Abstract

A background-limited optical system is to view an object through a turbulent medium. The threshold systems for detecting and resolving specific details of the object are derived and their probabilities of correct detection and resolution evaluated. The variance of a maximum-likelihood estimate of the position of the object is calculated for large signal-to-noise ratio. Specific numerical results are given for turbulence that produces a gaussian mutual-coherence function in light from a point object. A model of such turbulence, depicting it as concentrated in one or more thin, homogeneous phase screens, is proposed.

© 1969 Optical Society of America

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References

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  1. A bibliography of early work in this field is given by W. C. Hoffman, Proc. Symp. Appl. Math. 16, 117 (1964).
    [Crossref]
  2. J. B. Keller, Proc. Symp. Appl. Math. 16, 145 (1964).
    [Crossref]
  3. R. E. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964).
    [Crossref]
  4. M. J. Beran, J. Opt. Soc. Am. 56, 1475 (1966).
    [Crossref]
  5. D. L. Fried, J. Opt. Soc. Am. 56, 1372 (1966).
    [Crossref]
  6. G. R. Heidbreder, J. Opt. Soc. Am. 57, 1477 (1967).
    [Crossref]
  7. W. P. Brown, J. Opt. Soc. Am. 57, 1539 (1967).
    [Crossref]
  8. C. W. Helstrom, J. Opt. Soc. Am. 59, 164 (1969). Herein referred to as J.
    [Crossref]
  9. E denotes the expected value.
  10. D. M. Chase, J. Opt. Soc. Am. 55, 1559 (1965).
    [Crossref]
  11. S. A. Bowhill, J. Res. Natl. Bur. Std. (U. S.) 65D, 275 (1961).
  12. R. P. Mercier, Proc. Cambridge Phil. Soc. 58, 382 (1962).
    [Crossref]
  13. E. E. Salpeter, Astrophys. J. 147, 433 (1967).
    [Crossref]
  14. W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw–Hill Book Co., New York, 1958), p. 81
  15. D. Middleton, An Introduction to Statistical Communication Theory (McGraw–Hill Book Co., New York, 1960), Chs. 18–22.
  16. C. W. Helstrom, Statistical Theory of Signal Detection (Pergamon Press, Ltd., Oxford, 1968), 2nd ed.
  17. The statistic U as written here differs from that in I by a term C that is independent of the data, but not of the autocovariance of the signal field. In detection it is absorbed into the decision level with which U is compared.
  18. Reference 16, Ch. V. §1, p. 148.
  19. D. Middleton, Trans. IEEE IT-12, 230 (1966).
  20. G. N. Watson, Theory of Bessel Functions (Cambridge Univ. Press, 1944), 2nd ed. p. 395, Eq. (1).
  21. This section was prompted by a reviewer’s remark.
  22. D. L. Fried, J. Opt. Soc. Am. 55, 1427 (1965).
    [Crossref]
  23. Reference 16, Ch. IX, §1, pp. 290–295.
  24. Reference 16, Ch. IX, §2, pp. 295–312.
  25. Reference 16, p. 256.
  26. R. A. Fisher, Phil. Trans. Roy. Soc. (London) 222A, 309 (1922).
  27. Reference 16, p. 258.
  28. Reference 16, p. 265.
  29. Reference 15, p. 1076, Eq. A. 1.31d.

1969 (1)

1967 (3)

1966 (3)

1965 (2)

1964 (3)

A bibliography of early work in this field is given by W. C. Hoffman, Proc. Symp. Appl. Math. 16, 117 (1964).
[Crossref]

J. B. Keller, Proc. Symp. Appl. Math. 16, 145 (1964).
[Crossref]

R. E. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964).
[Crossref]

1962 (1)

R. P. Mercier, Proc. Cambridge Phil. Soc. 58, 382 (1962).
[Crossref]

1961 (1)

S. A. Bowhill, J. Res. Natl. Bur. Std. (U. S.) 65D, 275 (1961).

1922 (1)

R. A. Fisher, Phil. Trans. Roy. Soc. (London) 222A, 309 (1922).

Beran, M. J.

Bowhill, S. A.

S. A. Bowhill, J. Res. Natl. Bur. Std. (U. S.) 65D, 275 (1961).

Brown, W. P.

Chase, D. M.

Davenport, W. B.

W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw–Hill Book Co., New York, 1958), p. 81

Fisher, R. A.

R. A. Fisher, Phil. Trans. Roy. Soc. (London) 222A, 309 (1922).

Fried, D. L.

Heidbreder, G. R.

Helstrom, C. W.

C. W. Helstrom, J. Opt. Soc. Am. 59, 164 (1969). Herein referred to as J.
[Crossref]

C. W. Helstrom, Statistical Theory of Signal Detection (Pergamon Press, Ltd., Oxford, 1968), 2nd ed.

Hoffman, W. C.

A bibliography of early work in this field is given by W. C. Hoffman, Proc. Symp. Appl. Math. 16, 117 (1964).
[Crossref]

Hufnagel, R. E.

Keller, J. B.

J. B. Keller, Proc. Symp. Appl. Math. 16, 145 (1964).
[Crossref]

Mercier, R. P.

R. P. Mercier, Proc. Cambridge Phil. Soc. 58, 382 (1962).
[Crossref]

Middleton, D.

D. Middleton, Trans. IEEE IT-12, 230 (1966).

D. Middleton, An Introduction to Statistical Communication Theory (McGraw–Hill Book Co., New York, 1960), Chs. 18–22.

Root, W. L.

W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw–Hill Book Co., New York, 1958), p. 81

Salpeter, E. E.

E. E. Salpeter, Astrophys. J. 147, 433 (1967).
[Crossref]

Stanley, N. R.

Watson, G. N.

G. N. Watson, Theory of Bessel Functions (Cambridge Univ. Press, 1944), 2nd ed. p. 395, Eq. (1).

Astrophys. J. (1)

E. E. Salpeter, Astrophys. J. 147, 433 (1967).
[Crossref]

J. Opt. Soc. Am. (8)

J. Res. Natl. Bur. Std. (U. S.) (1)

S. A. Bowhill, J. Res. Natl. Bur. Std. (U. S.) 65D, 275 (1961).

Phil. Trans. Roy. Soc. (London) (1)

R. A. Fisher, Phil. Trans. Roy. Soc. (London) 222A, 309 (1922).

Proc. Cambridge Phil. Soc. (1)

R. P. Mercier, Proc. Cambridge Phil. Soc. 58, 382 (1962).
[Crossref]

Proc. Symp. Appl. Math. (2)

A bibliography of early work in this field is given by W. C. Hoffman, Proc. Symp. Appl. Math. 16, 117 (1964).
[Crossref]

J. B. Keller, Proc. Symp. Appl. Math. 16, 145 (1964).
[Crossref]

Trans. IEEE (1)

D. Middleton, Trans. IEEE IT-12, 230 (1966).

Other (14)

G. N. Watson, Theory of Bessel Functions (Cambridge Univ. Press, 1944), 2nd ed. p. 395, Eq. (1).

This section was prompted by a reviewer’s remark.

Reference 16, p. 258.

Reference 16, p. 265.

Reference 15, p. 1076, Eq. A. 1.31d.

E denotes the expected value.

Reference 16, Ch. IX, §1, pp. 290–295.

Reference 16, Ch. IX, §2, pp. 295–312.

Reference 16, p. 256.

W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw–Hill Book Co., New York, 1958), p. 81

D. Middleton, An Introduction to Statistical Communication Theory (McGraw–Hill Book Co., New York, 1960), Chs. 18–22.

C. W. Helstrom, Statistical Theory of Signal Detection (Pergamon Press, Ltd., Oxford, 1968), 2nd ed.

The statistic U as written here differs from that in I by a term C that is independent of the data, but not of the autocovariance of the signal field. In detection it is absorbed into the decision level with which U is compared.

Reference 16, Ch. V. §1, p. 148.

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Figures (10)

Fig. 1
Fig. 1

Light propagates from object plane O through the turbulence, indicated by cross-hatching, to the aperture A. I is an instrument to process the electromagnetic field ψa(r,t) at plane A.

Fig. 2
Fig. 2

Transmission through a single phase screen P.

Fig. 3
Fig. 3

Transmission through several phase screens P1, P2, ⋯, Pn.

Fig. 4
Fig. 4

Spatial factor F for detection of uniform circular object of radius b; observation over a circular aperture of radius a; α = kab/R, γ = a/L.

Fig. 5
Fig. 5

Spatial factor F for detection of a point object; observation over a circular aperture of radius a; γ = a/L.

Fig. 6
Fig. 6

Ratio Do/De of effective signal-to-noise ratios for detection in the presence and in the absence of turbulence; detection of uniform circular object of radius b by observation over an infinite aperture; ρ = R/kb.

Fig. 7
Fig. 7

Ratio D/D0 of effective signal-to-noise ratios for resolution and detection of two point objects separated by ; observation over a circular aperture of radius a; γ = a/L.

Fig. 8
Fig. 8

Ratio D/D0 of effective signal-to-noise ratios for resolution and detection of two uniform circular objects of radius b separated by ; observation over an infinite aperture; ρ = R/kb.

Fig. 9
Fig. 9

Spatial factor ℋ for variance of estimate of position of uniform circular object of radius b by observation over an infinite aperture; μ = R/kL, ρ = R/kb.

Fig. 10
Fig. 10

Spatial factor ℋ′ for variance of estimate of position of a point object by observation over an aperture of radius a; γ = a/L.

Equations (74)

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1 2 E [ ψ b ( r 1 , R ; t 1 ) ψ b * ( r 2 , R ; t 2 ) ] = φ b ( r 1 , R ; r 2 , R ; t 1 , t 2 ) = N δ ( r 1 - r 2 ) δ ( t 1 - t 2 ) .
φ b ( r 1 , r 2 ; t 1 , t 2 ) = N δ ( r 1 - r 2 ) δ ( t 1 - t 2 ) .
Rl ψ s ( r , z ; t ) e i Ω t ,
1 2 E [ ψ s ( u 1 , R ; t 1 ) ψ s * ( u 2 , R ; t 2 ) ] = φ s ( u 1 , R ; u 2 , R ; t 1 , t 2 ) = π k - 2 B ( u 1 ) δ ( u 1 - u 2 ) χ ( t 1 - t 2 ) ,
O B ( u ) d 2 u / 4 π
ψ s a ( r , t ) = ψ s ( r , 0 ; t ) = O S ( r , u ; t ) ψ s ( u , R ; t ) d 2 u ,
1 2 E [ ψ s a ( r 1 , t 1 ) ψ s a * ( r 2 , t 2 ) ] = φ s ( r 1 , r 2 ; t 1 , t 2 ) = O O S ( r 1 , u 1 ; t 1 ) S * ( r 2 , u 2 ; t 2 ) × φ s ( u 1 , R ; u 2 , R ; t 1 , t 2 ) d 2 u 1 d 2 u 2 = π k - 2 χ ( t 1 - t 2 ) × O S ( r 1 , u ; t 1 ) S * ( r 2 , u ; t 2 ) B ( u ) d 2 u .
φ s ( r 1 , r 2 ; t 1 , t 2 ) = φ s ( r 1 , r 2 ) χ ( t 1 - t 2 ) ,
φ s ( r 1 , r 2 ) = π k - 2 O S ( r 1 , u ; t 1 ) S * ( r 2 , u ; t 2 ) B ( u ) d 2 u .
T ( r 1 , r 2 ; u ) = S ( r 1 , u ; t ) S * ( r 2 , u ; t )
T ( r 1 , r 2 ; u ) = ( k 2 / 4 π 2 R 2 ) p ( r 1 - r 2 ) × exp [ i k 2 R ( r 1 2 - r 2 2 ) - i k R u · ( r 1 - r 2 ) ] , p ( 0 ) = 1.
p ( r ) = exp ( - r 2 / 4 L 2 ) ,
A exp [ - i k R i ( ξ - ξ i ) · r ] p ( r ) d 2 r , ξ i / R i = - u / R ,
φ s ( r 1 , r 2 ) = ( 4 π R 2 ) - 1 p ( r 1 - r 2 ) β ( r 1 - r 2 ) × exp [ i k ( r 1 2 - r 2 2 ) / 2 R ] ,
β ( r ) = O B ( u ) exp ( - i k r · u / R ) d 2 u
S ( r , u ; t ) = ( i k 2 π R ) ( i k 2 π R ) e i k R exp [ i k 2 R v - u 2 + i k 2 R r - v 2 + i θ ( v ; t ) ] d 2 v
θ ( v 1 ) θ ( v 2 ) = Θ ( v 1 - v 2 ) .
exp [ i θ ( v 1 ) - i θ ( v 2 ) ] = exp [ - 1 2 θ ( v 1 ) - θ ( v 2 ) 2 ] = exp [ - 1 2 D ( v 1 - v 2 ) ] ,
D ( v 1 - v 2 ) = 2 [ Θ ( 0 ) - Θ ( v 1 - v 2 ) ]
S ( r 1 , u 1 ) S * ( r 2 , u 2 ) = ( k 2 / 4 π 2 R R ) 2 exp [ i k 2 R ( v 1 - u 1 2 - v 2 - u 2 2 ) + i k 2 R ( r 1 - v 1 2 - r 2 - v 2 2 ) - 1 2 D ( v 1 - v 2 ) ] × d 2 v 1 d 2 v 2 .
v 1 = y + 1 2 ω ,             v 2 = y - 1 2 ω ,
( 4 π 2 R e 2 / k 2 ) δ ( x 1 - x 2 - ω ) ,
x i = ( R u i + R r i ) / R ,             i = 1 , 2 ,             R e = R R / R ,
S ( r 1 , u 1 ) S * ( r 2 , u 2 ) = ( k 2 / 4 π 2 R 2 ) exp [ ( i k / 2 R ) ( u 1 - r 1 2 - u 2 - r 2 2 ) - 1 2 D ( ( R / R ) ( u 1 - u 2 ) + ( R / R ) ( x 1 - x 2 ) ) ] .
S ( r 1 , u 1 ) S * ( r 2 , u 2 ) = ( k 2 / 4 π R 2 ) exp [ i k 2 R ( u 1 - r 1 2 - u 2 - r 2 2 ) - 1 2 j = 1 n D j ( R j R ( u 1 - u 2 ) + R j R ( r 1 - r 2 ) ) ] .
p ( r ) = exp [ - 1 2 j = 1 n D j ( R j r / R ) ] .
D ( v ) = D 1 v 2 + D 2 v 4 + ,
ln Λ { ψ a } = U = 1 2 N - 2 A A d 2 r 1 d 2 r 2 0 T 0 T d t 1 d t 2 × ψ a * ( r 1 , t 1 ) φ s ( r 1 , r 2 ; t 1 , t 2 ) ψ a ( r 2 , t 2 ) - C ,
Λ { ψ a } = e U ,
U θ = 1 2 N - 2 A A d 2 r 1 d 2 r 2 0 T 0 T d t 1 d t 2 × ψ a * ( r 1 , t 1 ) φ s ( r 1 , r 2 ) χ ( t 1 - t 2 ) ψ a ( r 2 , t 2 ) ,
Q 0 = erfc x = ( 2 π ) - 1 2 x exp ( - t 2 / 2 ) d t , Q d = erfc ( x - D 0 ) ,
D 0 2 = [ E ( U θ H 1 ) - E ( U θ H 0 ) ] 2 / Var 0 U θ ,
D 0 = ( E / N ) ( T W ) - 1 2 F ,
F = | A φ s ( r , r ) d 2 r | - 1 × [ A A φ s ( r 1 , r 2 ) 2 d 2 r 1 d 2 r 2 ] 1 2 .
F = A - 1 β ( 0 ) - 1 · | A A p ( r 1 - r 2 ) 2 β ( r 1 - r 2 ) 2 d 2 r 1 d 2 r 2 | 1 2 = A - 1 β ( 0 ) - 1 [ - - I A ( 2 ) ( r ) p ( r ) 2 β ( r ) 2 d 2 r ] 1 2 ,
I A ( v ) = { 1 , v A 0 , v A
I A ( 2 ) ( v ) = - - I A ( r ) I A ( r - v ) d 2 r .
F 2 = 16 π α 2 0 1 [ J 1 ( 2 α q ) ] 2 exp ( - 2 γ 2 q 2 ) × [ cos - 1 q - q ( 1 - q 2 ) 1 2 ] d q / q , α = k a b / R ,             γ = a / L .
F ( γ ) = 2 1 2 γ - 1 { 1 - exp ( - γ 2 ) [ I 0 ( γ 2 ) + I 1 ( γ 2 ) ] } 1 2
F = A - 1 2 β ( 0 ) - 1 [ - - p ( r ) 2 β ( r ) 2 d 2 r ] 1 2 ,
D 0 2 / D e 2 = - - p ( r ) 2 β ( r ) 2 d 2 r × [ - - β ( r ) 2 d 2 r ] - 1 ,
D 0 2 / D e 2 = 2 0 [ J 1 ( u ) ] 2 exp [ - 1 2 ( ρ / L ) 2 u 2 ] d u / u = 1 - exp ( - L 2 / ρ 2 ) [ I 0 ( L 2 / ρ 2 ) + I 1 ( L 2 / ρ 2 ) ] ,
d F / d x = 2 0 u e - x u 2 [ J 1 ( u ) ] 2 d u = x - 1 e - 1 / 2 x I 1 ( 1 / 2 x )
Λ { ψ a } = exp [ Ξ F ( η ) - C ] , F ( η ) = 0 T 0 T d t 1 d t 2 ν * ( t 1 ; η ) χ ( t 1 - t 2 ) ν ( t 2 ; η ) ,
ν ( t ; η ) = A exp ( - i η · r - i k 2 R r 2 ) ψ a ( r , t ) d 2 r ,
Ξ = ( 8 π R 2 N 2 ) - 1 O B ( u ) d 2 u ,
1 2 [ B ( u + 1 2 ɛ ) + B ( u - 1 2 ɛ ) ] ,
φ s ( r 1 , r 2 ) = 1 2 π k - 2 O T ( r 1 , r 2 ; u ) × [ B ( u + 1 2 ɛ ) + B ( u - 1 2 ɛ ) - 2 B ( u ) ] d 2 u .
P c = ζ ( 1 - Q 0 ) + ( 1 - ζ ) Q d , Q 0 = erfc x ,             Q d = erfc ( x - D ) ,
D = ( E / N ) ( T W ) - 1 2 F ( ɛ )
F ( ɛ ) = [ 3 2 G ( 0 ) + 1 2 G ( ɛ ) - 2 G ( 1 2 ɛ ) ] 1 2 ,
G ( ɛ ) = A - 2 β ( 0 ) - 2 - - - exp ( i k ɛ · r / R ) × p ( r ) 2 β ( r ) 2 I A ( 2 ) ( r ) d 2 r .
G ( ɛ ) = 16 π 0 1 q J 0 ( 2 η q ) e - 2 γ 2 q 2 [ cos - 1 q - q ( 1 - q 2 ) 1 2 ] d q = n = 0 Γ ( 3 2 + n ) Γ ( 3 2 ) Γ ( 3 ) Γ ( 3 + n ) ( - 2 γ 2 ) n ( n + 1 ) ! L n ( - η 2 / 2 γ 2 ) ,             η = k a ɛ / R ,             γ = a / L ,
D / D 0 = F ( ɛ ) [ G ( 0 ) ] - 1 2 ,
G ( ɛ ) A - 1 [ O B ( u ) d 2 u ] 2 O P 2 ( u - ɛ ) B 2 ( u ) d 2 u ,
P 2 ( u ) = p ( r ) 2 exp ( - i k u · r / R ) d 2 r
B 2 ( u ) = O B ( v ) B ( v - u ) d 2 v
G ( ɛ ) ( 32 L 2 / A ) exp ( - 2 / 2 μ 2 ) × 0 1 q exp ( - 2 b 2 q 2 / μ 2 ) I 0 ( 2 b q / μ 2 ) × [ cos - 1 q - q ( 1 - q 2 ) 1 2 ] d q = ( 2 π L 2 / A ) exp ( - 2 / 2 μ 2 ) × n = 0 Γ ( 3 2 + n ) Γ ( 3 2 ) Γ ( 3 ) Γ ( 3 + n ) ( - 2 L 2 / ρ 2 ) n ( n + 1 ) ! L n ( 2 / 2 μ 2 ) , μ = R / k L ,             ρ = R / k b ,             b / μ = L / ρ ,             = ɛ .
ln Λ ( ψ a ) U θ ( θ ) - C ( θ ) ,
E [ ln Λ { ψ a } H 0 ] = 0 , C ( θ ) = N - 2 A A d 2 r 1 d 2 r 2 0 T 0 T d t 1 d t 2 × φ s ( r 1 , r 2 ; θ ) χ ( t 1 - t 2 ) φ b ( r 1 , r 2 ; t 1 , t 2 ) = N - 1 A d 2 r 0 T d t χ ( 0 ) φ s ( r , r ; θ ) = E ( θ ) / N ,
Var θ ˆ E { [ U θ ( θ ) - C ( θ ) ] 2 H 0 } × { E [ U θ ( θ ) - C ( θ ) ] H 1 } - 2 ,
H ( θ 1 , θ 2 ) = A A φ s ( r 1 , r 2 ; θ 1 ) φ s ( r 2 , r 1 ; θ 2 ) d 2 r 1 d 2 r 2 ,
Var θ ˆ D 0 - 2 H ( θ , θ ) H 12 ( θ , θ ) / [ H 11 ( θ , θ ) ] 2 ,
H i j ( θ , θ ) = 2 θ i θ j H ( θ 1 , θ 2 ) | θ 1 = θ 2 = θ
H ( θ 1 , θ 2 ) = π 2 k - 4 A A O O B ( u ; θ 1 ) B ( u ; θ 2 ) × T ( r 1 , r 2 ; u 1 ) T ( r 2 , r 1 ; u 2 ) d 2 r 1 d 2 r 2 d 2 u 1 d 2 u 2 = ( 4 π R 2 ) - 2 A A β ( r 1 - r 2 ; θ 1 ) β * ( r 1 - r 2 ; θ 2 ) × p ( r 1 - r 2 ) 2 d 2 r 1 d 2 r 2 ,
Var x = Var y = D 0 - 2 ( 2 R 2 / k 2 ) ( h 0 / h 1 ) , h f = - - r 2 j I A ( 2 ) ( r ) p ( r ) 2 β ( r ) 2 d 2 r , j = 0 , 1 ,
Var x = Var y μ 2 D 0 - 2 - 1 ,
= ρ 2 2 L 2 0 x exp ( - ρ 2 x 2 / 2 L 2 ) [ J 1 ( x ) ] 2 d x × [ 0 exp ( - ρ 2 x 2 / 2 L 2 ) [ J 1 ( x ) ] 2 d x / x ] - 1 = I 1 ( z ) [ e z - I 0 ( z ) - I 1 ( z ) ] - 1 ,             z = L 2 / ρ 2 ,
Var x = Var y D 0 - 2 μ 2 / 2 ,
= [ exp γ 2 - I 0 ( γ 2 ) - 2 I 1 ( γ 2 ) ] / [ exp γ 2 - I 0 ( γ 2 ) - I 1 ( γ 2 ) ] , γ = a / L .
Var x = Var y D 0 - 2 ρ 2 ,             ρ = R / k a .
f ( a , y ) = 16 π 0 1 q e - a q 2 I 0 ( 2 y q ) [ cos - 1 q - q ( 1 - q 2 ) 1 2 ] d q = 16 π r = 0 y 2 τ ( r ! ) 2 0 1 q 2 r + 1 e - a q 2 [ cos - 1 q - q ( 1 - q 2 ) 1 2 ] d q = r = 0 ( - y 2 ) r ( r ! ) 2 d r d a r f ( a , 0 ) ,
f ( a , 0 ) = 16 π 0 1 q e - a q 2 [ cos - 1 q - q ( 1 - q 2 ) 1 2 ] d q = 4 a { 1 - e - a / 2 [ I 0 ( a / 2 ) + I 1 ( a / 2 ) ] } = 4 a [ 1 - F 1 1 ( 1 2 , 2 ; - a ) ] = n = 0 Γ ( 3 2 + n ) Γ ( 3 2 ) Γ ( 3 ) Γ ( 3 + n ) ( - a ) n ( n + 1 ) !
f ( a , y ) = r = 0 ( - y 2 / a ) r ( r ! ) 2 n = r Γ ( 3 2 + n ) Γ ( 3 2 ) Γ ( 3 ) Γ ( 3 + n ) × ( - a ) n ( n + 1 ) ( n - r ) ! = n = 0 Γ ( 3 2 + n ) Γ ( 3 2 ) Γ ( 3 ) Γ ( 3 + n ) ( - a ) n ( n + 1 ) ! r = 0 n ( n r ) ( - y 2 / a ) r r ! = n = 0 Γ ( 3 2 + n ) Γ ( 3 2 ) Γ ( 3 ) Γ ( 3 + n ) ( - a ) n ( n + 1 ) ! L n ( y 2 / a )