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chpp17 - 17 Helmholtz 2 u k2 u = 0 1 2u 2u 2 k2 u = 0 r2 2...

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17 Helmholtz 2 u + k 2 u = 0 (1) ( r, θ, z ) 1 r ∂r r ∂u ∂r + 1 r 2 2 u ∂θ 2 + 2 u ∂z 2 + k 2 u = 0 (2) u ( r, θ, z ) = R ( r )Θ( θ ) Z ( z ) , 1 r d d r r d R d r + k 2 - λ - μ r 2 R = 0 (3) d 2 Θ d θ 2 + μ Θ = 0 (4) d 2 Z d z 2 + λZ = 0 (5) μ , λ . , k 2 - λ = 0. x = k 2 - λr y ( x ) = R ( r ) μ = ν 2 , 1 x d d x x d y d x + 1 - ν 2 x 2 y ( x ) = 0 (6) ν Bessel . , : 1. ν = , Bessel J ± ν ( x ) = k =0 ( - 1) k k !Γ( k ± ν + 1) x 2 2 k ± ν (7) ± ν Bessel . 2. ν = , J n ( x ) Bessel , J - n ( x ) = ( - 1) n J n ( x ) , Neumann , . 17.1 Bessel Bessel . 1. Bessel J - n ( x ) = ( - 1) n J n ( x ) (8) 2. J n ( - x ) = ( - 1) n J n ( x ) (9) 1

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3. , exp x 2 t - 1 t = n = -∞ J n ( x ) t n (10) Bessel . 4. t = e i θ e i x sin θ = n = -∞ J n ( x )e i (11) J n ( x ) = 1 2 π π - π e i x sin θ ( e i ) * d θ = 1 2 π π - π [cos( x sin θ - ) +i sin( x sin θ - )] d θ = 1 2 π π - π cos( x sin θ - )d θ (12) J n ( x ) . 5. t = ie i θ e i x cos θ = n = -∞ J n ( x )i n e i = J 0 ( x ) + n =1 i n J n ( x )e i + i - n J - n ( x )e - i = J 0 ( x ) + n =1 i n J n ( x )e i + i - n ( - 1) n J n ( x )e - i = J 0 ( x ) + 2 n =1 i n J n ( x ) cos (13) 2 u ∂t 2 - v 2 2 u = 0 ( ω 2 = v 2 ( k 2 x + k 2 y + k 2 z )) u ( r, t ) = u ( x, y, z, t ) = X ( x ) Y ( y ) Z ( z ) T ( t ) = e i k x x e i k y y e i k z z e - i ωt = e i( k · r - ωt ) ( ω 2 = v 2 ( k 2 r + k 2 z )) u ( r, t ) = u ( r, θ, z, t ) = R ( r )Θ( θ ) Z ( z ) T ( t ) = J n ( k r r )e i e i k z z e - i ωt 2
x z e i( k · r - ωt ) = e i( k r r cos θ - ωt ) e i( k r r cos θ - ωt ) = n = -∞ i n J n ( k r r )e i e - i ωt = J 0 ( k r r )e - i ωt + 2 n =1 i n J n ( k r r ) cos e - i ωt Bessel 6. Bessel J ν ( x ) J - ν ( x ) d d x [ x ν J ν ( x )] = x ν J ν - 1 ( x ) (14) d d x x - ν J ν ( x ) = - x - ν J ν +1 ( x ) (15) Bessel d d x [ x ν J ν ( x )] = d d x k =0 ( - 1) k k !Γ( k + ν + 1) x 2 k +2 ν 2 2 k + ν = k =0 ( - 1) k k !Γ( k + ν ) x 2 k +2 ν - 1 2 2 k + ν - 1 = x ν J ν - 1 ( x ) d d x x - ν J ν ( x ) = d d x k =0 ( - 1) k k !Γ( k + ν + 1) x 2 k 2 2 k + ν = k =1 ( - 1) k ( k - 1)!Γ( k + ν + 1) x 2 k - 1 2 2 k + ν - 1 = k =0 ( - 1) k +1 k !Γ( k + ν + 2) x 2 k +1 2 2 k + ν +1 = - x - ν J ν +1 ( x ) J ν ( x ) J ν ( x ), J ν - 1 ( x ) - J ν +1 ( x ) = 2 J ν ( x ) (16) J ν - 1 ( x ) + J ν +1 ( x ) = 2 ν x J ν ( x ) (17) (17), Bessel , J 0 ( x ), J 1 ( x ), J 2 ( x ), J 3 ( x ), ... J n ( x ) J 0 ( x ) J 1 ( x ) . (16), J n ( x ) J 0 ( x ) J 1 ( x ) . 3

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7. Bessel x 0 J ν ( x ) = 1 Γ( ν + 1) x 2 ν + O ( x ν +2 ) (18) x → ∞ ( ) J ν ( x ) 2 πx cos x - νπ 2 - π 4 | arg x | < π (19) 8. Bessel J ν ( x ) J - ν ( x ) Wronski W [ J ν ( x ) , J - ν ( x )] = J ν ( x ) J - ν ( x ) J ν ( x ) J - ν ( x ) J ν Bessel, W [ J ν ( x ) , J - ν ( x )] = A exp - z p ( ξ )d ξ Bessel , p ( x ) = 1 x , W [ J ν ( x ) , J - ν ( x )] = A exp[ - ln x + B ] = C x x 0, W [ J ν ( x ) , J - ν ( x )] = 1 Γ( ν +1) ( x 2 ) ν 1 Γ(1 - ν ) ( x 2 ) - ν 1 Γ( ν ) ( x 2 ) ν - 1 1 2 1 Γ( - ν ) ( x 2 ) - ν - 1 1 2 = 1 2 x 2 - 1 1 Γ( ν + 1)Γ( - ν ) - 1 Γ( ν )Γ(1 - ν ) = - 2 x 1 Γ( ν )Γ(1 - ν ) = - 2 x sin πν π , C = - 2 π sin πν . x → ∞ , | arg x | < π , W [ J ν ( x ) , J - ν ( x )] = 2 πx cos ( x - νπ 2 - π 4 ) 2 πx cos ( x - νπ 2 + π 4 ) - 2 πx sin ( x - νπ 2 - π 4 ) - 2 πx sin ( x + νπ 2 - π 4 ) = - 2 πx sin πν 9. ( ν ) Bessel ν > - 1 , (a) J ν , (b) , (c) . 4
(a) x → ∞ J ν ( x ) 2 πx cos x - νπ 2 - π 4 . Bessel , x = n + 1 2 π + νπ 2 + π 4 . (b) ν > - 1 ( a 2 - b 2 ) x 0 tJ ν ( at ) J ν ( bt ) dt = x J ν ( ax ) d J ν ( bx ) d x - J ν ( bx ) d J ν ( ax ) d x (20) 1 t d d t t d J ν ( at ) d t + a 2 - ν 2 t 2 J ν ( at ) = 0 1 t d d t t d J ν ( bt ) d t + b 2 - ν 2 t 2 J ν ( bt ) = 0 tJ ν ( bt ) tJ ν ( at ) , , ( a 2 - b 2 ) x 0 tJ ν ( at ) J ν ( bt ) dt = tJ ν ( at ) d J ν ( bt ) d t - tJ ν ( bt ) d J ν ( at ) d t t = x t =0 J ν ( z ) , ν > - 1 , t = 0 0.

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chpp17 - 17 Helmholtz 2 u k2 u = 0 1 2u 2u 2 k2 u = 0 r2 2...

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