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7/26/2019 PV2_Grupo4
1/6
GRUPO 4 (PROBLEMAS 14,15,16,17)
CURSO: MATEMTICAS AVANZADAS
PROFESOR: RAUL P. CASTROL VIDAL
INTEGRANTES CODIGOS
AIME LAURA MARTIN. 1123210058
BLANCO MENDEZ JOS A. 062585C
FLORES MEJA HCTOR CLEMENTE 1023220343
MARTINEZ SANCHEZ PERC A !"0"15F
SANDO#AL LOPEZ JHON F. 1023220414
#IRRAREAL MAL#ACEDA DIE$O. 1123220555
PROBLEMA N14:
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Sea f(x )={1x2si|x|1
0 si|x|>1 , Obtener la integral de Fourier
y estudie su on!ergenia en xo=0 "
SOLUCION%
L& '()*+,&- * F/,'*, * *%
1
0
[A ( ) cosx+B () sinx ] d D/(*%
A()=
f(t) cos (t)dt
A()=1
1
(1t2 )cos (t)dt
A()=1
1
cos (t)dt1
1
t2cos (t)dt
A()=sen (t)
| 1
1 1
3
[(t)2 sen (t)+2cos (t)2 sen (t)]| 11
A()=4
3(sencos )
B ( )=1
1
(1t2 ) sen (t)dt
B ( )=1
1
sen (t)dt1
1
t2sen (t)dt
B ( )=cos (t)
| 11=13[(t)2cos (t)+2sen (t)2cos (t)]| 1
1
B ( )=00 ; B ()=0
1
0
[ 4
3(sencos )
]d
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PROBLEMA N1#:
a$ obtener la integral de Fourier de f(x )={cosx si|x|0 si|x|>
b$ estudiar la on!ergenia de la %F en xo=0 , x1=
SOL&'%(N:
L& '()*+,&- * F/,'*, * *%
1
0
[A ( ) cosx+B () sinx ] d
D/(*%
A()=
f(t) cos (t)dt
A()=
cos ( t)cos (t)dt
A()=2sen ()
212
B ( )=
f(t)sen (t)dt
B ( )=
cos ( t)sen (t)dt
B ( )=0
1
0
[2sen ()212 ]d
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PROBLEMA N 1):
a$Obtener %F"
cos ( )
0
xf x
=
*
x
x
>
IF%
0
1( ) ( ) cos ( )f x f t w x t dtdw
=
0
1( ) cost cos ( )f x w x t dtdw
= .. 1
I()*+,&7/ /, &,)* *(. 1
0
1 4 ( ) 4 ( )( )
4( 1)( 1) 4( 1)( 1)
wsenw x wsenw xf x dw
w w w w
+= + +
[ ]0
1( ) ( ) ( )
( 1)( 1)
wf x senw x senw x dw
w w
= +
+ A*79 -& ),&(/,7& * F/,'*,%
2( ) ( ) cos
( 1)( 1)
jwx jwx wF w f x e dx e dx sen ww w
= = = +
2( )
( 1)( 1)
wsen wF w
w w
=
+
PROBLEMA N 1+:
Si -.$ es una uni/n 0ar on integral de Fourier:
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0
( ) (w) Cos(wx) dwf x A
= e2uestre 3ue:
0
1(a ) ( ) Cos(wx) dw, 0
wf x A a
a a
= >
SOL&'%(N
* &, :/( '()*+,&- * F/,'*,%
0
( ) (w) Cos(wx) dw.......(1)f x A
=
P*,/ S&;*7/ &(/ 3 *( 4%
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0
1 1 1( ) ( ) ( ) ( ) ( )
w wf ax F Cos wx dw F Cos wx dw
a a a a
= =
P*,/ * ?%
0
( ) ( ) (
1( ) ( ) ( )
w wA w F w A F
a a
lqqd
wf ax A Cos wx dw
a a
= =
=