8/16/2019 Fourier Sumasinfinitas

1/14

 

!

" # $%# !

& %## ' ( # # %# )#

*# (+# , (+ (#

( -(# #. / ,+ # ( #

#(* 0 ( #, # *# ,#

 (#. 12(# 1#( ' # # %#

+ # . " #( ((

# %# )# % ## %# /

* # ,# # ## %## # # /

 + * # # (# # # %#

(# (# % #3. "# ((# #

*# 4(# # ' (( ) ##

## # # # (#. & %## ' # (

# 5#6 # 7(# $%# (# ## , (2

,#. # 0 ## ((# / # # (

# ## / ,### ( % 56.

8/16/2019 Fourier Sumasinfinitas

2/14

 

  ! * ( ,    0 ( (#

(!. *# # ( *   0 ## ( ,

 (#, 0 (     !=  + ! / # 4# -   (#, 0 (

   # (   !=  + ! #  ≥ . + , # 0

#  8   # (#,. " * *      # ,#

 ( *   =98 .

" * # # ( 0 ( ( (# (

* / 0 ## (## (# # + # # (#,# 0

(# .

# / # -# 4(# # # 4(# / 4(

# # ( ( (  ! 0 (   # )# #

 ( * ! # * 4(π    !   0

#   . # ( ( #, 0 ( 0

4(α  ! # ( (   #

( ) ( )( )   ( ) ( ) 4( 4( 4( 4(  α α α α  = + =   .

0 # 0 α    # # -( π . :# #

## ( # # 4(# 0 * ( (

*    # ## 4(# #.

: # 4(# * ( 4(## # 4(# # 4( '.

!

(# # 0 * * *     # ( #

( ) ( )4(

   π  +∞

=−∞

= ∑   9!

8/16/2019 Fourier Sumasinfinitas

3/14

"# # *# # # 4( '   !

#( *# '   !. ;# * ( # *#

(+# ,   (# (    # # # ( *

! ( 0 # ((# ( #<

( ) ( ) ( ) ( )

=

9

 

    

= ∫ i   !

# > # , ( *

/ ## + (+. "# *# 4(π    ! #

# #( # ( # #

( ) ( ) ( )( )

94( 4( 4(

 

  

π π π δ  = − =∫ i   ?!

δ  # # @A 0 # 9 # = / #

. > ( +( 0 # * # ( 4(# #

*# # 4( ' # *# # ## #

)# . :%# # #3   ! # ( 4(# 9!

#

( ) ( ) ( ) ( )

94( 4( 4(

 

   

π π π δ  

+∞ +∞

=−∞ =−∞

− = = ∑ ∑∫    i   B

( ) ( )

94(

 

  

π  = −∫    !

# *# # # ' # #

5(/6 *   ! # # 5,#6 4(π    !. # ,# *

+ ( . C (# 0 *

# (0 , ## # # ## D

"# $

 C# %# 0 ( *# ( # * (

# (+! # # #- ! # ( / 0

( ) ( ) ( )

9

 

  

= ∫ i  

; (# E  !E  ( #% #, #

 ((! *   ! )# # *#. ; +(

 ( ) #% # (( ,+ / )

8/16/2019 Fourier Sumasinfinitas

4/14

. & ;#, # ( (

* 4(#% ,)# ## *# '. ; #

0 # * # 4(# 9! #

( ) ( ) ( ) ( )= = 4( 4(

  π π δ 

∞ ∞ ∞ ∞

=−∞ =−∞ =−∞ =−∞

= =   ∑ ∑ ∑ ∑

i i   B

( )

 

9

 

   

+∞

=−∞

= = ∑∫    F!

( * # 0, #

 ( # # # * ! # # # *

# 4(# # ' (# # ( # #

# ## (# #( # *# #

4(π     !.

# # ( # ## *#B

 # *   ! 0 # *# ((#.

%&

*   ! # 4(# 9! #  / − # (+#

+#<

( ) ( ) ( ) ( ) ( )= = = 4( 4( 4(

 π π π  +∞ +∞ +∞

−

=−∞ =−∞ =−∞

= − = = =∑ ∑ ∑   .

* # ( #   != − ! (  ! # =−

# 0 # # ( #   !=− − !! (  ! #

=−−<

( ) ( ) ( ) ( ) ( ) 4( 4( 4(

 π π π  +∞ +∞ +∞

−

= −∞ = −∞ =−∞

− = − = = ± = ±∑ ∑ ∑   .

"# (# # # ## ) # # (

( ,.

" # # # *# '   ! #  !<

( ) ( ) 4(

  +∞ +∞

=−∞ =−∞

= =∑ ∑   !

# ) ( , # > ( # ##

*# , #3#   ! 0 # # *# ((#.

8/16/2019 Fourier Sumasinfinitas

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G( , *   # , ( #3

 ( # ( = !<

( ) ( ) ( )

9 94(

 

   

= =∫ ∫    H!

*   ! *# '  # ( τ  # #

*# ' * # 4(−π   τ !  ((

!<

( ) ( )( )   ( ) ( ) 4( 4( 4(

 τ π τ π τ π    +∞ +∞

=−∞ =−∞

− = − = − ∑ ∑  

:( )# # #3 # #   + 8!=−  ! (

 ! # # *# ' ( # #<

( ) ( ) ( ) ( ) ( ) 9 4( 4(

    π π  

+∞ +∞

=−∞ =−∞

+ = − = − −

  ∑ ∑   .

0 # # 0 #  # ( =− 0 # ( . &

( ( *% 0 ( ) # .

' # # *# '   ! # # *# '

,   ! #(   # π    (( ,!.

( ) ( ) ( ) ( ) 4( 4(

     

π  

π π  

+∞ +∞

=−∞ =−∞

= = ∑ ∑   .

&4# # ((# # *# ' ( #% ,# (

# #.

'! (

I# # # 0 4# *#   # 0 *   !

##* 9!. " ( 0 # #( , # J( 0) *#

 (# ( *    # ( 9!K " #(# # ( #

# 0# 0 ( # # 0 #<

9!  " * # # ( # ,

#   ! # # * ,# (+#!

(.

!  " * - * %4# / # (.

?!  " * - * ## ( # #

# *.

8/16/2019 Fourier Sumasinfinitas

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## # # # ( # ' , *

# 0 ( 0 4# * / #

- )# # <

( ) ( )

9 4(   

  

  

    

π  

+

→+∞=−

− =∑∫    .

:%# 9! # *# # ! # 4

(   #, # (##! (# ## #   ! # 0

# ' , ( , ( # # #

  ! ( >0. # # # # / 0 )# #

, (+ # ## L FL # #

$%#! / ( ( ( #*, (

# #.

;# ) ( 0 + # *# 4( π    ! #

## + *# % ( # # 0

##* # # / # # ,# #!.

# + # *# 4(π    ! # #

.

) * +

&4# ## 0 ##* # # ( # ( #

4(## # '. +( # # ( (

## δ  ! ) # (## #! #(#

  * 0 # #) #  =. : # #

(## # 5( 6 / # (   !<

( ) ( )   δ 

+∞

=−∞

= −

∑  .

" # (# # ((# 0 # (

# 0,#<

9!  : , * ># # , (# (

# # ( ( #

* # # ( ( #

( >0 * ( #.

8/16/2019 Fourier Sumasinfinitas

7/14

!  " # δ  ! # ( ,  

()# %# 0

( ) ( ) ( )

δ    =∫   

 ( , ! 0 # .

> # # (( # , ( 0 # *# '

5( 6 #

( ) ( )  ( )

4( 9 94(

 

 

   

δ π 

+

−

= − = =∫    .

, -

./ $ 0

/ # ( ( ( # ## ' % ##

*# ## #3# /# *# # ((#. ;

+( (# # ( / (

* 9 #    =

 =

9! / (# %* #<

, ( ( 0 ## *# ' #

8/16/2019 Fourier Sumasinfinitas

8/14

( ) ( ) ( )9

9

4( 4( 9 9

 

π π  

π π π  −

− − − − −

= + =   .

# −98 # , 98! #

# ( 0 , 9  = / /# *# '

#

( ) ( )9 9 9 94(

π  

π π  

− − − − = =

  .

 C# 0 ## *# # #  # ( /  # ( (#,

,# # ((# 98< 9=8π   ?=−8?π  !

F=8Fπ  !D 7, =− ( #3 # (. ; (

! # ( ( , #<

( )9

9 ...

? F

  π π π  

+∞ +∞

= −∞ =

= = = = − + −

∑ ∑   B

9 9 99 ...

? F H

π  

− + − + =   .

0$ / $

# ## % ## *<

9

9 

 

+∞

=

∑  

#    (#, 0. & ( (# #, 0

# ( # - α  # α  

, /   (#, ( / ## # α   # #(

#

9 9 9

 ( (  

9 9 9 9

9

 

   

α α 

α α 

+∞ +∞ +∞ +∞ +∞

= = = = =

= = = ⋅ =

−

∑ ∑ ∑ ∑ ∑   9

 ( 

9     

 

+∞

=

= ∑   .

# * (# > *# # /#

*#   # ,# ((#    ( # #.

G# ) 0 # # * /# *# #

,# ((#   / )# # , # *

# # - % ( (( , *# ,#

 ((#  +9

. " ( #  9 (# *% / 0

# ,# / ( #% # 9

8/16/2019 Fourier Sumasinfinitas

9/14

# / ( ) # 9 # ( ! *# ,#

 ((# <

9 9

 ( (  

M 9 M9

  π π π  

+∞ +∞

=−∞ =

= = =∑ ∑   B

99

 ( 

9

M

 

π  +∞

=

= =∑   B

9 9 9

9

9

π  +∞

=

= = =

−

∑   .

; # ## # # # * *#<

( )

( )

( )

( )

9

9

9

9 9

9 9

9

9 .

 

 

 

 

 

   

 

−

−

−

− + − ≤

8/16/2019 Fourier Sumasinfinitas

10/14

" ( ## #3# # (, ## ## (

( #- 0   # ( ( #(,

( )  ( )

( )

99 9

9   9 9

9 9

 

 

 

    

 

−

−  −

= − = =

− −

∫   .

"#    # *# ' #3    !. "# # #

, * * / # (#

( ) 9

9

 

π  −

−

= −     (

( )   ( )( ) 9

9 9 9

 

π π  −

− −−

= − −     (.

; +( / 0 # ( / # , #

  ! / # ( #  9 ! #<

( )( )9

 

9 9

π π  

− −−

= = −   .

 C# 0 4# ( * 8 ( = ( #

#, ( 0 # = / 0   ! # .

&# # *

9

 ( 

9 9 ?

? 

  π π  

+∞ +∞

=−∞ =

= = = =∑ ∑   B

9

 ( 

9

N

 

π  +∞

=

= =∑   B

9

9

9 N

π  +∞

=

= = =

−

∑   .

; # ## ( ,# /#   # ( # # ##

# # * # ((# , / / #

*# / # 5( 6   !<

( ) 9

9

 

π  −

−

=     (( )   ( )9

9   9 9 

π π  

−

−   − −

= +     (.

:# (  =? / #(, #

( ) ( )?   ? ?

9 9 9 9M

π π  

− − − −

= −   B( ) ( )

 

9 9 9 9M

π π  

− − − −

= − +   B

N NH ?

? ? 9

? ?  

9 9

F ?

   

  π π π π    

+∞

=−∞

= = = − + = −

∑   B

? N 

  π  

=

  .

8/16/2019 Fourier Sumasinfinitas

11/14

N 99 F 99

?   M M

? ? ? ?9 N

H F

    

π π π π    

+∞

=−∞

⋅⋅ ⋅ ⋅

= = = − + = −∑   BM

9H

99M 

  π  

=   .

? ? ?9

9

9 ? F H NF

π π  +∞

=

= = = =

− ⋅ ⋅

∑   M M M

M M ? 9

9

9 ? F H NF

π π  +∞

=

= = = =

− ⋅ ⋅ ⋅

∑   .

; ,# /#    # ( > # ( # 0 #

( * %( ( ,> %# ##  

0 , )# ># % # E  E.

0$ / $

 ====1

# # ## # *# ,# (#   #

  !=−9 # - ( / 0 # , # #

# # , *  =. ;# (#

( (( 0 # ( , * *

## *# ' > %# 0 # ( *

> 0 −= / 0 # # *# (# # # ( # #    !

#)# <

9

 ( 

9

+∞ +∞

=−∞ =

− = =

∑ ∑   .

> %# # # #,# # ( #  

# 0 (  (

( )( ) ( )

9 9

 

π π  −

− − −

= − −    

π  = −   .

" # # #, # ( #  

 (# 4 # ,# # ## ##. , #

( ( 0 (  (

( )  ( )

( )

  9

9 O9

O

   

    

 

 π  =

−

= −

−

∑   .

&*, (  =9 #

( )  ( )

( )

9

9 O   9

O 

π π  

−

− = − =

−

# 0 # # # (   #

8/16/2019 Fourier Sumasinfinitas

12/14

( ) ( ) ( )

( )  ( )

( )( )( )

( )   ( )( )

P 9P

P P P

P9P

P PP 9

9 9

9 O 9 O9

P O O

9 O9

P O

   

  

   

  

 

 

 

 

 

π π  

π π  

π  

+

+

=

+

=

+ +

= − −

+ − + −

= − −

+ − + −

+ −

= −

+ −

∑

∑

 

# > ,  P= +9. 0

( )  ( )

( )  ( ) ( )

  9 9 9

 ( 

9 O O 9 9

           

     

   

 π π  

+∞

= = =

− − = = − = −

∑ ∑ ∑   B

# ( # *# ## #, .

; #(* ## %# (# *

( )

O

 

  

 

 

π  

=   ( )   ( ) 9 9

9 9

    

 

 

 

 

−

=

− = − −

∑  

9 9

O 9

 

π  

= =   ? O 9

?  

π  

= =   ? ? FO 9

 

π  

= =   M HMO 9

?  

π  

= =   .

; +( (#

( )   ( )   ( )

N 9 N 9

F F

9

? F H M

9 9 ? 99 ?9 F 9 9

F F N F 9 F ? F HN F ?9B

   

 

 

 

−

=

− = ⋅ ⋅ ⋅ ⋅ = + − −

= − ⋅ + ⋅ ⋅ − ⋅ ⋅ + ⋅ ⋅ = ⋅ ⋅

∑ 

F

F

  =   B

N 9   9

FF   9

9

9

9O N?FFF

 

π     π  +∞

=

= = =∑   .

( )   ( )   ( )F

99 9 99 9

9

N

9

9 9 ? F H 9? 9 9

? ? 99 ? FF ? HH ? N?F ? 9HF

? N9B

   

 

 

 

−

=

− = ⋅ ⋅ ⋅ ⋅ ⋅ = − − − −

= − ⋅ + ⋅ ⋅ − ⋅ ⋅ + ⋅ ⋅ − ⋅ ⋅ + ⋅ ⋅

= ⋅ ⋅

∑

 

N9

H?   =   B

99 9   9

  9

9

9 N9

9O ?MF9MHF

 

π     π  +∞

=

= = =∑   .

:# # % 0 #   # # ,# ##

# -#    *# + # (#   !

<

( )

9 O

 

 

   

∞

=

=−  ∑   B ( )

9

   

 

→

= =   −

.

8/16/2019 Fourier Sumasinfinitas

13/14

# :* IQ> % *# # RR /

 ((# RR # 0 ( ≥

( )

( )

( )

4( O

   

    

π  

π  

+∞

=−∞

≠

= − ∑   .

0 ( =  ( (( ,  = #

( )  ( ) ( )

( )

( ) ( )     9

9

9 O 9 O9

 

      

    

  π  π  

+∞

−

=

− −

= = − = −∑   B

( )( )

  ( )( )

9 9 9

9

O O

     

 

 

 

 

π  π  

−  −−

−

= =   .

"# -# #% # ( ,# / #   7

F # ( ,  F!. 7#, # ,# ## S9T

 (# # <

9? 9

99

9 H

9O  

π  +∞

=

⋅ ⋅

=

⋅

∑   9F 9

99

9 ?9H

F9 9O  

π  +∞

=

⋅ ⋅

=

⋅

∑   9H 9M

9M9

9 ?MH

HNM 9MO  

π  +∞

=

⋅ ⋅

=

⋅

∑  

9N

9

9 9H99

?? O  

π  +∞

=

⋅ ⋅

=

⋅

∑   9

9

9 MFF9?

9?M O  

π  +∞

=

⋅ ⋅

=

⋅

∑   .

 C# * 0 # -# / *

> G , ( +( S9T!

( ) ( )( )

9

  9

9 O

 

     

    ζ 

π 

−

−

−

=  

 (# ( # # 0 #% ( #

#<

( )

9

9

 

 

ζ ∞

=

=∑   .

2/

S9T (

8/16/2019 Fourier Sumasinfinitas

14/14

RReevviissttaa EEssccoollaar r  ddee llaa OOlliimmppííaaddaa IIbbeer r ooaammeer r iiccaannaa ddee MMaatteemmááttiiccaa 

hhttttpp::////wwwwww..ccaammppuuss--ooeeii..oor r gg//ooiimm//r r eevviissttaaooiimm// 

EEddiittaa:: 

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