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§7. òÑÄÙ æÕÒØÅ 53
ÓÈÏÄÉÔÓÑ Ë f(x
0
).
f(x
0
) =
a
0
2
+
∞
X
n=1
a
n
cos
nπx
0
l
+ b
n
sin
nπx
0
l
.
-
6
0 x
y
3
−2
−π
π
ðÒÉÍÅÒ 1. îÁÊÔÉ ÒÁÚÌÏÖÅÎÉÅ × ÒÑÄ æÕÒØÅ
ÆÕÎËÃÉÉ f (x) ÎÁ ÉÎÔÅÒ×ÁÌÅ (−π; π).
f(x) =
−2, ÅÓÌÉ −π < x < 0,
3, ÅÓÌÉ 0 6 x < π.
òÅÛÅÎÉÅ: úÁÄÁÎÎÁÑ ÆÕÎËÃÉÑ f(x) ÕÄÏ×ÌÅ-
Ô×ÏÒÑÅÔ ÕÓÌÏ×ÉÑÍ ÔÅÏÒÅÍÙ Ï ÒÁÚÌÏÖÉÍÏÓÔÉ ×
ÒÑÄ æÕÒØÅ, ÔÁË ËÁË ÎÁ ÉÎÔÅÒ×ÁÌÅ (−π; π) ÆÕÎË-
ÃÉÑ ÉÍÅÅÔ ÏÄÎÕ ÔÏÞËÕ ÒÁÚÒÙ×Á ÐÅÒ×ÏÇÏ ÒÏÄÁ
(ÐÒÉ x = 0), Á ×Ï ×ÓÅÈ ÄÒÕÇÉÈ ÔÏÞËÁÈ ÜÔÏÇÏ ÉÎ-
ÔÅÒ×ÁÌÁ ÏÎÁ ÄÉÆÆÅÒÅÎÃÉÒÕÅÍÁ. óÌÅÄÏ×ÁÔÅÌØÎÏ, ÄÌÑ ÄÁÎÎÏÊ ÆÕÎËÃÉÉ ÓÐÒÁ-
×ÅÄÌÉ×Ï ÒÁ×ÅÎÓÔ×Ï
f(x) =
a
0
2
+
∞
X
n=1
(a
n
cos nx + b
n
sin nx) .
þÔÏÂÙ ÎÁÊÔÉ ËÏÜÆÆÉÃÉÅÎÔ a
0
, ÐÒÉÍÅÎÑÅÍ ÆÏÒÍÕÌÕ (5) ÐÒÉ n = 0.
a
0
=
1
π
π
Z
−π
f(x) dx =
1
π
0
Z
−π
−2 dx +
π
Z
0
3 dx
=
=
1
π
[−2x]
0
−π
+ [3x]
π
0
=
1
π
(−2π + 3π) = 1.
ôÅÐÅÒØ ÎÁÈÏÄÉÍ ËÏÜÆÆÉÃÉÅÎÔÙ a
n
(n = 1, 2, 3, . . .) ÐÏ ÆÏÒÍÕÌÅ (5).
a
n
=
1
π
0
Z
−π
−2 cos nx dx +
π
Z
0
3 cos nx dx
=
=
1
π
−2 sin nx
n
0
−π
+
3 sin nx
n
π
0
!
= 0.
§7. òÑÄÙ æÕÒØÅ                                                                                         53

ÓÈÏÄÉÔÓÑ Ë f (x0).
                                             ∞
                                 a0 X          nπx0          nπx0 
                        f (x0) =   +     an cos      + bn sin        .
                                 2   n=1
                                                  l             l


  ðÒÉÍÅÒ 1. îÁÊÔÉ ÒÁÚÌÏÖÅÎÉÅ × ÒÑÄ æÕÒØÅ                                             y6
ÆÕÎËÃÉÉ f (x) ÎÁ ÉÎÔÅÒ×ÁÌÅ (−π; π).
                                                                                     3
                      
                          −2, ÅÓÌÉ −π < x < 0,
        f (x) =
                           3, ÅÓÌÉ 0 6 x < π.
                                                                            −π                         -
                                                                                         0          π x
   òÅÛÅÎÉÅ: úÁÄÁÎÎÁÑ ÆÕÎËÃÉÑ f (x) ÕÄÏ×ÌÅ-
Ô×ÏÒÑÅÔ ÕÓÌÏ×ÉÑÍ ÔÅÏÒÅÍÙ Ï ÒÁÚÌÏÖÉÍÏÓÔÉ ×               −2
ÒÑÄ æÕÒØÅ, ÔÁË ËÁË ÎÁ ÉÎÔÅÒ×ÁÌÅ (−π; π) ÆÕÎË-
ÃÉÑ ÉÍÅÅÔ ÏÄÎÕ ÔÏÞËÕ ÒÁÚÒÙ×Á ÐÅÒ×ÏÇÏ ÒÏÄÁ
(ÐÒÉ x = 0), Á ×Ï ×ÓÅÈ ÄÒÕÇÉÈ ÔÏÞËÁÈ ÜÔÏÇÏ ÉÎ-
ÔÅÒ×ÁÌÁ ÏÎÁ ÄÉÆÆÅÒÅÎÃÉÒÕÅÍÁ. óÌÅÄÏ×ÁÔÅÌØÎÏ, ÄÌÑ ÄÁÎÎÏÊ ÆÕÎËÃÉÉ ÓÐÒÁ-
×ÅÄÌÉ×Ï ÒÁ×ÅÎÓÔ×Ï
                                                 ∞
                                    a0 X
                            f (x) =   +     (an cos nx + bn sin nx) .
                                    2   n=1


þÔÏÂÙ ÎÁÊÔÉ ËÏÜÆÆÉÃÉÅÎÔ a0 , ÐÒÉÍÅÎÑÅÍ ÆÏÒÍÕÌÕ (5) ÐÒÉ n = 0.

            Zπ                      Z0                Zπ
                                                                
        1                     1
 a0 =            f (x) dx =              −2 dx +           3 dx =
        π                     π
            −π                    −π                  0
                                                     1                   1
                                                 =     [−2x]0−π + [3x]π0 = (−2π + 3π) = 1.
                                                     π                    π
ôÅÐÅÒØ ÎÁÈÏÄÉÍ ËÏÜÆÆÉÃÉÅÎÔÙ an (n = 1, 2, 3, . . .) ÐÏ ÆÏÒÍÕÌÅ (5).

                 Z0                      Zπ
                                                           
        1
 an =                 −2 cos nx dx +          3 cos nx dx =
        π
                 −π                      0
                                                                           0              π !
                                                       1        −2 sin nx         3 sin nx
                                                     =                          +                   = 0.
                                                       π            n        −π       n       0

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