Fractional Laplace transform,” Signal, Image and Video Processing 4, 377–379 (2009).
[CrossRef]

Shen, F.

Sheridan, J. T.

Siegman, A. E.

Stern, A.

A. Stern, “Sampling of linear canonical transformed signals,” Signal Process. 86, 1421–1425 (2006).
[CrossRef]

Tao, R.

J. Zhao, R. Tao, and Y. Wang, “Sampling rate conversion for linear canonical transform,” Signal Process. 88, 2825–2832 (2008).
[CrossRef]

B.-Z. Li, R. Tao, and Y. Wang, “New sampling formulae related to linear canonical transform,” Signal Process. 87, 983–990 (2007).
[CrossRef]

B. Deng, R. Tao, and Y. Wang, “Convolution theorems for the linear canonical transform and their applications,” Science in China Ser. F 49, 592–603 (2006).
[CrossRef]

Wang, A.

Wang, D.

C. Liu, D. Wang, and Y. Zhang, “Comparison and verification of numerical reconstruction methods in digital holography,” Opt. Eng. 48, 105802 (2009).
[CrossRef]

Wang, Y.

J. Zhao, R. Tao, and Y. Wang, “Sampling rate conversion for linear canonical transform,” Signal Process. 88, 2825–2832 (2008).
[CrossRef]

B.-Z. Li, R. Tao, and Y. Wang, “New sampling formulae related to linear canonical transform,” Signal Process. 87, 983–990 (2007).
[CrossRef]

B. Deng, R. Tao, and Y. Wang, “Convolution theorems for the linear canonical transform and their applications,” Science in China Ser. F 49, 592–603 (2006).
[CrossRef]

Wolf, K. B.

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, 1979).

Yura, H. T.

Zhang, Y.

C. Liu, D. Wang, and Y. Zhang, “Comparison and verification of numerical reconstruction methods in digital holography,” Opt. Eng. 48, 105802 (2009).
[CrossRef]

Zhao, J.

J. Zhao, R. Tao, and Y. Wang, “Sampling rate conversion for linear canonical transform,” Signal Process. 88, 2825–2832 (2008).
[CrossRef]

Appl. Opt. (2)

IEEE Signal Process. Lett. (1)

F. Oktem and H. M. Ozaktas, “Exact relation between continuous and discrete linear canonical transforms,” IEEE Signal Process. Lett. 16, 727–730 (2009).
[CrossRef]

IEEE Trans. Signal Process. (2)

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (7)

Opt Lett. (1)

H. M. Ozaktas, A. Koç, I. Sari, and M. A. Kutay, “Efficient computation of quadratic-phase integrals in optics,” Opt Lett. 31, 35–37 (2006).
[CrossRef] [PubMed]

Opt. Commun. (2)

H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293–297 (1981).
[CrossRef]

Opt. Eng. (1)

C. Liu, D. Wang, and Y. Zhang, “Comparison and verification of numerical reconstruction methods in digital holography,” Opt. Eng. 48, 105802 (2009).
[CrossRef]

Opt. Lett. (3)

Science in China Ser. F (1)

B. Deng, R. Tao, and Y. Wang, “Convolution theorems for the linear canonical transform and their applications,” Science in China Ser. F 49, 592–603 (2006).
[CrossRef]

Signal Process. (4)

B.-Z. Li, R. Tao, and Y. Wang, “New sampling formulae related to linear canonical transform,” Signal Process. 87, 983–990 (2007).
[CrossRef]

J. J. Healy and J. T. Sheridan, “Sampling and discretization of the linear canonical transform,” Signal Process. 89, 641–648(2009).
[CrossRef]

J. Zhao, R. Tao, and Y. Wang, “Sampling rate conversion for linear canonical transform,” Signal Process. 88, 2825–2832 (2008).
[CrossRef]

A. Stern, “Sampling of linear canonical transformed signals,” Signal Process. 86, 1421–1425 (2006).
[CrossRef]

Signal, Image and Video Processing (1)

K. K. Sharma, “Fractional Laplace transform,” Signal, Image and Video Processing 4, 377–379 (2009).
[CrossRef]

Other (4)

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, 1979).

J.-J. Ding, “Research of fractional Fourier transform and linear canonical transform,” Ph.D. dissertation (National Taiwan University, 2001).

F. Jia, “Study on the principle and applications of digital holography,” Master’s dissertation (Northwest University, 2008).

J. W. Brown and R. V. Churchill, Complex Variables and Applications (McGraw- Hill, 2004).

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Figures (8)

Fig. 1
Fig. 1

Illustration for the complex LCT of samples of an input function. ξ I and ξ R represent the imaginary and real axis of the complex plane. The radius of the circle is r. (a)  g ( ξ ) is considered to be zero outside the circular area. (b) Shows the periodicity of the shifted replicas in the complex plane.

Fig. 2
Fig. 2

Optical Fourier transformation system with a Gaussian apodized limiting aperture (LA) of radius σ placed immediately to the right of the transmission lens.

Fig. 3
Fig. 3

Relationship between the sequence number and the coordinates.

Fig. 4
Fig. 4

Flow chart of the FFT-DI. The operation * indicates an inner product of the two data vectors, i.e., their entry-wise product.

Fig. 5
Fig. 5

Results of simulations using the different methods: (a) the zero order by the FFT-DI, (b) the analytical formula, (c) the difference between the results presented in (a) and (b). (d) The first replica by the FFT-DI.

Fig. 6
Fig. 6

Deviations between the different methods calculated for results along the imaginary axis of the complex plane. (a) The differences between the FFT-DI and the DI results. (b) The deviation of the FFT-DI from the analytical formula.

Fig. 7
Fig. 7

Outputs of an Fourier transformation system with lossy Gaussian apodized apertures of different widths. (a)  σ = 4.0 mm and (b)  σ = 0.4 mm .

Fig. 8
Fig. 8

Discrete FST results calculated using the FFT-DI with α = γ = 1 / λ d . (a) The magnitude and (b) the corresponding phase predictions.

Equations (26)

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g ( ξ ) = β exp ( j π / 4 ) f ( x ) exp [ j π ( α x 2 2 β x ξ + γ ξ 2 ) ] d x ,
h ( η ) = h ( β ξ ) = f ( x ) exp ( + j π α x 2 ) exp ( j 2 π x β ξ ) d x ,
h ( β ξ ) = g ( ξ ) / [ β exp ( j π / 4 ) exp ( + j π γ ξ 2 ) ] .
g d ( ξ ) = β exp ( j π / 4 ) exp ( + j π γ ξ 2 ) m = + f ( m Δ x ) exp [ + j π α ( m Δ x ) 2 ] exp ( j 2 π m Δ x β ξ ) Δ x = β exp ( j π / 4 ) exp ( + j π γ ξ 2 ) + f ( x ) exp ( + j π α x 2 ) comb ( x Δ x ) exp ( j 2 π x η ) d x = β exp ( j π / 4 ) exp ( + j π γ ξ 2 ) [ h ( η ) Δ x comb ( Δ x η ) ] = β exp ( j π / 4 ) exp ( + j π γ ξ 2 ) m = h ( η m Δ x ) = β exp ( j π / 4 ) exp ( + j π γ ξ 2 ) m = h ( β ξ m Δ x ) = exp ( + j π γ ξ 2 ) m = g ( ξ m β Δ x ) / exp [ + j π γ ( ξ m β Δ x ) 2 ] ,
P = 1 β Δ x .
g ( ξ ) = 0 + f ( x ) exp ( ξ x ) d x ,
g ( ξ ) = ( β exp ( j π / 4 ) f ( x ) exp [ j π ( 2 β x ξ ) ] d x ) / ( β exp ( j π / 4 ) ) ,
1 | β | Δ x 2 r .
Δ x 1 2 r | β | .
T = exp ( x 2 σ 2 ) ,
M = ( 1 λ f 0 1 ) ( 1 0 j / π σ 2 1 ) ( 1 0 1 / λ f 1 ) ( 1 λ f 0 1 ) = ( j λ f π σ 2 λ f ( 1 + j λ f π σ 2 ) 1 λ f + j π σ 2 j λ f π σ 2 ) .
α = A B = 1 λ f j π σ 2 ,
β = 1 B = 1 λ f ( 1 + j λ f π σ 2 ) ,
γ = D B = 1 λ f j π σ 2 .
θ = arctan ( π σ 2 λ f ) .
Δ x { 1 / [ r | β | cot 2 ( θ ) + 1 ] , π / 4 θ 3 π / 4 1 / [ 2 r | β | | cos ( θ ) | ] , otherwise .
x = ( m [ M / 2 ] ) Δ x + x 0 , m [ 0 M 1 ] ,
ξ = ( n [ N / 2 ] ) Δ ξ + ξ 0 , n [ 0 N 1 ] ,
g ( n ) = c ( n ) m = 0 M 1 a ( m ) b ( n m ) ,
c ( n ) = Δ x β exp ( j π / 4 ) exp { + j π [ α ( [ M 2 ] Δ x + x 0 ) 2 + γ ( [ N 2 ] Δ ξ + ξ 0 ) 2 ] } × exp { j 2 β π ( x 0 ξ 0 + [ M 2 ] [ N 2 ] Δ x Δ ξ [ M 2 ] ξ 0 Δ x [ N 2 ] x 0 Δ ξ ) } × exp { + j π n 2 ( γ Δ ξ 2 β Δ x Δ ξ ) } × exp { + j 2 π n [ γ Δ ξ ( ξ 0 [ N 2 ] Δ ξ ) β ( x 0 Δ ξ [ M 2 ] Δ x Δ ξ ) ] } ,
a ( m ) = f ( m ) exp { + j π m 2 ( α Δ x 2 β Δ x Δ ξ ) } × exp { + j 2 π m [ α Δ x ( x 0 [ M 2 ] Δ x ) β ( ξ 0 Δ x [ N 2 ] Δ x Δ ξ ) ] } ,
b ( m ) = exp ( + j π β m 2 Δ x Δ ξ ) .
O [ ( M + N ) log ( M + N ) ] ,
g ( ξ ) = 0 1 exp ( ξ x ) d x = { [ 1 exp ( ξ ) ] / ξ , ξ 0 1 , ξ = 0 .
g ( ξ ) = g ( c x ) = c x [ 1 exp ( x / c ) ] .
| g ( ξ ) | k | g ( ξ ) | max ,

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