Abstract

Mellin-transform techniques are used to evaluate an integral involving the product of three Bessel functions and a power. This integral appears in a number of results concerning the statistics of propagation through turbulence. The solution takes the form of a generalized hypergeometric function, which is expressible as a series that converges rapidly for many cases of interest that pertain to atmospheric turbulence.

© 1990 Optical Society of America

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References

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  1. P. H. Hu, J. Stone, A. T. Stanley, “Applications of Zernike polynomials to atmospheric propagation problems,” J. Opt. Soc. Am. A 6, 1595–1608 (1989), Eq. (A7).
    [CrossRef]
  2. G. A. Tyler, “Evaluation of an integral involving the product of three Bessel functions,” Rep. TR-931 (The Optical Sciences Company, Placentia, Calif., 1988).
  3. O. I. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions (Halsted, New York, 1982).
  4. R. J. Sasiela, A Unified Approach to Electromagnetic Wave Propagation in Turbulence and the Evaluation of Multiparameter Integrals, Tech. Rep. 807 (MIT Lincoln Laboratory, Lexington, Mass., 1988).
  5. I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, Orlando, Fla., 1980), Eq. (6.574.2).
  6. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Wiley, New York, 1972), Eq. (11.4.16).
  7. Ref. 6, Eq. (6.1.18).
  8. A. Erdelyi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. II, Eq. (19.3.7).

1989 (1)

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Wiley, New York, 1972), Eq. (11.4.16).

Erdelyi, A.

A. Erdelyi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. II, Eq. (19.3.7).

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, Orlando, Fla., 1980), Eq. (6.574.2).

Hu, P. H.

Marichev, O. I.

O. I. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions (Halsted, New York, 1982).

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, Orlando, Fla., 1980), Eq. (6.574.2).

Sasiela, R. J.

R. J. Sasiela, A Unified Approach to Electromagnetic Wave Propagation in Turbulence and the Evaluation of Multiparameter Integrals, Tech. Rep. 807 (MIT Lincoln Laboratory, Lexington, Mass., 1988).

Stanley, A. T.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Wiley, New York, 1972), Eq. (11.4.16).

Stone, J.

Tyler, G. A.

G. A. Tyler, “Evaluation of an integral involving the product of three Bessel functions,” Rep. TR-931 (The Optical Sciences Company, Placentia, Calif., 1988).

J. Opt. Soc. Am. A (1)

Other (7)

G. A. Tyler, “Evaluation of an integral involving the product of three Bessel functions,” Rep. TR-931 (The Optical Sciences Company, Placentia, Calif., 1988).

O. I. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions (Halsted, New York, 1982).

R. J. Sasiela, A Unified Approach to Electromagnetic Wave Propagation in Turbulence and the Evaluation of Multiparameter Integrals, Tech. Rep. 807 (MIT Lincoln Laboratory, Lexington, Mass., 1988).

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, Orlando, Fla., 1980), Eq. (6.574.2).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Wiley, New York, 1972), Eq. (11.4.16).

Ref. 6, Eq. (6.1.18).

A. Erdelyi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. II, Eq. (19.3.7).

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Figures (2)

Fig. 1
Fig. 1

Contour of integration for β/2 ≤ 1. The contour runs from u = ci∞ to u = c + i ∞ along a line that is parallel to the imaginary axis and is closed in the left half-plane by using a semicircle of infinite radius. The squares represent poles that are typical of Eq. (23), the triangles represent poles that are typical of Eq. (24), and the circles represent poles that are typical of Eq. (25). It is assumed that the contour is traversed in the counterclockwise direction.

Fig. 2
Fig. 2

Contour of integration for β/2 ≥ 1. The contour runs from u = c + i ∞ to u = ci ∞ along a line that is parallel to the imaginary axis and is closed in the right half-plane by using a semicircle of infinite radius. The squares represent poles that are typical of Eq. (26). It is assumed that the contour is traversed in the counterclockwise direction.

Equations (41)

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I = 0 d x x α J l ( a x ) J m ( a x ) J n ( 2 b x ) .
I = a α 1 G l m n ( a , β )
G l m n ( α , β ) = 0 d τ τ α J l ( τ ) J m ( τ ) J n ( β τ ) ,
τ = a x ,
β = 2 b / a .
Re ( α ) < 3 / 2 ,
Re ( α + l + m + n + 1 ) > 0 ,
G l m n ( α , β ) = 0 d τ τ α H 1 ( τ ) H 2 ( β τ ) ,
H ( s ) = 0 d τ τ s 1 H ( τ ) .
H ( τ ) = 1 2 π i γ i γ + i d s τ s H ( s ) ,
G l m n ( α , s ) = 0 d β β s 1 G l m n ( α , β ) .
G l m n ( α , s ) = H 1 ( α s + 1 ) H 2 ( s ) .
G l m n ( α , β ) = 1 2 π i γ i γ + i d s β s H 1 ( α s + 1 ) H 2 ( s ) .
H 1 ( τ ) = J l ( τ ) J m ( τ ) ,
H 2 ( τ ) = J n ( τ ) .
H 1 ( α s + 1 ) = Γ ( s α ) Γ ( l + m + α s + 1 2 ) 2 s α Γ ( l + m + 1 + s α 2 ) Γ ( l + m + 1 + s α 2 ) Γ ( l m + 1 + s α 2 ) ,
H 2 ( s ) = 2 s 1 Γ ( n + s ) 2 ) Γ ( n + 2 s ) 2 ) .
G l m n ( α , β ) = 2 α 1 2 π i γ i γ + i d s β s Γ ( α + l + m + 1 s 2 ) Γ ( s + n 2 ) Γ ( s α l + m + 1 2 ) Γ ( s α + l + m + 1 2 ) Γ ( s α + l m + 1 2 ) Γ ( n + 2 s 2 ) .
s = 2 u
c = γ / 2
Γ ( 2 z ) = ( 2 π ) 1 / 2 2 2 z ( 1 / 2 ) Γ ( z ) Γ ( z + ( 1 / 2 ) .
G l m n ( α , β ) = i 4 π 3 / 2 c i c + i d u ( β / 2 ) 2 u Γ ( u α 2 ) Γ ( u α 2 + 1 2 ) Γ ( α + l + m + 1 2 u ) Γ ( u + n 2 ) Γ ( u + α l + m + 1 2 ) Γ ( u + α + l + m + 1 2 ) Γ ( u + α + l m + 1 2 ) Γ ( n + 2 2 u ) .
u = α 2 p ,
u = α 1 2 p ,
u = n 2 p ,
u = α + l + m + 1 2 + p ,
p = 0 , 1 , 2 , .
lim | u | I ( u ) = ( β / 2 ) 2 u u α 3 / 2 ,
I ( u ) = ( β 2 ) 2 u Γ ( u α 2 ) Γ ( u α 2 + 1 2 ) Γ ( α + l + n + 1 2 u ) Γ ( u + u 2 ) Γ ( u + α l + m + 1 2 ) Γ ( u + α + l + m + 1 2 ) Γ ( u + α + l m + 1 2 ) Γ ( u + 2 2 u ) .
G m n ( α , β ) = i 4 π 3 / 2 c 1 d u ( β / 2 ) 2 u Γ ( u α 2 ) Γ ( u α 2 + 1 2 ) Γ ( α + l + m + 1 2 u ) Γ ( u + n 2 ) Γ ( u + α l + m + 1 2 ) Γ ( u + α + l + m + 1 2 ) Γ ( u + α + l m + 1 2 ) Γ ( n + 2 2 u ) .
C 1 d z G ( z ) Γ ( z ) = 2 π i p = 0 ( 1 ) p p ! G ( p ) ,
G l m n ( α , β ) = 1 2 π ½ [ p = 0 ( 1 ) p p ! ( β 2 ) α + 2 p Γ ( 1 2 p ) Γ ( p + l + m + 1 2 ) Γ ( α + n 2 p ) Γ ( l + m + 1 2 p ) Γ ( l + m + 1 2 p ) Γ ( l m + 1 2 p ) Γ ( p + α + n + 2 2 ) + p = 0 ( 1 ) p p ! ( β 2 ) α + 2 p + 1 Γ ( 1 2 p ) Γ ( p + l + m + 2 2 ) Γ ( α + n 1 2 p ) Γ ( l + m 2 p ) Γ ( l + m 2 p ) Γ ( l m 2 p ) Γ ( p + α + n + 3 2 ) + p = 0 ( 1 ) 2 p ! ( β 2 ) 2 p + n Γ ( n α 2 p ) Γ ( α n + 1 2 p ) Γ ( p + α + l + m + n + 1 2 ) Γ ( α l + m n + 1 2 p ) Γ ( α + l + m n + 1 2 p ) Γ ( α + l m n + 1 2 p ) Γ ( p + n + 1 ) ] .
p = 0 z p p ! i = 1 N a Γ ( p + a i ) i = 1 N b Γ ( b i p ) i = 1 N c Γ ( p + c i ) i = 1 N d Γ ( d i p ) = i = 1 N a Γ ( a i ) i = 1 N b Γ ( b i ) i = 1 N c Γ ( c i ) i = 1 N d Γ ( d i ) × N a + N d F N c + N b [ a 1 , a 2 , , a N a , d 1 + 1 , d 2 + 1 , , d N d + 1 ; c 1 , c 2 , , c N c , b 1 + 1 , b 2 + 1 , , b N b + 1 ; ( 1 ) N b N d z ] .
Γ ( ½ ) = π 1 / 2 ,
Γ ( ½ ) = 2 π ½ ,
Γ ( z ) Γ ( z + ½ ) = 2 1 2 z π 1 / 2 Γ ( 2 z )
G l m n ( α , β ) = Γ ( α + n 2 ) ( β / 2 ) α 2 Γ ( l + m + 1 2 ) Γ ( l m + 1 2 ) Γ ( α + n + 2 2 ) × F 4 3 ( l + m + 1 2 , l m + 1 2 , l m + 1 2 , l + m + 1 2 ; α + n + 2 2 , 1 2 , α n + 2 2 ; β 2 / 4 ) Γ ( l + m + 2 2 ) Γ ( α + n 1 2 ) ( β / 2 ) α + 1 Γ ( l + m 2 ) Γ ( l + m 2 ) Γ ( l m 2 ) Γ ( α + n + 3 2 ) × F 4 3 ( l + m + 2 2 , l m + 2 2 , l m + 2 2 , l + m + 2 2 ; α + n + 3 2 , 3 2 , α n + 3 2 ; β 2 / 4 ) + 2 α + n Γ ( α n ) Γ ( α + l + m + n + 1 2 ) ( β / 2 ) n Γ ( α l + m n + 1 2 ) Γ ( α + l + m n + 1 2 ) Γ ( α + l m n + 1 2 ) Γ ( n + 1 ) × F 4 3 ( α + l + m + n + 1 2 , α + l m + n + 1 2 , α l m + n + 1 2 , α l + m + n + 1 2 ; n + 1 , n + α + 2 2 , n + α + 1 2 ; β 2 / 4 ) .
G l m n ( α , β ) = i 4 π 3 / 2 C 2 d u ( β / 2 ) 2 u Γ ( u α 2 ) Γ ( u α 2 + 1 2 ) Γ ( α + l + m + 1 2 u ) Γ ( u + n 2 ) Γ ( u + α l + m + 1 2 ) Γ ( u + α + l + m + 1 2 ) Γ ( u + α + l m + 1 2 ) Γ ( n + 2 2 u ) .
C 2 d z G ( z ) Γ ( z ) = 2 π i p = 0 ( 1 ) p p ! G ( p ) ,
G l m n ( α , β ) = 1 2 π 1 / 2 p = 0 ( 1 ) p p ! ( β / 2 ) α l m 1 2 p Γ ( p + l + m + 1 2 ) Γ ( p + l + m + 2 2 ) Γ ( p + α + l + m + n + 1 2 ) Γ ( p + m + 1 ) Γ ( p + l + m + 1 ) Γ ( p + l + 1 ) Γ ( α l m + n + 1 2 p ) .
G l m n ( α , β ) = 2 l m 1 Γ ( α + l + m + n + 1 2 ) ( β / 2 ) α l m 1 Γ ( m + 1 ) Γ ( l + 1 ) Γ ( α l m + n + 1 2 ) × F 4 3 ( l + m + 1 2 , l + m + 2 2 , α + l + m + n + 1 2 , α + l + m n + 1 2 ; m + 1 , l + m + 1 , l + 1 ; 4 / β 2 ) .

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