Abstract

Two related basic transition operators, T1 and T2, are found that transform arbitrary solutions of the parabolic equation of the paraxial approximation into exact monochromatic solutions of the scalar wave equation or of the corresponding Helmholtz equation. The operators realize different boundary conditions. The operator T1 preserves the transverse field distribution of the paraxial approximation in the plane z = 0 for the obtained exact solution. The method is applied to calculate the complete corrections to the paraxial approximation of the fundamental Gaussian beam solutions of the n-dimensional wave equation. The lowest-order correction to the paraxial approximation in the three-dimensional case is found to be in agreement with the result of Agrawal and Pattanayak [ J. Opt. Soc. Am. 69, 575 ( 1979)]. The complete series of corrections on the axis is summed up to a transcendental function and discussed for the three-dimensional case.

© 1992 Optical Society of America

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References

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  1. H. Kogelnik, T. Li, “Laser beams and resonators,” Appl. Opt. 10, 1550–1567 (1966).
    [CrossRef]
  2. S. Solimeno, B. Crosignani, P. Di Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, New York, 1986).
  3. M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
  4. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
    [CrossRef]
  5. G. P. Agrawal, D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,”J. Opt. Soc. Am. 69, 575–578 (1979).
    [CrossRef]
  6. G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
    [CrossRef]
  7. M. Couture, P. A. Bélanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
    [CrossRef]
  8. B. T. Landesman, H. H. Barrett, “Gaussian amplitude functions that are exact solutions to the scalar Helmholtz equation,” J. Opt. Soc. Am. A 5, 1610–1619 (1988).
    [CrossRef]
  9. B. T. Landesman, “Geometrical representation of the fundamental mode of a Gaussian beam in oblate spheroidal coordinates,” J. Opt. Soc. Am. A 6, 5–17 (1989).
    [CrossRef]
  10. A. WiInsche, “Generalized Gaussian beam solutions of paraxial optics and their connection to a hidden symmetry,” J. Opt. Soc. Am. A 6, 1320–1329 (1989).
    [CrossRef]
  11. H. Bateman, A. Erdélyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 2.
  12. G. C. Sherman, H. J. Bremermann, “Generalization of the angular spectrum of plane waves and the diffraction transform,”J. Opt. Soc. Am. 59, 146–156 (1969).
    [CrossRef]
  13. G. C. Sherman, “Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves,”J. Opt. Soc. Am. 59, 697–711 (1969).
    [CrossRef]
  14. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, Natl. Bur. Stand. (U.S.) Appl. Math. Ser. 55, (1964).
  15. E. T. Whittaker, G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, Cambridge, 1952).
  16. G. M. Fikhtengolts, Course of Differential and Integral Calculus (in Russian) (Fizmatgiz, Moscow, 1959), Vol. II.
  17. I. S. Gradshteyn, I. M. Ryshik, Tables of Integrals, Series and Products (Academic, New York, 1965).
  18. A. Wünsche, “Grenzbedingungen der Elektrodynamik für Medien mit räiumlicher Dispersion und Übergangsschichten,” Ann. Phys. (Leipzig) 37, 121–142 (1980).
  19. E. M. Lifshits, P. L. Pitayevskyi, Physical Kinetics, Vol. 10 of Course on Theoretical Physics, L. D. Landau, E. M. Lifshits, eds. (Pergamon, Oxford, 1981) [Fizicheskaya Kinetika(Nauka, Moscow, 1979)].

1989

1988

1983

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

1981

M. Couture, P. A. Bélanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[CrossRef]

1980

A. Wünsche, “Grenzbedingungen der Elektrodynamik für Medien mit räiumlicher Dispersion und Übergangsschichten,” Ann. Phys. (Leipzig) 37, 121–142 (1980).

1979

1975

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

1969

1966

1964

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, Natl. Bur. Stand. (U.S.) Appl. Math. Ser. 55, (1964).

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, Natl. Bur. Stand. (U.S.) Appl. Math. Ser. 55, (1964).

Agrawal, G. P.

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

G. P. Agrawal, D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,”J. Opt. Soc. Am. 69, 575–578 (1979).
[CrossRef]

Barrett, H. H.

Bateman, H.

H. Bateman, A. Erdélyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 2.

Bélanger, P. A.

M. Couture, P. A. Bélanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[CrossRef]

Bremermann, H. J.

Couture, M.

M. Couture, P. A. Bélanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[CrossRef]

Crosignani, B.

S. Solimeno, B. Crosignani, P. Di Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, New York, 1986).

Davis, L. W.

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

Di Porto, P.

S. Solimeno, B. Crosignani, P. Di Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, New York, 1986).

Erdélyi, A.

H. Bateman, A. Erdélyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 2.

Fikhtengolts, G. M.

G. M. Fikhtengolts, Course of Differential and Integral Calculus (in Russian) (Fizmatgiz, Moscow, 1959), Vol. II.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryshik, Tables of Integrals, Series and Products (Academic, New York, 1965).

Kogelnik, H.

Landesman, B. T.

Lax, M.

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Li, T.

Lifshits, E. M.

E. M. Lifshits, P. L. Pitayevskyi, Physical Kinetics, Vol. 10 of Course on Theoretical Physics, L. D. Landau, E. M. Lifshits, eds. (Pergamon, Oxford, 1981) [Fizicheskaya Kinetika(Nauka, Moscow, 1979)].

Louisell, W. H.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

McKnight, W. B.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Pattanayak, D. N.

Pitayevskyi, P. L.

E. M. Lifshits, P. L. Pitayevskyi, Physical Kinetics, Vol. 10 of Course on Theoretical Physics, L. D. Landau, E. M. Lifshits, eds. (Pergamon, Oxford, 1981) [Fizicheskaya Kinetika(Nauka, Moscow, 1979)].

Ryshik, I. M.

I. S. Gradshteyn, I. M. Ryshik, Tables of Integrals, Series and Products (Academic, New York, 1965).

Sherman, G. C.

Solimeno, S.

S. Solimeno, B. Crosignani, P. Di Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, New York, 1986).

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, Natl. Bur. Stand. (U.S.) Appl. Math. Ser. 55, (1964).

Watson, G. N.

E. T. Whittaker, G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, Cambridge, 1952).

Whittaker, E. T.

E. T. Whittaker, G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, Cambridge, 1952).

WiInsche, A.

Wünsche, A.

A. Wünsche, “Grenzbedingungen der Elektrodynamik für Medien mit räiumlicher Dispersion und Übergangsschichten,” Ann. Phys. (Leipzig) 37, 121–142 (1980).

Ann. Phys. (Leipzig)

A. Wünsche, “Grenzbedingungen der Elektrodynamik für Medien mit räiumlicher Dispersion und Übergangsschichten,” Ann. Phys. (Leipzig) 37, 121–142 (1980).

Appl. Opt.

Handbook of Mathematical Functions

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, Natl. Bur. Stand. (U.S.) Appl. Math. Ser. 55, (1964).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Phys. Rev. A

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

M. Couture, P. A. Bélanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[CrossRef]

Other

E. T. Whittaker, G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, Cambridge, 1952).

G. M. Fikhtengolts, Course of Differential and Integral Calculus (in Russian) (Fizmatgiz, Moscow, 1959), Vol. II.

I. S. Gradshteyn, I. M. Ryshik, Tables of Integrals, Series and Products (Academic, New York, 1965).

E. M. Lifshits, P. L. Pitayevskyi, Physical Kinetics, Vol. 10 of Course on Theoretical Physics, L. D. Landau, E. M. Lifshits, eds. (Pergamon, Oxford, 1981) [Fizicheskaya Kinetika(Nauka, Moscow, 1979)].

H. Bateman, A. Erdélyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 2.

S. Solimeno, B. Crosignani, P. Di Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, New York, 1986).

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Equations (87)

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L f = 0 ,
M g = 0.
M = L + B .
g = T f .
M g = ( L + B ) T f = ( [ L , T ] + B T ) f = 0 ,
[ L , T ] + B T = 0
[ L , T ] + B T = C L ,
( 2 + 2 z 2 - 1 c 2 2 t 2 ) u ( r , z , t ) = 0
u ( r , z , t ) = exp [ i ( k 0 z - ω 0 t ) ] g ( r , z ) + c . c ,             k 0 = ± ω 0 / c
( 2 + i 2 k 0 z + z 2 ) g ( r , z ) = 0.
u ( 0 ) ( r , z , t ) = exp [ i ( k 0 z - ω 0 t ) ] f ( r , z ) + c . c . ,
( 2 + i 2 k 0 z ) f ( r , z ) = 0 ,
L 2 + i 2 k 0 z ,             B 2 z 2
T 1 m = 0 ( i 2 k 0 ) m 1 m ! z 2 m z 2 m z m - 1 = m = 0 ( i 2 k 0 ) m 1 m ! ( 2 m z 2 m z m - 2 m 2 m - 1 z 2 m - 1 z m - 1 ) = ( 1 - i k 0 z ) T 2
T 2 m = 0 ( i 2 k 0 ) m 1 m ! 2 m z 2 m z m .
T 1 = 1 + z m = 1 ( i 2 k 0 ) m × r = 1 m 2 ( 2 m - 1 ) ! ( r - 1 ) ! ( m - r ) ! ( m + r ) ! z r - 1 m + r z m + r ,
T 2 = m = 1 ( i 2 k 0 ) m r = 0 m ( 2 m ) ! r ! ( m - r ) ! ( m + r ) ! z r m + r z m + r .
T = λ 1 T 1 + λ 2 T 2
u ( r , z , t ) = exp [ i ( k 0 z - ω 0 t ) ] T f ( r , z ) + c . c .
z u ( r , z , t ) = exp [ i ( k 0 z - ω 0 t ) ] ( i k 0 + z ) T f ( r , z ) + c . c .
( i k 0 + z ) T 1 = i k 0 + m = 0 ( i 2 k 0 ) m × r = 0 m + 1 ( 2 m ) ! ( m + 1 - 2 r 2 ) r ! ( m + 1 - r ) ! ( m + 1 + r ) ! z r m + 1 + r z m + 1 + r
( i k 0 + z ) T 2 = i k 0 T 1 = i k 0 - z m = 0 ( i 2 k 0 ) m × r = 1 m + 1 ( 2 m + 1 ) ! ( r - 1 ) ! ( m + 1 - r ) ! ( m + 1 - r ) ! z r - 1 m + 1 + r z m + 1 + r .
u 1 ( r , z , t ) = exp [ i ( k 0 z - ω 0 t ) ] T 1 f ( r , z ) + c . c .
u 1 ( r , 0 , t ) = exp ( - i ω 0 t ) f ( r , 0 ) + c . c . ,
u 1 z ( r , 0 , t ) = exp ( - i ω 0 t ) [ i k 0 f ( r , 0 ) + m = 0 ( i 2 k 0 ) m ( 2 m ) ! m ! ( m + 1 ) ! m + 1 f z m + 1 ( r , 0 ) ] + c . c .
u 1 z ( r , 0 , t ) = exp ( - i ω 0 t ) i k 0 n = 0 ( - 1 ) n + 1 ( 2 n - 3 ) ! ! 2 n n ! × ( 2 k 0 2 ) n f ( r , 0 ) + c . c . = exp ( - i ω 0 t ) i ( k 0 2 + 2 ) 1 / 2 f ( r , 0 ) + c . c .
u 1 ( r , z , t ) = exp ( - i ω 0 t ) exp [ i ( k 0 2 + 2 ) 1 / 2 z ] f ( r , 0 ) + c . c .
u 2 ( r , z , t ) = exp [ i ( k 0 z - ω 0 t ) ] T 2 f ( r , z ) + c . c .
u 2 ( r , 0 , t ) = exp ( - i ω 0 t ) m = 0 ( i 2 k 0 ) m ( 2 m ) ! m ! m ! m f z m ( r , 0 ) + c . c . ,
u 2 z ( r , 0 , t ) = exp ( - i ω 0 t ) i k 0 f ( r , 0 ) + c . c .
u 2 ( r , 0 , t ) = exp ( - i ω 0 t ) m = 0 ( - 1 ) m ( 2 m - 1 ) ! ! 2 m m ! × ( 2 k 0 2 ) m f ( r , 0 ) + c . c . = exp ( - i ω 0 t ) k 0 ( k 0 2 + 2 ) - 1 / 2 f ( r , 0 ) + c . c .
u 2 ( r , z , t ) = exp ( - i ω 0 t ) k 0 exp [ i ( k 0 2 + 2 ) 1 / 2 z ] ( k 0 2 + 2 ) 1 / 2 f ( r , 0 ) + c . c .
g z ( r , z ) = i k 0 [ + ( 1 + 2 k 0 2 ) 1 / 2 - 1 ] g ( r , z ) = i 2 2 k 0 ( 1 - 2 4 k 0 2 + ) g ( r , z ) ,
f z ( r , z ) = i 2 2 k 0 f ( r , z ) .
[ z + i k 0 + i k 0 ( 1 + 2 k 0 2 ) 1 / 2 ] × [ z + i k 0 - i k 0 ( 1 + 2 k 0 2 ) 1 / 2 ] g ( r , z ) = 0.
g 1 ( r , 0 ) = f ( r , 0 ) ,
g 2 ( r , 0 ) + 1 i k 0 g 2 z ( r , 0 ) = f ( r , 0 ) .
f ( r , z ) = A 0 ( k 0 w 0 2 k 0 w 0 2 + i 2 z ) ( n - 1 ) / 2 exp ( - k 0 r 2 k 0 w 0 2 + i 2 z ) .
f 0 1 k 0 w 0 ,             l 0 1 2 k 0 w 0 2 = w 0 2 f 0 = 1 2 f 0 2 k 0 .
v l 0 l 0 + i z ,             z i l 0 v - 1 v ,             z - i 1 l 0 v 2 v .
f ( r , i l 0 v - 1 v ) = A 0 v ( n - 1 ) / 2 exp ( - r 2 w 0 2 v ) .
g 1 ( r , i l 0 v - 1 v ) = T 1 f ( r , i l 0 v - 1 v ) = f ( r , i l 0 v - 1 v ) ( 1 + ( v - 1 ) × m = 1 1 ( 2 k 0 l 0 v ) m m ! { v - 1 + [ m - ( n - 1 ) / 2 ] × [ v 2 ( v - r 2 w 0 2 ) ] 2 m v 1 - [ m - ( n - 1 ) / 2 ] } × ( v - 1 ) m - 1 ) .
g 1 ( r , z ) = f ( r , z ) { 1 - i z l 0 m = 1 ( f 0 l 0 l 0 + i z ) 2 m ( 2 m ) ! m ! × k = 0 m - 1 ( m - 1 ) ! k ! ( m - 1 - k ) ! ( - i z l 0 ) k L m + 1 + k - [ ( 3 - n ) / 2 + k ] ( r 2 w 0 2 l 0 l 0 + i z ) } .
g 2 ( r , z ) = f ( r , z ) m = 0 ( f 0 l 0 l 0 + i z ) 2 m ( 2 m ) ! m ! × k = 0 m m ! k ! ( m - k ) ! ( - i z l 0 ) k L m + k - [ ( 3 - n ) / 2 + k ] ( r 2 w 0 2 l 0 l 0 + i z ) .
L m + 1 + k - [ ( 3 - n ) / 2 + k ] ( u ) = j = 0 m + 1 + k ( - 1 ) j [ m + ( n - 1 ) / 2 ] ! j ! ( m + 1 + k - j ) ! [ j - ( 3 - n ) / 2 - k ] ! u j , L m + k - [ ( 3 - n ) / 2 + k ] ( u ) = j = 0 m + k ( - 1 ) j [ m - ( 3 - n ) / 2 ] ! j ! ( m + k - j ) ! [ j - ( 3 - n ) / 2 - k ] ! u j .
g 1 ( r , z ) = f ( r , z ) [ 1 - i z l 0 ( f 0 l 0 l 0 + i z ) 2 × 2 L 2 0 ( r 2 w 0 2 l 0 l 0 + i z ) ] , L 2 0 ( u ) 1 - 2 u + ½ u 2 L 2 ( u ) .
g 1 ( r , z ) = f ( r , z ) [ 1 - i z l 0 ( f 0 l 0 l 0 + i z ) 2 × 2 L 2 - 1 / 2 ( r 2 w 0 2 l 0 l 0 + i z ) ] , L 2 - 1 / 2 ( u ) 3 8 - 3 2 u + 1 2 u 2 1 32 H 4 ( u 1 / 2 ) ,
g ( 0 , z ) = T f ( 0 , z ) .
f ( 0 , z ) = A 0 ( l 0 l 0 + i z ) ( n - 1 ) / 2 .
g 1 ( 0 , z ) = f ( 0 , z ) { 1 - i z l 0 [ m = 0 ( 2 m ) ! m ! ( f 0 l 0 l 0 + i z ) 2 m - 1 ] } ,
g 2 ( 0 , z ) = f ( 0 , z ) m = 0 ( 2 m ) ! m ! ( f 0 l 0 l 0 + i z ) 2 m .
φ ( ζ ) m = 0 ( 2 m ) ! m ! 1 ( 2 ζ ) 2 m = m = 0 ( 2 m - 1 ) ! ! 2 m 1 ζ 2 m
ζ ξ + i η ,             ξ = 1 2 f 0 = k 0 w 0 2 ,             η z 2 f 0 l 0 = z w 0 .
φ ( ζ ) = i π 1 / 2 ζ exp ( - ζ 2 ) ϕ ( - i ζ ) = 2 ζ F ( ζ ) ,
F ( ζ ) exp ( - ζ 2 ) 0 ζ d τ exp ( + τ 2 ) .
φ ( ζ ) = 2 ζ 2 exp ( - ζ 2 ) k = 0 ζ 2 k k ! ( 2 k + 1 ) = 2 ζ 2 l = 0 ( - 1 ) l 2 l ( 2 l + 1 ) ! ! ζ 2 l .
g 1 ( 0 , z ) = f ( 0 , z ) [ 1 + i 2 f 0 2 k 0 z × m = 0 ( - 1 ) m 2 m ( 2 m - 1 ) ! ! ( 1 2 f 0 + i f 0 k 0 z ) 2 m ] ,
g 2 ( 0 , z ) = f ( 0 , z ) m = 0 ( - 1 ) m 2 m + 1 ( 2 m + 1 ) ! ! ( 1 2 f 0 + i f 0 k 0 z ) 2 ( m + 1 ) .
f = T g
[ L , T ] - T B = 0 ,
T 1 m = 0 ( - i 2 k 0 ) m 1 m ! z m 2 m z 2 m ,
T 2 m = 0 ( - i 2 k 0 ) m 1 m ! z m - 1 2 m z 2 m z = m = 0 ( - i 2 k 0 ) m 1 m ! ( z m 2 m z 2 m + 2 m z m - 1 2 m - 1 z 2 m - 1 ) = T 1 ( 1 - i k 0 z ) ,
T 1 T 1 = T 1 ( 1 - i k 0 z ) T 2 = T 2 T 2 = 1 ,
T 1 1 = T 1 exp ( - i 2 k 0 z ) = 1 ,             T 1 1 = 1
T 2 1 = T 2 [ - exp ( - i 2 k 0 z ) = 1 ,             T 2 1 = 1.
p z p z q = r = 0 { p , q } p ! q ! r ! ( p - r ) ! ( q - r ) ! z q - r p - r z p - r .
z p q z q = s = 0 { p , q } ( - 1 ) s p ! q ! s ! ( p - s ) ! ( q - s ) ! q - s z q - s z p - s .
p z p z q = p ! z q - p N [ L p q - p ( - z z ) ] = q ! z p - q N [ L q p - q ( - z z ) ] ,
L n α ( u ) r = 0 n ( α + n ) ! r ! ( n - r ) ! ( α + n - r ) ! ( - u ) n - r .
T 1 ( 1 - i k 0 z ) T 2 ,             T 2 A [ exp ( i 2 k 0 z 2 z 2 ) ] ,
T 1 N [ exp ( - i 2 k 0 z 2 z 2 ) ] ,             T 2 T 1 ( 1 - i k 0 z ) .
v - ( 1 + α ) ( v 2 v ) n v 1 + α = v - α ( v v v ) n v α = ( v 1 - α v v 1 + α ) n = j = 0 n n ! ( α + n ) ! j ! ( n - j ) ! ( α + j ) ! v n + j j v j = n ! v n N [ L n α ( - v v ) ] .
( v 1 - α v v 1 + α ) n = v n - α n v n v n + α .
[ v 1 - α ( v - μ ) v 1 + α ] n = j = 0 n n ! j ! L n - j α + j ( μ v ) v n + j j v j .
ζ n δ ( m ) ( ζ ) = ( - 1 ) n m ! ( m - n ) ! δ ( m - n ) ( ζ ) ,
n = 0 a n ζ - 1 - n n = 0 [ a n ( - 1 ) n n ! ( P 1 ζ ) ( n ) + c n δ ( n ) ( ζ ) ] ,
f ( ζ ) = n = 0 f n δ ( n ) ( ζ ) ,             f n = ( - 1 ) n n ! - + d ζ f ( ζ ) ζ n .
1 π 1 / 2 a exp ( - ζ 2 a 2 ) = exp ( a 2 4 2 ζ 2 ) δ ( ζ ) = m = 0 1 m ! ( a 2 ) 2 m δ ( 2 m ) ( ζ ) .
φ ( ζ ) = 1 π 1 / 2 - + d u exp ( - u 2 ) m = 0 u 2 m ζ 2 m .
1 ζ 2 m ( - 1 ) 2 m - 1 ( 2 m - 1 ) ! ( P 1 ζ ) ( 2 m - 1 ) ,             m 1 ,
φ ( ζ ) = ζ 2 π 1 / 2 - + d u exp ( - u 2 ) ( P 1 ζ - u + P 1 ζ + u ) .
φ ( ζ ) = ζ π 1 / 2 - + d u exp ( - u 2 ) 1 i 2 0 + d t { exp [ i ( ζ - u ) t ] - exp [ - i ( ζ - u ) t ] } = ζ i 2 exp ( - ζ 2 ) 0 + d t { exp [ - ( t - i 2 ζ ) 2 4 ] - exp [ - ( t + i 2 ζ ) 2 4 ] } ,
1 π 1 / 2 - + d u exp [ - ( u ± i t 2 ) 2 ] = 1.
φ ( ζ ) = - i ζ exp ( - ζ 2 ) - i ζ + i ζ d v exp ( - v 2 ) = i π 1 / 2 ζ exp ( - ζ 2 ) ϕ ( ± i ζ ) ,
φ ( ξ + i η ) = ξ + i η i η exp ( - ξ 2 ) n = 0 ( i η ) n 1 π 1 / 2 - + d v v n × exp ( 2 ξ v - v 2 ) = ξ + i η i η n = 0 ( i 2 η ) n exp ( - ξ 2 ) n ξ n exp ( + ξ 2 ) .
φ ( ξ + i η ) = ( 1 - i ξ η ) [ m = 0 H 2 m ( i ξ ) 2 2 m η 2 m + m = 0 H 2 m + 1 ( i ξ ) 2 2 m + 1 η 2 m + 1 ] .
φ ( ξ + i η ) = ( 1 + i η ξ ) [ m = 0 ( - 1 ) m H 2 m ( η ) 2 2 m ξ 2 m - i m = 0 ( - 1 ) m H 2 m + 1 ( η ) 2 2 m + 1 ξ 2 m + 1 ] .

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