Специальные функции. Мицик М.Ф. - 9 стр.

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( )
( )
( )
.
1!
2
1
2
0
2
∑
∞
=
−
−
+−Γ
−
=
k
k
k
kk
x
x
xJ
ν
ν
ν (26)
 ,    (25) (26)   
 , 
(
)
xJ
ν
   

0
=
x
, 
( ) ( )
.0,00,10
0
>
=
=
ν
ν
JJ

( )
xJ
ν−

( )
....,2,1, =≠ nnν
 

0
=
x
, 
ν
:
( )
( )
( )
0,
1
2
~ →≠
−Γ
−
xn
x
xJ
ν
ν
ν
ν
.
 ,  
( )
....,2,1, =≠ nnν

( )
xJ
ν

( )
xJ
ν−
     ,
:
( ) ( ) ( ) ( )
....,2,1,
21
=≠+=
−
nnxJCxJCxy ν
νν
. (27)
, 
=
ν
, 
( )
xJ
ν

( )
−
−
xJ
ν

 (27).
.
2
  
(
)
Rba ⊂;
 

( ) ( ) ( ) ( )
{
}
....,,,,...,
2101
xxxx ϕϕϕϕ
−
.

( ) ( )
n
n
n
zxzx
∑
∞
∞−=
=ϕω ;

z
, 
( )
bax ;
∈
,  
( )
zx;
ω
  

( )
{ }
x
n
ϕ
.
       

( )
xJ
ν
.

( )
xJ
n

( )
−
=
z
z
x
ezx
1
2
;ω
,

                                                                                 2k

                                                  −ν ∞   (− 1)  x   k

                                              x               2 .
                                 J −ν (x ) =   ∑
                                              2  k =0 k!Γ(k − ν + 1)
                                                                                                             (26)
                             ,                                                 (25)       (26)
                     ,                                       Jν ( x )
                                                                                           x = 0,
                    J 0 (0) = 1, Jν (0) = 0 ,                    ν > 0.
                     J −ν (x )                             ν ≠ n, (n = 1, 2, ....)
                                                        x = 0,
ν:

                   J −ν ( x ) ~                 (ν ≠ n, x → 0) .
                                        2
                                     x Γ(1 −ν )
                                       ν


                         ,                  ν ≠ n, (n = 1, 2, ....)                              Jν ( x )   J −ν (x )
                                                                                                                    ,
                                                                                      :
                   y( x ) = C1 Jν ( x) + C2 J −ν ( x),                      ν ≠ n (n = 1, 2,....) .          (27)
                         ,                        ν = n,          Jν ( x )       J −ν ( x ) −
                                                                                (27).
           2   .                            (a ; b) ⊂ R
                   { ...,ϕ−1(x), ϕ0 (x),ϕ1( x), ϕ2 (x),....} .
                                        ∞
                   ω (x; z ) =         ∑ϕ ( x ) z
                                              n
                                                       n

                                      n=−∞

                                                                                                      z,
x ∈ (a ; b) ,                                 ω (x; z )
                                    {ϕn (x)}.
                                 Jν ( x ) .

J n (x)
                                                   x 1 
                                                     z− 
                             ω ( x; z ) = e        2 z 
                                                             ,

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