Abstract

Phase distortions in optical systems induced by atmospheric turbulence are investigated with the use of Zernike polynomial decompositions. An analytic solution for the variances of the Zernike coefficients is found for the case of Kolmogorov turbulence with a finite outer-scale length. It is shown that the effect of finite outer scale is to attenuate low-order Zernike components, even when the outer-scale length is much larger than the optical aperture. Effects are investigated for constant outer-scale size and for height-dependent outer scales. It is found that seeing effects on large telescopes are dependent more on the magnitude of the outer scale than on the shape of the outer-scale vertical profile.

© 1991 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergammon, New York, 1980).
  2. S. N. Bezdid’ko, “The use of Zernike polynomials in optics,” Sov. J. Opt. Technol. 41, 425–429 (1974).
  3. D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am. 55, 1427–1435 (1965).
    [CrossRef]
  4. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  5. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).
  6. P. H. Hu, J. Stone, T. Stanley, “Application of Zernike polynomials to atmospheric propagation problems,” J. Opt. Soc. Am. A 6, 1595–1608 (1989).
    [CrossRef]
  7. G. M. B. Bouricius, S. F. Clifford, “Experimental study of atmospherically induced phase fluctuations in an optical signal,” J. Opt. Soc. Am. 60, 1484–1489 (1970).
    [CrossRef]
  8. S. F. Clifford, G. M. B. Bouricius, G. R. Ochs, M. H. Ackley, “Phase variations in atmospheric optical propagation,” J. Opt. Soc. Am. 61, 1279–1284 (1971).
    [CrossRef]
  9. A. K. Blackadar, “The vertical distribution of wind and turbulent exchange in a neutral atmosphere,” J. Geophys. Res. 67, 3095–3102 (1962).
    [CrossRef]
  10. D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–67 (1967).
    [CrossRef]
  11. See, for example, M.-H. Ulrich, ed., Proceedings of the ESO Conference on Very Large Telescopes and Their Instrumentation (European Southern Observatory, Garching, 1988);L. Barr, ed., Advanced Technology Optical Telescopes IV, Proc. Soc. Photo-Opt. Instrum. Eng.1236 (1990).
  12. A. Consortini, L. Ronchi, E. Moroder, “Role of the outer scale of turbulence in atmospheric degradation of optical images,”J. Opt. Soc. Am. 63, 1246–1248 (1973).
    [CrossRef]
  13. G. C. Valley, “Long- and short-term Strehl ratios for turbulence with finite inner and outer scales,” Appl. Opt. 18, 984–987 (1979).
    [CrossRef] [PubMed]
  14. R. J. Sasiela, “A unified approach to electromagnetic wave propagation in turbulence and the evaluation of multiparameter integrals,” MIT Lincoln Laboratory Tech. Rep. 807 (Massachusetts Institute of Technology, Cambridge, Mass., 1988).
  15. D. M. Winker, “Zernike decomposition for a Kolmogorov turbulence spectrum with finite outer scale,” inDigest of the International Conference on Optical and Millimeter Wave Propagation and Scattering in the Atmosphere (University of Florence, Florence, 1986).
  16. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2.
  17. Consider the limit of the ratio of successive terms in the series,limm→∞|um+1um|=limm→∞|(a+m)(b+m)(c+m)(d+m)(e+m)z2m+1|.
  18. D. L. Fried, “Limiting resolution looking down through the atmosphere,” J. Opt. Soc. Am. 56, 1380–1384 (1966).
    [CrossRef]
  19. R. E. Hufnagel, “Variations of atmospheric turbulence,” in Optical Propagation through Turbulence (Optical Society of America, Washington, D.C., 1974), pp. WA1-1–WA1-4.
  20. A. E. Gur’yanov, “Astronomical image quality and the vertical distribution of turbulent optical interference in the night atmosphere,” Sov. Astron. 28, 343–350 (1984).
  21. F. F. Forbes, National Optical Astronomy Observatories, 950 North Cherry Avenue, Tucson, Ariz. (personal communication).
  22. D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE. 55, 57–67 (1967).
    [CrossRef]
  23. D. M. Winker, G. A. Ameer, S. L. Brown, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoff, G. A. Tyler, “Characteristics of turbulence measured on a large aperture,” in Optical, Infrared, and Millimeter Wave Propagation Engineering, W. B. Miller, N. S. Kopeika, eds.,Proc. Soc. Photo-Opt. Instrum. Eng.926, 360–366 (1988).
    [CrossRef]
  24. G. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, New York, 1944), Subsec. 13.6.
  25. I. S. Gradshteyn, I. M. Rhyzhik, Tables of Integrals, Series and Products (Academic, New York, 1965).
  26. O. I. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions (Halsted, New York, 1983).

1989

1984

A. E. Gur’yanov, “Astronomical image quality and the vertical distribution of turbulent optical interference in the night atmosphere,” Sov. Astron. 28, 343–350 (1984).

1979

1976

1974

S. N. Bezdid’ko, “The use of Zernike polynomials in optics,” Sov. J. Opt. Technol. 41, 425–429 (1974).

1973

1971

1970

1967

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–67 (1967).
[CrossRef]

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE. 55, 57–67 (1967).
[CrossRef]

1966

1965

1962

A. K. Blackadar, “The vertical distribution of wind and turbulent exchange in a neutral atmosphere,” J. Geophys. Res. 67, 3095–3102 (1962).
[CrossRef]

Ackley, M. H.

Ameer, G. A.

D. M. Winker, G. A. Ameer, S. L. Brown, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoff, G. A. Tyler, “Characteristics of turbulence measured on a large aperture,” in Optical, Infrared, and Millimeter Wave Propagation Engineering, W. B. Miller, N. S. Kopeika, eds.,Proc. Soc. Photo-Opt. Instrum. Eng.926, 360–366 (1988).
[CrossRef]

Bezdid’ko, S. N.

S. N. Bezdid’ko, “The use of Zernike polynomials in optics,” Sov. J. Opt. Technol. 41, 425–429 (1974).

Blackadar, A. K.

A. K. Blackadar, “The vertical distribution of wind and turbulent exchange in a neutral atmosphere,” J. Geophys. Res. 67, 3095–3102 (1962).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergammon, New York, 1980).

Bouricius, G. M. B.

Brown, S. L.

D. M. Winker, G. A. Ameer, S. L. Brown, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoff, G. A. Tyler, “Characteristics of turbulence measured on a large aperture,” in Optical, Infrared, and Millimeter Wave Propagation Engineering, W. B. Miller, N. S. Kopeika, eds.,Proc. Soc. Photo-Opt. Instrum. Eng.926, 360–366 (1988).
[CrossRef]

Clifford, S. F.

Cochran, G. C.

D. M. Winker, G. A. Ameer, S. L. Brown, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoff, G. A. Tyler, “Characteristics of turbulence measured on a large aperture,” in Optical, Infrared, and Millimeter Wave Propagation Engineering, W. B. Miller, N. S. Kopeika, eds.,Proc. Soc. Photo-Opt. Instrum. Eng.926, 360–366 (1988).
[CrossRef]

Consortini, A.

Dueck, R.

D. M. Winker, G. A. Ameer, S. L. Brown, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoff, G. A. Tyler, “Characteristics of turbulence measured on a large aperture,” in Optical, Infrared, and Millimeter Wave Propagation Engineering, W. B. Miller, N. S. Kopeika, eds.,Proc. Soc. Photo-Opt. Instrum. Eng.926, 360–366 (1988).
[CrossRef]

Forbes, F. F.

F. F. Forbes, National Optical Astronomy Observatories, 950 North Cherry Avenue, Tucson, Ariz. (personal communication).

Fried, D. L.

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE. 55, 57–67 (1967).
[CrossRef]

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–67 (1967).
[CrossRef]

D. L. Fried, “Limiting resolution looking down through the atmosphere,” J. Opt. Soc. Am. 56, 1380–1384 (1966).
[CrossRef]

D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am. 55, 1427–1435 (1965).
[CrossRef]

D. M. Winker, G. A. Ameer, S. L. Brown, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoff, G. A. Tyler, “Characteristics of turbulence measured on a large aperture,” in Optical, Infrared, and Millimeter Wave Propagation Engineering, W. B. Miller, N. S. Kopeika, eds.,Proc. Soc. Photo-Opt. Instrum. Eng.926, 360–366 (1988).
[CrossRef]

Gradshteyn, I. S.

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

Gur’yanov, A. E.

A. E. Gur’yanov, “Astronomical image quality and the vertical distribution of turbulent optical interference in the night atmosphere,” Sov. Astron. 28, 343–350 (1984).

Hu, P. H.

Hufnagel, R. E.

R. E. Hufnagel, “Variations of atmospheric turbulence,” in Optical Propagation through Turbulence (Optical Society of America, Washington, D.C., 1974), pp. WA1-1–WA1-4.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2.

Lussier, D. M.

D. M. Winker, G. A. Ameer, S. L. Brown, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoff, G. A. Tyler, “Characteristics of turbulence measured on a large aperture,” in Optical, Infrared, and Millimeter Wave Propagation Engineering, W. B. Miller, N. S. Kopeika, eds.,Proc. Soc. Photo-Opt. Instrum. Eng.926, 360–366 (1988).
[CrossRef]

Marichev, O. I.

O. I. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions (Halsted, New York, 1983).

Moretti, W.

D. M. Winker, G. A. Ameer, S. L. Brown, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoff, G. A. Tyler, “Characteristics of turbulence measured on a large aperture,” in Optical, Infrared, and Millimeter Wave Propagation Engineering, W. B. Miller, N. S. Kopeika, eds.,Proc. Soc. Photo-Opt. Instrum. Eng.926, 360–366 (1988).
[CrossRef]

Moroder, E.

Noll, R. J.

Ochs, G. R.

Rhyzhik, I. M.

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

Roberts, P. H.

D. M. Winker, G. A. Ameer, S. L. Brown, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoff, G. A. Tyler, “Characteristics of turbulence measured on a large aperture,” in Optical, Infrared, and Millimeter Wave Propagation Engineering, W. B. Miller, N. S. Kopeika, eds.,Proc. Soc. Photo-Opt. Instrum. Eng.926, 360–366 (1988).
[CrossRef]

Ronchi, L.

Sasiela, R. J.

R. J. Sasiela, “A unified approach to electromagnetic wave propagation in turbulence and the evaluation of multiparameter integrals,” MIT Lincoln Laboratory Tech. Rep. 807 (Massachusetts Institute of Technology, Cambridge, Mass., 1988).

Stanley, T.

Steinhoff, K. E.

D. M. Winker, G. A. Ameer, S. L. Brown, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoff, G. A. Tyler, “Characteristics of turbulence measured on a large aperture,” in Optical, Infrared, and Millimeter Wave Propagation Engineering, W. B. Miller, N. S. Kopeika, eds.,Proc. Soc. Photo-Opt. Instrum. Eng.926, 360–366 (1988).
[CrossRef]

Stone, J.

Tatarski, V. I.

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

Tyler, G. A.

D. M. Winker, G. A. Ameer, S. L. Brown, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoff, G. A. Tyler, “Characteristics of turbulence measured on a large aperture,” in Optical, Infrared, and Millimeter Wave Propagation Engineering, W. B. Miller, N. S. Kopeika, eds.,Proc. Soc. Photo-Opt. Instrum. Eng.926, 360–366 (1988).
[CrossRef]

Valley, G. C.

Watson, G.

G. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, New York, 1944), Subsec. 13.6.

Winker, D. M.

D. M. Winker, “Zernike decomposition for a Kolmogorov turbulence spectrum with finite outer scale,” inDigest of the International Conference on Optical and Millimeter Wave Propagation and Scattering in the Atmosphere (University of Florence, Florence, 1986).

D. M. Winker, G. A. Ameer, S. L. Brown, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoff, G. A. Tyler, “Characteristics of turbulence measured on a large aperture,” in Optical, Infrared, and Millimeter Wave Propagation Engineering, W. B. Miller, N. S. Kopeika, eds.,Proc. Soc. Photo-Opt. Instrum. Eng.926, 360–366 (1988).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergammon, New York, 1980).

Appl. Opt.

J. Geophys. Res.

A. K. Blackadar, “The vertical distribution of wind and turbulent exchange in a neutral atmosphere,” J. Geophys. Res. 67, 3095–3102 (1962).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Proc. IEEE

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–67 (1967).
[CrossRef]

Proc. IEEE.

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE. 55, 57–67 (1967).
[CrossRef]

Sov. Astron.

A. E. Gur’yanov, “Astronomical image quality and the vertical distribution of turbulent optical interference in the night atmosphere,” Sov. Astron. 28, 343–350 (1984).

Sov. J. Opt. Technol.

S. N. Bezdid’ko, “The use of Zernike polynomials in optics,” Sov. J. Opt. Technol. 41, 425–429 (1974).

Other

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergammon, New York, 1980).

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

See, for example, M.-H. Ulrich, ed., Proceedings of the ESO Conference on Very Large Telescopes and Their Instrumentation (European Southern Observatory, Garching, 1988);L. Barr, ed., Advanced Technology Optical Telescopes IV, Proc. Soc. Photo-Opt. Instrum. Eng.1236 (1990).

R. E. Hufnagel, “Variations of atmospheric turbulence,” in Optical Propagation through Turbulence (Optical Society of America, Washington, D.C., 1974), pp. WA1-1–WA1-4.

R. J. Sasiela, “A unified approach to electromagnetic wave propagation in turbulence and the evaluation of multiparameter integrals,” MIT Lincoln Laboratory Tech. Rep. 807 (Massachusetts Institute of Technology, Cambridge, Mass., 1988).

D. M. Winker, “Zernike decomposition for a Kolmogorov turbulence spectrum with finite outer scale,” inDigest of the International Conference on Optical and Millimeter Wave Propagation and Scattering in the Atmosphere (University of Florence, Florence, 1986).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2.

Consider the limit of the ratio of successive terms in the series,limm→∞|um+1um|=limm→∞|(a+m)(b+m)(c+m)(d+m)(e+m)z2m+1|.

F. F. Forbes, National Optical Astronomy Observatories, 950 North Cherry Avenue, Tucson, Ariz. (personal communication).

D. M. Winker, G. A. Ameer, S. L. Brown, G. C. Cochran, R. Dueck, D. L. Fried, D. M. Lussier, W. Moretti, P. H. Roberts, K. E. Steinhoff, G. A. Tyler, “Characteristics of turbulence measured on a large aperture,” in Optical, Infrared, and Millimeter Wave Propagation Engineering, W. B. Miller, N. S. Kopeika, eds.,Proc. Soc. Photo-Opt. Instrum. Eng.926, 360–366 (1988).
[CrossRef]

G. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, New York, 1944), Subsec. 13.6.

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

O. I. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions (Halsted, New York, 1983).

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Figures (5)

Fig. 1
Fig. 1

Turbulence power spectrum versus spatial wave number k for various values of ko = 1/Lo.

Fig. 2
Fig. 2

Residual phase variance versus Rko (solid curves) compared with residual phase variance for infinite outer scale (dashed lines): a, Z1, piston-removed variance; b, Z3, piston- and tilt-removed variances; c, Z6, residual variance with modes through astigmatism removed.

Fig. 3
Fig. 3

Normalized Zernike variances versus Rko.

Fig. 4
Fig. 4

Normalized Zernike variances for infinite outer scale and for L o = 4 z with various maximum values: a, infinite outer scale; b, Lo profile with no max0imum; c, Lo ≤ 100 m; d, Lo ≤ 10 m.

Fig. 5
Fig. 5

Normalized Zernike variances for infinite outer scale and for Lo =0.4z with various maximum values: a, infinite outer scale; b, Lo profile with no maximum; c, Lo ≤ 100 m; d, Lo ≤ 10 m.

Equations (27)

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Φ n ( k , z ) = 0.00969 C n 2 ( z ) k 11 / 3
Φ n ( k , z ) = 0.00969 C n 2 ( z ) ( k 2 + k o 2 ) 11 / 6 .
Φ ϕ ( k ) = κ 2 0 d z C n 2 ( z ) Φ n ( k ) cos 2 ( k 2 z 2 κ ) ,
ϕ 2 = j = 1 | a j 2 | ,
a j * a j = d k d k Q j * ( k ) Φ ϕ ( k R , k R ) Q j ( k ) ,
Φ ϕ ( k R , k R ) = 0.00969 κ 2 R 5 / 3 0 d z C n 2 ( z ) ( k 2 + R 2 k o 2 ) 11 / 6 × δ ( k k ) cos 2 ( k 2 z 2 κ ) .
Q j ( k ) = μ J μ ( 2 π k ) π k ( 1 ) ( μ 1 ) / 2 ,
Q even j ( k , θ ) Q odd j ( k , θ ) } = μ J μ ( 2 π k ) π k ( 1 ) ( μ 1 m ) / 2 i m 2 { cos m θ sin m θ .
| a j | 2 = 0.00969 κ 2 R 5 / 3 μ 2 π 0 d z C n 2 ( z ) × 0 d k J μ 2 ( 2 π k ) k ( k 2 + R 2 k o 2 ) 11 / 6 cos 2 ( k 2 z 2 κ ) .
0 d k J μ 2 ( 2 π k ) k ( k 2 + R 2 k o 2 ) 11 / 6 .
| a j | 2 = 0.00969 κ 2 R 5 / 3 μ 0 d z C n 2 ( z ) × { ( R k o ) 2 μ 11 / 3 π 2 μ 1 Γ [ ( 11 / 6 ) μ ] μ Γ ( 11 / 6 ) Γ ( μ + 1 ) × F 2 3 [ μ , μ + 1 2 ; μ + 1 , 2 μ + 1 , μ 5 6 ; ( 2 π R k o ) 2 ] + ( 2 π ) 11 / 3 π 3 / 2 Γ [ ( μ ( 11 / 6 ) ] Γ ( 7 / 3 ) Γ ( 17 / 6 ) Γ [ μ + ( 17 / 6 ) ] × F 2 3 [ 11 6 , 7 3 ; 17 3 μ , 17 6 , μ + 17 6 ; ( 2 π R k o ) 2 ] } .
lim k o 0 | a j | 2 = 0.7554 ( D r o ) 5 / 3 μ Γ [ ( μ ( 11 / 6 ) ] Γ [ ( μ + ( 17 / 6 ) ] ,
ϕ 2 = 0 2 π d θ 0 k d k Φ ϕ ( k R ) ,
ϕ 2 = 0.0863 L o 5 / 3 2.91 6.88 κ 2 0 d z C n 2 ( z ) .
Δ J ( R k o ) = ϕ 2 j = 1 J | a j | 2 ,
A j = | a j | 2 [ D 5 / 3 2.91 6.88 κ 2 0 d z C n 2 ( z ) ] 1 , Z J ( R k 0 ) = Δ J ( R k o ) [ D 5 / 3 2.91 6.88 κ 2 0 d z C n 2 ( z ) ] 1 .
A j = | a j | 2 ( D / r o ) 5 / 3 , Z J ( 0 ) = Z Δ J ( 0 ) ( D / r o ) 5 / 3 .
C n 2 ( z ) = 2.15 × 10 14 ( z / 10 ) 2
Φ ϕ ( k ) = κ 2 0 d z exp ( i 2 π k 3 z ) Φ n ( k , k 3 , | z | ) .
f ( x , x 0 ) = 0 d x J μ 2 ( 2 π x ) x ( x 2 + x o 2 ) 11 / 6
J μ ( z ) J ν ( z ) = 2 π 0 π / 2 d θ J μ + ν ( 2 z cos θ ) cos ( μ ν ) θ
J ν ( x ) = 1 2 π i i + i d s Γ ( s ) ( x / 2 ) ν + 2 s Γ ( ν + s + 1 ) .
f ( x , x 0 ) = 1 π 2 i i i d s Γ ( s ) Γ ( 2 μ + s + 1 ) 0 d x ( 2 π x ) 2 μ + 2 s x ( x 2 + x o 2 ) 11 / 6 × 0 π / 2 d θ ( cos θ ) 2 μ + 2 s .
f ( x , x o ) = x o 11 / 3 1 2 π i i i d s ( 2 π x o ) 2 μ + 2 s 2 π Γ ( 11 / 6 ) × Γ ( s ) Γ [ μ + s + ( 1 / 2 ) ] Γ ( μ + s ) Γ [ ( 11 / 6 ) μ s ] Γ ( μ + s + 1 ) Γ ( 2 μ + s + 1 ) .
f ( x , x o ) = 1 2 π Γ ( 11 / 6 ) { x o 2 μ 11 / 3 ( 2 π ) 2 μ m = 0 ( 1 ) m m ! ( 2 π x o ) 2 m Γ [ μ + m + ( 1 / 2 ) ] Γ ( μ + m ) Γ [ ( 11 / 6 ) μ m ] Γ ( μ + m + 1 ) Γ ( 2 μ + m + 1 ) + ( 2 π ) 11 / 3 m = 0 ( 1 ) m m ! ( 2 π x o ) 2 m Γ [ ( 11 / 6 ) + μ m ] Γ [ ( 7 / 3 ) + m ] Γ [ ( 11 / 6 ) + m ] Γ [ ( 17 / 6 ) + m ] Γ [ μ + ( 17 / 6 ) + m ] } .
f ( x , x o ) = π 2 { x o 2 μ 11 / 3 π 2 μ 1 Γ [ ( 11 / 6 ) μ ] μ Γ ( 11 / 6 ) Γ ( μ + 1 ) × 2 F 3 [ μ , μ + 1 2 ; μ + 1 , 2 μ + 1 , μ 5 6 ; ( 2 π x o ) 2 ] + ( 2 π ) 11 / 3 π 3 / 2 Γ [ μ ( 11 / 6 ) ] Γ ( 7 / 3 ) Γ ( 17 / 6 ) Γ [ μ + ( 17 / 6 ) ] × 2 F 3 [ 11 6 , 7 3 ; 17 6 μ , 17 6 , μ + 17 6 ; ( 2 π x o ) 2 ] } .
limm|um+1um|=limm|(a+m)(b+m)(c+m)(d+m)(e+m)z2m+1|.

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