Специальные функции. Мицик М.Ф. - 20 стр.

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 ,     

( )
.01
22
=+
′
−
′′
− ynyxyx
(66)
 (62)      .
(62) , 
( )
0=xT
n

( )
0arccoscos =xn
(67)
  
( )
π
;0
. 
 (67) :
( )
.1...,,2,1,0,
2
12
cos −=
+
= nk
n
k
k
α
(68)
,  (68).
.

[ ]
1;1
−


( )
2
1
1
x
xw
−
=
.
.
( ) ( )
≠=
==
≠
=
−
∫
−
.0,
2
;0,
;,0
1
2
1
1
mn
mn
mn
x
dx
xTxT
mn
π
π
(69)
 (69) 
. 
.
,  (69) 
 – , 

 (, ) 
.

( )
xT
n
,

n
.
                                           ,

                              (1 − x ) y′′ − x y′ + n
                                           2                      2
                                                                      y = 0.                                     (66)
                   (62)                                                                                                       .
(62)                  ,            Tn ( x ) = 0
                                               cos(n arccos x ) = 0                                              (67)
                                                                                                     (0 ;π ) .
               (67)                             :

                              α k = cos
                                                    (2k + 1)π ,        k = 0, 1, 2, ..., n − 1.                  (68)
                                                         2n
 ,                                                                                               (68).

                                                                                                 .

                                                                                                             [−1;1]

                              w( x ) =
                                                     1
                                                              .
                                                    1 − x2
                                                                                      .
                                            
                                             0 , n ≠ m;
                                            
     1

     ∫ T (x)T (x )
                                    dx
           n       m                      = π , n = m = 0 ;
     −1                            1 − x 2 π                                                                    (69)
                                             2 , n = m ≠ 0.
                                                     (69)
                                       .
     .
                          ,                                           (69)
       –                       ,

               (                   ,                                                             )
           .
                                                                                      Tn ( x )                        ,
                                                                      n                                                   .

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