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:

7/26/2019 PV2_Grupo4

2/6

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

7/26/2019 PV2_Grupo4

3/6

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

7/26/2019 PV2_Grupo4

4/6

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:

7/26/2019 PV2_Grupo4

5/6

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%

7/26/2019 PV2_Grupo4

6/6

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

= =

=