Специальные функции. Мицик М.Ф. - 13 стр.

UptoLike

ВУЗ: 

ИСОиП ДГТУ | Шахты

Составители: 

Рубрика: 


10
<
<δ
.    (38) 
( )
zx;ω


( )
xP
n
.
  (39)    ,

( )
.121
2
1
2
ztzzx −=+−
−
(40)

z

t
, :
( )
( )
( )
( )
.
1
122
,
12
1
21,
1
2
2
2
2
2
2
2
1
2
2
dt
t
xtt
zd
xtt
t
zzx
t
xt
z
−
+−
−=
+−
−
=+−
−
−
=
−
 ,   
z
 
δ=z
, 

t

C
, 
x
t
. 
 (40) 
( )
xP
n
:
( )
( )
( )
.
2
1
2
1
1
2
2
dt
xt
t
i
xP
C
n
n
n
∫
+
−
−
=
π
(41)
:
( )
( )
xt
n
n
n
n
n
t
dt
d
n
xP
=
−= 1
!2
1
2

( )
( )
....,2,1,0,1
!2
1
2
=−= nx
dx
d
n
xP
n
n
n
n
n
(42)
 ,  
( )
xP
n
 . 
.
      . 
 (42) :
( ) ( ) ( )
( )
( )
( )
( )
( )
( )
( )
.157063
8
1
,33035
8
1
,35
2
1
,13
2
1
,,1
35
5
24
3
3
3
2
210
xxxxPxxxP
xxxPxxPxxPxP
+−=+−=
−=−===

. 
( )
xP
n
:
( )
(
)
(
)
( ) ( )
[
]
.
!2!!2
!221
2
2
0
kn
n
k
n
k
n
x
knknk
kn
xP
−
=
∑
−−
−−
=
(43)
0 < δ <1.                                                      (38)                  ω ( x; z )
                       Pn ( x ) .
                         (39)                                                                                  ,


                      (1 − 2 x z + z )
                                               1
                                            2 −2
                                                     = 1 − zt .                                         (40)
      z                t,                   :
            2(t − x )                                                  1− t2
                        (1 − 2 x z + z )
                                                               1
                                                            2 −2
          z= 2        ,                                            = 2           ,
             t −1                                                   t − 2t x + 1
                 2(t − 2 t x + 1)
                            2
          dz = −                  dt .
                    (t − 1)         2   2


              ,                                                     z                             z =δ ,
          t                                     C,                                   t = x.
  (40)                                                                                     Pn ( x ) :

                      Pn ( x ) =
                                  1         t2 −1   (         )2


                                 2π i C∫ 2n (t − x )n+1
                                                        dt .                                            (41)

                                                                                                   :
                                 1  dn 2   n
                      Pn ( x ) = n  n t − 1 
                                2 n!  dt
                                                        (          )
                                             t = x

                     1 dn 2
          Pn ( x ) = n
                    2 n! dx n
                              x − 1
                                    n
                                            (
                                      , n = 0,1, 2, ....)                                               (42)

                                ,                             Pn ( x )                                  .
                                                                         .
                                                                                                        .
    (42)                    :

P0 ( x ) = 1, P1 ( x ) = x, P2 ( x ) =
                                                          1 2
                                                          2
                                                              (              )           (
                                                              3x − 1 , P3 ( x ) = 5 x 3 − 3x ,
                                                                                  1
                                                                                  2
                                                                                                        )
P3 ( x ) =
              1
              8
                  (
                35x 4 − 30 x 2 + 3 ,            )                        (
                                                        P5 ( x ) = 63x 5 − 70 x 3 + 15x .
                                                                  1
                                                                  8
                                                                                              )

                  .                             Pn ( x )                         :

          Pn ( x ) = ∑ n
                            (− 1) (2n − 2k )!
                        [n 2 ]    k
                                                  x n−2 k .
                     k =0 2 k !(n − k )!(n − 2k )!
                                                                                                        (43)

Страницы