Abstract

Atmospheric Karhunen–Loève functions are the optimal set of basis functions for modal atmospheric compensation. They are seldom applied in practice on account of their nonanalytical nature. A pseudoanalytical set of these functions is constructed with a least-squares-fitting procedure. To produce an analytical expression for the optical resolution of modal atmospheric compensation, a modified form of structure functions is used and applied to the compensated wave front. This results in analytical residual phase structure functions for Zernike polynomials and pseudoanalytical residual phase structure functions for Karhunen–Loève functions. With these structure functions it is found that the modulation transfer function (MTF) after modal compensation is the product of the telescope MTF and the uncompensated atmospheric MTF under the assumption of isotropic compensating phase. Comparison with an accurate numerical method shows that the approximated analytical method developed is much faster and gives reasonably accurate results, especially for high-order compensations.

© 1995 Optical Society of America

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References

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  1. F. Roddier, “The effect of atmospheric turbulence in optical astronomy,” in Progress in Optics XIX, E. Wolf ed. (North-Holland, Amsterdam, 1981), pp. 281–376.
    [CrossRef]
  2. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980).
    [CrossRef]
  3. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), Sect. 9.2.
  4. G.-m. Dai, “Wavefront simulation for atmospheric turbulence,” in Image Reconstruction and Restoration, T. J. Schulz, D. L. Snyder, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2302, 62–72 (1994).
    [CrossRef]
  5. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
    [CrossRef]
  6. J. Y. Wang, J. K. Markey, “Modal compensation of atmospheric turbulence phase distortion,” J. Opt. Soc. Am. 68, 78–86 (1978).
    [CrossRef]
  7. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), Chap. 1, p. 10.
  8. D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am. 55, 1427–1435 (1965).
    [CrossRef]
  9. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  10. This definition differs fromp Noll’s definition [Eq. (2) in Ref. 9] by a factor of only n+1; however, a difference of the same factor appears in the definition of the triangular functions, so that the two definitions are equivalent.
  11. It can be shown that the complete atmospheric wave-front phase variance is infinite. This infinity is just matched by the infinity of the piston variance, so that the wave-front residual error after piston correction is finite.
  12. D. L. Fried, “Probability of getting a lucky short-exposure image through turbulence,” J. Opt. Soc. Am. 68, 1651–1658 (1978).
    [CrossRef]
  13. N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
    [CrossRef]
  14. J. Y. Wang, “Optical resolution through a turbulent medium with adaptive phase compensation,” J. Opt. Soc. Am. 67, 383–390 (1977).
    [CrossRef]
  15. E. L. O’Neill, Introduction to Statistical Optics (Dover, New York, 1991), Chap. 7, p. 87.
  16. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, New York, 1988), Chap. 2, p. 52.

1990

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

1980

1978

1977

1976

1966

1965

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), Sect. 9.2.

Dai, G.-m.

G.-m. Dai, “Wavefront simulation for atmospheric turbulence,” in Image Reconstruction and Restoration, T. J. Schulz, D. L. Snyder, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2302, 62–72 (1994).
[CrossRef]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, New York, 1988), Chap. 2, p. 52.

Fried, D. L.

Markey, J. K.

Noll, R. J.

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Dover, New York, 1991), Chap. 7, p. 87.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, New York, 1988), Chap. 2, p. 52.

Roddier, F.

F. Roddier, “The effect of atmospheric turbulence in optical astronomy,” in Progress in Optics XIX, E. Wolf ed. (North-Holland, Amsterdam, 1981), pp. 281–376.
[CrossRef]

Roddier, N.

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

Southwell, W. H.

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), Chap. 1, p. 10.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, New York, 1988), Chap. 2, p. 52.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, New York, 1988), Chap. 2, p. 52.

Wang, J. Y.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), Sect. 9.2.

J. Opt. Soc. Am.

Opt. Eng.

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

Other

E. L. O’Neill, Introduction to Statistical Optics (Dover, New York, 1991), Chap. 7, p. 87.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, New York, 1988), Chap. 2, p. 52.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), Sect. 9.2.

G.-m. Dai, “Wavefront simulation for atmospheric turbulence,” in Image Reconstruction and Restoration, T. J. Schulz, D. L. Snyder, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2302, 62–72 (1994).
[CrossRef]

F. Roddier, “The effect of atmospheric turbulence in optical astronomy,” in Progress in Optics XIX, E. Wolf ed. (North-Holland, Amsterdam, 1981), pp. 281–376.
[CrossRef]

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), Chap. 1, p. 10.

This definition differs fromp Noll’s definition [Eq. (2) in Ref. 9] by a factor of only n+1; however, a difference of the same factor appears in the definition of the triangular functions, so that the two definitions are equivalent.

It can be shown that the complete atmospheric wave-front phase variance is infinite. This infinity is just matched by the infinity of the piston variance, so that the wave-front residual error after piston correction is finite.

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Figures (8)

Fig. 1
Fig. 1

Relative gain in terms of the RMS residual phase for modal compensation with K–L functions compared with that with Zernike polynomials.

Fig. 2
Fig. 2

Coordinate system for the modified definition of structure functions. Point O is the center of a unit circle, and point A is the center of a circle with radius u. The two points P(x′, θ′) and Q(x, θ) are considered for structure function calculation. u varies within the interval [0, 1], and u′ varies within [0, 1 − u], and both ξ and ξ′ vary within [0, 2π].

Fig. 3
Fig. 3

Analytical curves for Zernike mode shape functions (solid curves) and cross-mode shape functions (dotted curves): A, d2(u) and d3(u); B, d4(u), 2d5(u), and 2d6(u); C, d7(u) and d8(u); D, d9(u) and d10(u); E, d11(u); F, c2,8(u) and c3,7(u); G, c4,11(u), 2c5,13(u), and 2c6,12(u).

Fig. 4
Fig. 4

Residual phase structure functions. The solid curves refer to K–L compensation, and the dotted curves refer to Zernike compensation. A, D 3(u); B, D 4(u); C, D 6(u).

Fig. 5
Fig. 5

MTF versus normalized spatial frequency u for D/r0 = 6 in the case of near-field approximation for modal Zernike correction. The numbers of corrected modes are indicated. The solid curves refer to WM’s results, and the dotted curves refer to results of this paper.

Fig. 6
Fig. 6

Fried’s resolution for tip/tilt correction, calculated with different numbers of cross-mode shape functions and different methods. A, Fried’s original curve. B, Curves from more accurate methods; solid curve: WM’s curve; dotted curve: only c2,8(u) and c3,7(u) are used in the MTF calculation; dashed curve: c2,8(u), c3,7(u), c2,16(u), and c3,17(u) are used in the MTF calculation. C, curve of overcorrection.

Fig. 7
Fig. 7

Strehl ratio versus D/r0 for different orders of correction in the case of near-field approximation. The numbers of corrected modes are indicated. The solid curves refer to K–L functions, and the dotted curves refer to Zernike polynomials.

Fig. 8
Fig. 8

Fried’s resolution versus D/r0 for different orders of correction in the case of near-field approximation. The numbers of corrected modes are indicated. The solid curves refer to K–L functions, and the dotted curves refer to Zernike polynomials.

Tables (4)

Tables Icon

Table 1 Zernike and K–L Variances and Residual Errors

Tables Icon

Table 2 Angular Dependence of MTF for Different N for Zernike Compensationa

Tables Icon

Table 3 Maximum Errors in Peak Resolutions Resulting from This Paper’s Approach Compared with Those Resulting from WM’s Numerical Approach

Tables Icon

Table 4 Numerically Fitted Coefficients of K–L Radial Polynomials

Equations (107)

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ψ ( R x ) = exp [ l ( R x ) + j φ ( R x ) ] ,
φ ( R x ) = i = 1 a i F i ( x ) ,
1 π W ( x ) F i ( x ) F i ( x ) d 2 x = δ i i ,
W ( x ) = { 1 x 1 0 otherwise .
a i = 1 π W ( x ) F i ( x ) φ ( R x ) d 2 x .
α i = a i ( D / r 0 ) - 5 / 6             ( rad ) .
Z i ( x ) = R n m ( x ) Θ m ( θ ) ,
R n m ( x ) = s = 0 ( n - m ) / 2 ( - 1 ) s n + 1 ( n - s ) ! x n - 2 s s ! [ ( n + m ) / 2 - s ] ! [ ( n - m ) / 2 - s ] !
Θ m ( θ ) = { 2 cos ( m θ ) m 0 even term 2 sin ( m θ ) m 0 odd term 1 m = 0 .
α i α i = c 0 ( - 1 ) ( n + n - 2 m ) / 2 [ ( n + 1 ) ( n + 1 ) ] 1 / 2 δ m m Γ [ ( n - n + 17 / 3 ) / 2 ] Γ ( n - n + 17 / 3 ) / 2 ] Γ [ ( n + n - 5 / 3 ) / 2 ] Γ [ ( n + n + 23 / 3 ) / 2 ] ,
Δ N Z = Δ 1 - i = 2 N a i 2             ( rad 2 ) ,
K i ( x ) = S p q ( x ) Θ q ( θ ) ,
0 1 Y q ( x , x ) S p q ( x ) d x = α i 2 S p q ( x ) ,
Y 2 ( x , x ) = - 0.345 x 0 2 π ( x 2 + x 2 - 2 x x cos β ) 5 / 6 × cos ( 2 β ) d β .
S p q ( x ) = x q s = 0 g s x 2 s ,
S 1 1 ( x ) 0.04385 x 5 - 0.24434 x 3 + 1.55446 x ,
S 1 0 ( x ) 0.03205 x 10 - 0.21918 x 8 + 1.36543 x 6 - 4.12302 x 4 + 6.45447 x 2 - 2.15576.
0 1 S p q ( x ) S p q ( x ) x d x = 1 2 δ p p ,
Δ N K = Δ 1 - i = 2 N a i 2             ( rad 2 ) .
D = UCU T
K l ( x , θ ) = i = 2 U i l Z i ( x , θ ) ,
S p q ( x ) = i = 2 U i l R n m ( x ) ,
S 1 1 ( x ) R 1 1 ( x ) - 0.032 R 3 1 ( x ) + 0.002 R 5 1 ( x ) ,
S 1 0 ( x ) 0.984 R 2 0 ( x ) - 0.178 R 4 0 ( x ) + 0.019 R 6 0 ( x ) .
G ( N ) = Δ N Z - Δ N K Δ N Z ,
D F ( x - x ) = F ( x ) - F ( x ) 2 ,
x 2 = u 2 + u 2 + 2 u u cos ξ ,
x 2 = u 2 + u 2 - 2 u u cos ξ ,
x cos θ = u cos ξ + u cos ( ξ + ξ ) ,
x cos θ = u cos ξ - u cos ( ξ + ξ ) ,
x sin θ = u sin ξ + u sin ( ξ + ξ ) ,
x sin θ = u sin ξ - u sin ( ξ + ξ ) .
D F ( u ) = w ( u ) [ F ( x , θ ) - F ( x , θ ) ] 2 d 3 u ,
w ( u ) = 1 2 π 2 ( 1 - u ) 2
D ( u ) = D ϕ ( u ) + D l ( u ) = 6.88 u 5 / 3 ,
D ϕ ( u ) = ( D / r 0 ) - 5 / 3 w ( u ) { i = 2 a i [ F i ( x ) - F i ( x ) ] } 2 d 3 u = i = 2 α i 2 w ( u ) [ F i ( x ) - F i ( x ) ] 2 d 3 u + 2 i = 2 i = i + 1 α i α i w ( u ) [ F i ( x ) - F i ( x ) ] [ F i ( x ) - F i ( x ) ] d 3 u .
d i ( u ) = w ( u ) [ F i ( x ) - F i ( x ) ] 2 d 3 u ,
c i , i ( u ) = w ( u ) [ F i ( x ) - F i ( x ) ] [ F i ( x ) - F i ( x ) ] d 3 u ,
D ϕ ( u ) = i = 2 α i 2 d i ( u ) + 2 i = 2 i = i + 1 α i α i c i , i ( u ) .
ϕ N ( R x ) = i = 2 N a i F i ( x ) ,
ϕ ( R x ) = φ ( R x ) - ϕ N ( R x ) = i = N + 1 a i F i ( x ) .
D N ( u ) = w ( u ) ( D / r 0 ) - 5 / 3 ϕ ( R x ) - ϕ ( R x ) 2 d 3 u .
D N ( u ) = D ϕ ( u ) - i = 2 N α i 2 d i ( u ) - 2 i = 2 N i = i + 1 α i α i c i , i ( u ) .
D N Z ( u ) = D ϕ ( u ) - i = 2 N α i 2 d i ( u ) - 2 i = 2 N i = i + 1 α i α i c i , i ( u ) .
D N K ( u ) = D ϕ ( u ) - i = 2 N α i 2 d i ( u ) .
c i , i ( u ) = 2 w ( u ) s = 0 ( n - m ) s = 0 ( n - m ) / 2 ( - 1 ) s + s ( n - s ) ! ( n - s ) ! b s ! s ! t 1 ! t 2 ! s 1 ! s 2 ! × I 1 ( u ) ,
I 1 ( u ) = [ ( x ) n - 2 s cos ( m θ - x n - 2 s cos ( m θ ) ] × [ ( x ) n - 2 s cos ( m θ ) - x n - 2 s cos ( m θ ) ] d 3 u .
c i , i ( u ) = ϒ A B C u 2 τ + ρ - 2 v ( 1 - u ) μ + 2 v - h - h ,
ϒ = ( l + h + t + 1 ) ( l + h + t + 1 ) × ( h + h + v ) ( η ) ,
A = ( - 1 ) s + s + k + k 2 t + t + 4 b μ + 2 v - h - h + 2 ,
B = ( n - s ) ! ( n - s ) ! ( m ! ) 2 τ ! s 1 ! s 2 ! ( t 1 - t ) ! ( t 2 - t ) ! s ! s ! E F ,
C = ( μ - 2 k - 2 k - v - 1 ) ! ! ( σ - 1 ) ! ! ( μ - h - h ) ! ! ρ ! ! × ( h + h + v - 1 ) ! ! ( η - 2 v - 1 ) ! ! ,
E = ( m - 2 k - l ) ! ( m - 2 k - l ) ! ( 2 k - h ) ! × ( 2 k - h ) ! ( t + t - v ) ! ( τ - v ) ! ,
F = t ! t ! l ! l ! h ! h ! v ! v ! ,
ρ = l + l + h + h + t + t ,
σ = 2 k + 2 k + v - h - h ,
η = l + l + t + t + v ,
μ = 2 m + t + t - l - l ,
( n ) = { 1 even n 0 otherwise .
d 2 ( u ) = 8 u 2 ,
d 4 ( u ) = 48 u 2 ( 1 - u ) 2 ,
d 8 ( u ) = 16 u 2 ( 42 u 4 - 96 u 3 + 84 u 2 - 36 u + 7 ) ,
d 10 ( u ) = 16 u 2 ( 4 u 4 - 12 u 3 + 18 u 2 - 12 u + 3 ) ,
d 11 ( u ) = 240 u 2 ( 1 - u ) 2 ( 34 u 4 - 56 u 3 + 32 u 2 - 8 u + 1 ) ,
c 2 , 8 ( u ) = 4 8 u 2 ( 6 u 2 - 6 u + 1 ) ,
c 4 , 11 ( u ) = 16 15 u 2 ( 1 - u ) 2 ( 10 u 2 - 8 u + 1 ) .
S p q ( x ) x q s = 0 M g s x 2 s .
d i ( u ) = 2 w ( u ) s = 0 M s = 0 M g s g s I 1 ( u ) ,
I 1 ( u ) = [ ( x ) 2 s + q cos ( q θ ) - x 2 s + q cos ( q θ ) ] × [ ( x ) 2 s + q cos ( q θ ) - x 2 s + q cos ( q θ ) ] d 3 u .
d i ( u ) = ϒ A B C u 2 s + 2 s + τ - 2 v ( 1 - u ) 2 q - τ + 2 v ,
A = ( - 1 ) k + k 2 t + t + 4 g s g s 2 q + 2 v - τ + 2 ,
B = s ! s ! ( s + s - t - t ) ! ( t + t ) ! ( q ! ) 2 ( s - t ) ! ( s - t ) ! ( s + s - t - t - v ) ! E F ,
C = ( μ - 2 k - 2 k - v - 1 ) ! ! ( σ - 1 ) ! ! ( μ - h - h ) ! ! ρ ! ! × ( h + h + v - 1 ) ! ! ( η - 2 v - 1 ) ! ! ,
E = ( q - 2 k - l ) ! ( q - 2 k - l ) ! ( 2 k - h ) ! × ( 2 k - h ) ! ( t + t - v ) ! ,
μ = 2 q + t + t - l - l .
τ ( u ) = 1 π ψ ( R x ) ψ * ( R x ) W ( u + u ) W ( u - u ) d 2 u ,
ψ ( R x ) = exp [ l ( R x ) + j φ ( R x ) - j ϕ N ( R x ) ] .
ψ ( R x ) ψ * ( R x ) = exp { [ l ( R x ) + l ( R x ) ] + j [ ϕ ( R x ) - ϕ ( R x ) ] } = exp { - 1 2 [ D l ( u ) + D N ( u ) ] ( D / r 0 ) 5 / 3 } .
τ ( u ) = τ 0 ( u ) exp { - 1 2 [ D l ( u ) + D N ( u ) ] ( D / r 0 ) 5 / 3 } = τ 0 ( u ) exp { - 1 2 D ( u ) ( D / r 0 ) 5 / 3 } × exp [ 1 2 i = 2 N a i 2 d i ( u ) + i = 2 N i = i + 1 a i a i c i , i ( u ) ] ,
τ 0 ( u ) = 2 π [ cos - 1 ( u ) - u 1 - u 2 ]             ( u 1 ) .
τ ( u , θ ) = 4 π exp [ - 1 2 D ( u ) ( D / r 0 ) 5 / 3 ] × θ π / 2 + θ d θ 0 L ( θ , θ ) u d u × exp [ 1 2 i = 2 N i = 2 N Q i i ( u , u ) + i = 2 N i = N + 1 Q i i ( u , u ) ] ,
L ( θ , θ ) = - u cos ( θ - θ ) + [ 1 - u 2 sin 2 ( θ - θ ) ] 1 / 2 ,
Q i i ( u , u ) = a i a i [ F i ( u + u ) - F i ( u - u ) ] × [ F i ( u + u ) - F i ( u - u ) ] .
D l ( u ) 0             ( near field ) ,
D l ( u ) 1 2 D ( u )             ( far field ) .
τ ( u ) n f = τ 0 ( u ) exp [ - 1 2 D ( u ) ( 1 - u 1 / 3 ) ] ,
τ ( u ) f f = τ 0 ( u ) exp [ - 1 2 D ( u ) ( 1 - 1 2 u 1 / 3 ) ] .
τ ( u ) n f = τ 0 ( u ) exp [ - 1 2 D ( u ) ( 1 - 1.043 u 1 / 3 ) ] ,
τ ( u ) f f = τ 0 ( u ) exp [ - 1 2 D ( u ) ( 1 - 0.522 u 1 / 3 ) ] ,
S = τ ( u ) d 2 u / τ 0 ( u ) d 2 u = 16 π 0 1 u [ cos - 1 ( u ) - u 1 - u 2 ] × exp { - 1 2 [ D l ( u ) + D N ( u ) ] ( D / r 0 ) 5 / 3 } d u .
R = 16 π ( D / r 0 ) 2 0 1 u [ cos - 1 ( u ) - u 1 - u 2 ] × exp { - 1 2 [ D l ( u ) + D N ( u ) ] ( D / r 0 ) 5 / 3 } d u ,
S n f = 16 π 0 1 u [ cos - 1 ( u ) - u 1 - u 2 ] × exp [ - 1 2 D N ( u ) ( D / r 0 ) 5 / 3 ] d u ,
R n f = 16 π ( D / r 0 ) 2 0 1 u [ cos - 1 ( u ) - u 1 - u 2 ] × exp [ - 1 2 D N ( u ) ( D / r 0 ) 5 / 3 ] d u .
S ^ p q ( x ) = x q s = 0 M g s x 2 s .
S ^ p q ( x ) = { x q s = 0 M ( - 1 ) s + p + 1 g s x 2 s ( q 0 ) s = 0 M ( - 1 ) s + p g s x 2 s ( q = 0 ) ,
f = { m / 2 even m ( m - 1 ) / 2 otherwise
cos ( m θ ) = k = 0 f ( - 1 ) k m ! ( cos θ ) m - 2 k ( sin θ ) 2 k ( m - 2 k ) ! ( 2 k ) ! ,
0 2 π sin m θ cos n θ d θ = 2 π ( m ) ( n ) ( m - 1 ) ! ! ( n - 1 ) ! ! ( m + n ) ! ! ,
( a + b ) n = i = 0 n n ! ( n - i ) ! i ! a n - i b i ,
I 1 ( u ) = [ ( x ) n - 2 s cos ( m θ ) - x n - 2 s cos ( m θ ) ] × [ ( x ) n - 2 s cos ( m θ ) - x n - 2 s cos ( m θ ) ] d 3 u = t h ( - 1 ) k + k ϒ 1 ( m ! ) 2 2 t + t + 2 t 1 ! t 2 ! ( t 1 - t ) ! ( t 2 - t ) ! ( m - 2 k - l ) ! ( m - 2 k - l ) ! × u ρ I 2 I 3 ( u ) ( 2 k - h ) ! ( 2 k - h ) ! t ! t ! l ! l ! h ! h ! ,
I 2 = 0 2 π 0 2 π ( sin ξ ) 2 k + 2 k - h - h ( cos ξ ) 2 m - 2 k - 2 k - l - l × [ sin ( ξ + ξ ) ] h + h [ cos ( ξ + ξ ) ] l + l ( cos ξ ) t + t d ξ d ξ = 4 π 2 v = 0 t + t ϒ 2 C ( t + t ) ! ( t + t - v ) ! v ! ,
I 3 ( u ) = 0 1 - u ( u ) μ - h - h + 1 ( u 2 + u 2 ) τ d u = v = 0 τ τ ! u 2 τ - 2 v ( 1 - u ) μ + 2 v - h - h + 2 ( τ - v ) ! v ! ( μ + 2 v - h - h + 2 ) .
I 1 ( u ) = t v ( - 1 ) k + k π 2 ϒ 2 t + t + 4 C τ ! t 1 ! t 2 ! ( m ! ) 2 u 2 τ + ρ - 2 v ( t 1 - t ) ! ( t 2 - t ) ! ( μ + 2 v - h - h + 2 ) E F × ( 1 - u ) μ + 2 v - h - h + 2 ,
I 1 ( u ) = [ ( x ) 2 s + q cos ( q θ ) - x 2 s + q cos ( q θ ) ] × [ ( x ) 2 s + q cos ( q θ ) - x 2 s + q cos ( q θ ) ] d 3 u = t h ( - 1 ) k + k ϒ 1 2 t + t + 2 ( q ! ) 2 s ! s ! ( s - t ) ! ( s - t ) ! ( q - 2 k - l ) ! ( q - 2 k - l ) ! × u ρ I 2 I 3 ( u ) ( 2 k - h ) ! ( 2 k - h ) ! t ! t ! l ! l ! h ! h ! ,
I 2 = 0 2 π 0 2 π ( sin ξ ) 2 k + 2 k - h - h ( cos ξ ) 2 q - 2 k - 2 k - l - l × [ sin ( ξ + ξ ) ] h + h [ cos ( ξ + ξ ) ] l + l ( cos ξ ) t + t d ξ d ξ = 4 π 2 v = 0 t + t ϒ 2 C ( t + t ) ! ( t + t - v ) ! v ! ,
I 3 ( u ) = 0 1 - u ( u ) μ - h - h ( u 2 + u 2 ) s + s - t - t u d u = v = 0 s + s - t - t ( s + s - t - t ) ! u 2 s + 2 s + τ - 2 v ( 1 - u ) 2 q + 2 v - τ + 2 ( s + s - t - t - v ) ! v ! ( 2 q + 2 v - τ + 2 ) .
I 1 ( u ) = t v ( - 1 ) k + k π 2 ϒ 2 t + t + 4 B C 2 q + 2 v - τ + 2 × u 2 s + 2 s + τ - 2 v ( 1 - u ) 2 q + 2 v - τ + 2 ,

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