Jx 0 ⬘ ( ) =− Jx 1 ( ) Bessel Functions of the Second Kind of Order n

27.7. Jx 0 ⬘ ( ) =− Jx 1 ( ) Bessel Functions of the Second Kind of Order n

27.8. Yx n () = ⎨ ⎪

This is also called Weber’s function or Neumann’s function [also denoted by N n (x)].

BESSEL FUNCTIONS

For n = 0, 1, 2, …, L’ Hospital’s rule yields

2 1 n − 1 ( n k 1 )!

27.9. Yx n () = {ln ( x / 2 ) + γ }() Jx

( x / 2 ) 2 kn −

( x / 2 ) 2 kk n +

(){() − 1 Φ k + Φ ( n + k )}

π ∑ k = 0 kn !( + k )!

where g = .5772156 … is Euler’s constant (see 1.20) and

27.11. Yx 0 () = {ln ( x / 2 ) + γ }() Jx

27.12. Y

− n ()() x =− 1 Yx n ()

n = 012 , , , ...

For any value n ⭌ 0, Jx n ( ) is bounded at x = 0 while Y n (x) is unbounded.

General Solution of Bessel’s Differential Equation

27.13. y = AJ x n () + BJ − n () x

n ≠ 012 , , , .. .

27.14. y = AJ x n () + BY x n ()

all n

27.15. y = AJ x n () + BJ x n ()

dx

∫ n 2 all

xJ x n ()

where A and B are arbitrary constants.

Generating Function for J n (x)

27.16. e xt ( − 1 // t ) 2 Jxt () = n n

n ∑ =−∞

Recurrence Formulas for Bessel Functions

27.17. 2 Jx n n + 1 () = Jx () J

x n − n − 1 () x

27.18. Jx n ′ () = 1 2 { J n − 1 () x − J n + 1 ( )} x

27.19. xJ x ′ n () = xJ n − 1 () x − nJ x n ()

27.20. xJ x ′ n () = nJ x n () − xJ n + 1 () x

BESSEL FUNCTIONS

27.21. { xJx n dx ( )} n xJ n = n −1 () x

The functions Y n (x) satisfy identical relations.

Bessel Functions of Order Equal to Half an Odd Integer In this case the functions are expressible in terms of sines and cosines.

2  cos x ⎞

27.23. J 12 / () x sin

+ sin =π x x

27.26. J − () x =

27.27. J 52 / x =

27.25. J 32 / () x =

− cos x 27.28. J − 52 / x

⎝⎜ x ⎠⎟ ⎭⎪ For further results use the recurrence formula. Results for Y 12 / ( ), xY 32 / ( ), ... x are obtained from 27.8.

Hankel Functions of First and Second Kinds of Order n

27.29. H n () 1 () x = Jx n () + iY x n ()

27.30. H n () 2 () x = Jx n ()

() − iY x n

Bessel’s Modified Differential Equation

27.31. xy 2 ′′ + ′ − xy ( x 2 + ny 2 ) = 0 n ⭌ 0

Solutions of this equation are called modified Bessel functions of order n.

Modified Bessel Functions of the First Kind of Order n

2 n Γ ( n 1 ) ⎨ + ⎩ 22 ( n + 2 + + ) 242 i ( n + 22 )( n + 4 ) + ⋅⋅⋅ ⎬ = ∑ ⎭ k = 0 k !( Γ n ++ k 1 )

27.33. I − n () x n = iJ − n () ix e ni π = /2 J − n () ix

Γ − ⎩ 222 ( − n ) 2422 i ( − n )( 42 − n )

⎭ k = 0 k !( Γ k +− 1 n )

27.34. I − n () x = Ix n () n = 012 , , , ...

BESSEL FUNCTIONS

If n ≠012 , , , ..., then I n (x) and I –n (x) are linearly independent. For n = 0, 1, we have

2 + 24 2 i + 2462468 2 i 2 i + 2 i 2 i 2 i + ⋅⋅⋅

27.37. Ix 0 ′ () = Ix 1 ()

Modified Bessel Functions of the Second Kind of Order n

27.38. Kx ⎪ n () = ⎨ ⎪ lim

For n = 0, 1, 2, …, L’ Hospital’s rule yields

27.39. Kx ( ) ( ) {ln ( n + 1 n =− 1 x / 2 ) + γ )() Ix +

()( 1 n k ∑ − n −− k 1 ))!( x / 2 ) 2 kn −

() 1 n ∞ ( x / 2 ) n + 2 − k +

2 ∑ k = 0 kn !( + k )! Φ k + Φ n + k

where Φ(p) is given by 27.10. For n = 0,

27.40. Kx 0 () =− {ln( x / 2 ) + γ }() Ix

27.41. K − n () x = Kx n ()

n = 012 , , , ...

General Solution of Bessel’s Modified Equation

27.42. y = AI x n () + BI − n () x

n ≠ 012 , , , ...

27.43. y = AI x n () + BK x n ()

all n

27.44. y = AI x n () + BI x n ()

∫ all n

dx

xI n 2 () x

where A and B are arbitrary constants.

Generating Function for I n (x)

27.45. e xt ( + 1 /) t / 2 Ixt () = n

=−∞ n

BESSEL FUNCTIONS

Recurrence Formulas for Modified Bessel Functions

27.46. I n + 1 () x = I n − 1 () x − x Ix n ()

27.52. K n + 1 () x = K n − 1 () x + Kx n x ()

27.47. Ix n ′ () = 1 2 { I n − 1 () x + I n + 1 ( )} x

27.53. Kx ()

1 { K () x K ( )} x n ′ =− 2 n − 1 + n + 1

27.48. xI x ′ n () = xI n − 1 () x − nI x n () 27.54. xK x n ′ () =− xK n − 1 () x − nK x n ()

27.49. xI x n ′ () = xI n + 1 () x + nI x n ()

27.55. xK x n ′ () = nK x n () − xK n + 1 () x

27.50. n { dx xIx n n ( )} = xI n n −1 () x

{ n ( )} () 27.57. { xKx − n ( )} =− xK − n () x

Modified Bessel Functions of Order Equal to Half an Odd Integer In this case the functions are expressible in terms of hyperbolic sines and cosines.

2 cosh x

27.58. I 12 / () x sinh x

27.59. I − 12 / () x = cosh x

27.62. I x =

⎝⎜ x x

⎠⎟ x ⎬

⎨ 2 + 1 sinh x − cosh

I () x ⎛ 3 ⎞

= ⎛ cosh x −

2 sinh x

⎨ 2 + 1 cosh x − sinh π x x ⎝⎜ x ⎠⎟ π x ⎩ ⎝⎜ x

27.63. I − 52 / () x =

⎠⎟ x ⎬

⎭ For further results use the recurrence formula 27.46. Results for K 1/2 (x), K 3/2 (x), … are obtained from

Ber and Bei Functions

The real and imaginary parts of J xe i n ( 3 π / 4 ) are denoted by Ber n (x) and Bei n (x) where

27.64. Ber n () x = ∑

Bei n () x = ∑

27.66. Ber () x =− 1 2! 2 + 4 ! 2 − ⋅⋅⋅

( x / 2 ) 10

27.67. Bei ()( x = x / 2 ) − 3! 2 + 5 ! 2 − ⋅⋅⋅

BESSEL FUNCTIONS

Ker and Kei Functions

The real and imaginary parts of e − ni π / 2 K xe π i n ( / 4 ) are denoted by Ker n (x) and Kei n (x) where

27.68. Ker n () x =− {ln ( x / 2 ) + γ } Ber n () x + 1 4 π Bei n () x

k )}cos

27.69. Kei n () x =− {ln ( x / 2 ) + γ } Bei n () x − 1 4 π Ber n () x

( n k )}sin

22 k = 0 kn !( + k )!

and Φ is given by 27.10. If n = 0,

27.70. Ker () x =− {ln( x / 2 ) + γ } Ber () x + Bei () x +− 1 ( 1 ++ 1 )

27.71. Kei () x =− {ln( x / 2 ) + γ } Bei () x − Ber( ) x + ( x / 2 ) 2 −

Differential Equation For Ber, Bei, Ker, Kei Functions

27.72. xy 2 ′′ + ′ − xy ( ix 2 + ny 2 ) = 0

The general solution of this equation is

27.73. y = { A Ber x n () + i Bei n ( )} x + B { Ker n () x + i Kei n ( )} x

Graphs of Bessel Functions

Fig. 27-1 Fig. 27-2

BESSEL FUNCTIONS

Fig. 27-3 Fig. 27-4

Fig. 27-5 Fig. 27-6

Indefinite Integrals Involving Bessel Functions

27.74. ∫ xJ x dx 0 () = xJ x 1 ()

27.75. x J x dx 2 0 () = xJx 2 1 () + xJ x 0 () − J x dx ∫ () ∫ 0

27.76. x J x dx m () xJx m ()( m 1 ) x m − 1 2 m − ∫ 2 0 = 1 + − Jx 0 ()( − m − 1 ) x JJ x dx ∫ 0 ()

Jx ()

Jx ()

0 ∫ () x ∫

27.77. 0 2 dx = Jx 1 ()

− J x dx

∫ dx

27.79. J x dx 1 () =− Jx 0 ∫ ()

27.80. xJ x dx ∫ () 1 =− xJ x 0 () + J x dx 0 ∫ ()

27.81. x J x dx m ()

xJx m

0 () + m x ∫ J x dx 0 ∫ ()

BESSEL FUNCTIONS

∫ Jx

Jx ()

27.82. 1 dx

=− 1 () + J x dx 0 ∫ ()

27.83. 1 Jx () m dx ∫ 0

Jx 1 () Jx () 1

=− mx m − 1 + m ∫ x m − 1 dx

27.84. xJ n n x dx xJx ∫ n − 1 () = n ()

27.85. xJ − n () x dx

xJx − ∫ n n + 1 =− n ()

27.86. x J x dx m ()

=− xJ n − 1 ()( x + m +− n 1 ) x m − 1 ∫ J ∫ n − 1 () x dx

α n β n ′ α − βα n n ′ n ∫ β

x { J ( xJ )( x )

J ( xJ ) ( x )}

27.87. xJ ( xJ )( x dx )

27.88. xJ n ( α x dx ) = {( J x )} 2 ⎛

21 n α x 2 )} ⎝⎜ α x ⎠⎟

− 22 {{ ( J

The above results also hold if we replace J n (x) by Y n (x) or, more generally, AJ n (x) + BY n (x) where A and B are constants.

Definite Integrals Involving Bessel Functions

27.89. e J bx dx ∫ ()

∞ − ax

27.90. e − ax J bx dx

27.91. cos ax J bx dx ()

27.92. J bx dx n ∫ ()

n >− 1

b ∞ J bx

n 123 ∫ , , , ...

() dx = ,

27.93. n

J b x dx ( ∫ )

27.94. e − ax

1 α J ()() β J ′ α − β J ()( α J ′ )) β

27.95. xJ n ( α xJ )( x dx )

27.96. xJ 2 n ( α x dx ) 1 2 1 2 2 ∫ 2

0 = 2 { ( )} J n ′ α + 2 ( 1 − n / α ){ ( ) J n α }}

27.97. xJ 0 ( α xI )( x dx ) βα 0 0 ′ βα − 0 ′ α 0 ∫ β

1 J ()() I J ()( I ))

BESSEL FUNCTIONS

Integral Representations for Bessel Functions

27.98. Jx 0 () = cos( sin )

π ∫ 0 x θθ d

27.99. Jx n () = cos(

π ∫ 0 n θ − x θθ

sin ) d n

= integer

27.100. Jx n () = n

cos( sin )cos x 2 θ n θθ d ,

n >− 1

27.101. Yx 0 () =−

π u du ∫

cos( cosh )

27.102. Ix 0 () = cosh( sin )

d = x sin ∫ θ θθ 2 ∫ e 0 d θ