Специальные функции. Мицик М.Ф. - 11 стр.

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  
( )
xJ
ν

( )
xN
ν
  
, :
( )
( ) ( ) ( )
,
1
xNixJxH
ννν
+=
( )
( ) ( ) ( )
.
2
xNixJxH
ννν
−=
      
.
.
  ,    
.
 (25) 
ν
x

x
:
( )
()( )
( )
( )
( )
( )
.
!
2
1
1!
2
1
1
0
122
0
122
xJx
kk
x
kk
x
k
xJx
dx
d
k
k
k
k
k
k
−
∞
=
−+
∞
=
−+
=
+Γ
−
=
=
++Γ
+−
=
∑
∑
ν
ν
ν
ν
ν
ν
ν
ν
ν

( ) ( )
.
1
xJxxJx
dx
d
−
=
ν
ν
ν
ν
(33)

( ) ( )
.
1
xJxxJx
dx
d
+
−−
−=
ν
ν
ν
ν
(34)
    (33) (34)  

ν±
x
, :
( )
( ) ( )
,
1
xJxJ
x
xJ
−
=+
′
ννν
ν
( ) ( ) ( )
.
1
xJxJ
x
xJ
+
−=−
′
ννν
ν
:
( ) ( )
( )
,
2
11
xJ
x
xJxJ
ννν
ν
=+
+−
(35)
.2
11
xJxJxJ
ννν
′
=−
+−
(36)
  
( )
xN
ν

( )
( )
xH
2,1
ν
 
 
( )
xJ
ν

( )
xJ
ν−
,   (33) – (36),
                                      Jν ( x )             Nν (x )
                                      ,                                                             :
Hν (x) = Jν ( x) + i Nν ( x) ,
  (1)
                                                   Hν (x) = Jν ( x) − i Nν ( x).
                                                     (2)


        .

                                                                                                    .

                                               ,
                                                                                 .
                 (25)            xν                                                      x:
                                                                   2 k + 2ν −1

                             ∞
                                  (− 1)k (k + ν ) x 
        d ν
           x Jν (x ) = ∑                           2                           =
        dx             k =0                k!Γ(k + ν + 1)
                             2 k + 2ν −1

          ∞
              (− 1)  x 
                   k


        =∑            2                  = xν Jν −1 (x ).
         k =0    k!Γ(k + ν )


                  x Jν ( x ) = xν Jν −1 ( x ).
               d ν
                                                                                                        (33)
               dx

                  x Jν ( x ) = − x −ν Jν +1 ( x ).
               d −ν
                                                                                                        (34)
               dx
                                                                            (33)         (34)
                                      ±ν
                                  x ,                                                :
                          ν
               Jν′ ( x ) +  Jν ( x ) = Jν −1 ( x ) ,
                          x
                          ν
               Jν′ ( x ) − Jν ( x ) = − Jν +1 ( x ).
                          x
                                           :
                                          2ν
              Jν −1 ( x ) + Jν +1 ( x ) =      Jν ( x ) ,                                               (35)
                                           x
              Jν −1 ( x ) − Jν +1 ( x ) = 2 Jν′ ( x ) .                                                 (36)
                                  Nν (x )                   Hν(1, 2 ) ( x )
                             Jν ( x )                J −ν (x ) ,                                (33) – (36),

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