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UNIVERSIDAD ANDINA DEL CUSCO P.A.P.INGENIERIA CIVIL
UNIVERSIDAD ANDINA DEL CUSCO
FACULTAD DE INGENIERIACARRERA PROFESIONAL DE INGENIERIA CIVIL
CÁLCULO III
DOCENTE: ABEL SARCCO USTO
INTEGRANTES:
• VALDIVIA HUMBERTO• VALER VERA JORGE JESÚS• PÉREZ MATHEUS JAN GABRIEL ELISBAN• SAAVEDRA BLANCO KIARA CHASKA
• FLÓREZ MASÍAS GUIDO ERMEREGILDO• RIOS YAÑAC SUNEM• CAVIEDES TOLEDO JOSÉ RODRIGO• SHERLOP MEDRANO YANQUE• ALPACA NINAN DANIEL• CORTÉZ VALLE YONATHAN FRANCYS• CCORIMANYA USCA ELIZABETH
C!"#$P%&'
CALCULO III Año-2015
SERIES DE FOURIER
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UNIVERSIDAD ANDINA DEL CUSCO P.A.P.INGENIERIA CIVIL
INTRODUCCION Y PRESENTACION
E( %) *&%!%(+% ,(-#&.% !% /%+0))0( /%1(,",#(%!2 +%#&%.0!3 %4%.*)#! *0&0 )0 "#.*&%!,5( /% )0! S%&,%! /% F#&,%&2( +%.0 ,.*#&+0(+% *0&0 %) "6)")# /% ."70!%"0",#(%! /% )0! ",%(",0! 8% !% -#&.)0( "#( /%&,90/0!*0&",0)%! 3 !% &%!%)9%(2 %( #"0!,#(%!2 /%!"#.*#(,%(/#)0 ,("5(,+0 %( !%&,%! ;!.0! ,(1(,+0!<= A!> +0.?,@( "#.#! 0*),"0",5( %( /,9%&!#! 6&%0! /% ",%(",0! % ,(%(,%&>0=
L0 ,/%0 ?6!,"0 /% )0! !%&,%! /% F#&,%& %! 8% +#/0-(",5( *%&,5/,"0 /% *%&,#/# T *%/% !%& %*&%!0/0"#.# (0 !.0 +&,#(#.@+&,"0 /% !%(#! 3 "#!%(#! /%).,!.# *%&,#/# T= E) *&#?)%.0 0*0&%"% (0+&0).%(+% %(0!+&#(#.>02 30 8% N%%?0%& ;< /%!"?&,5 8% )#!B0?,)#(,#! +,),0&#( (0 -#&.0 *&,.,+,90 /% )0! !%&,%! /%
F#&,%& %( )0 *&%/,"",5( /% ",%&+#! %9%(+#! "%)%!+,0)%!=
CALCULO III Año-2015
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UNIVERSIDAD ANDINA DEL CUSCO P.A.P.INGENIERIA CIVIL
GENERALIDADESDEFINICION=$
U(0 !%&,% /% F#&,%& %! (0 !%&,% ,(1(,+0 8% "#(9%&%*(+0).%(+% 0 (0 -(",5( *%&,5/,"0 3 "#(+,(0 0 +&##!;# *#& *0&+%!<=
L0! !%&,%! /% F#&,%& "#(!+,+3%( )0 7%&&0.,%(+0.0+%.6+,"0 ?6!,"0 /%) 0(6),!,! /% F#&,%& %.*)%0/# *0&00(0),0& -(",#(%! *%&,5/,"0! 0 +&09@! /% )0/%!"#.*#!,",5( /% /,"70 -(",5( %( (0 !.0 ,(1(,+0 /%-(",#(%! !,(!#,/0)%! ."7# .6! !,.*)%! ;"#.#"#.?,(0",5( /% !%(#! 3 "#!%(#! "#( -&%"%(",0!%(+%&0!<=
HISTORIA.-
E) (#.?&% !% /%?% 0) .0+%.6+,"# -&0("@! J%0($ B0*+,!+% J#!%*7 F#&,%& 8% /%!0&&#))5 )0 +%#&>0 "0(/# %!+/,0?0)0 %"0",5( /%) "0)#&= F% %) *&,.%&# 8% %!+/,5 +0)%!!%&,%! !,!+%.6+,"0.%(+%2 3 *?),"5 !! &%!)+0/#! ,(,",0)%!
%( 3 = E!+0 6&%0 /% ,(9%!+,0",5( !% ))0.00)(0! 9%"%! A(6),!,! 0&.5(,"#=
CALCULO III Año-2015
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UNIVERSIDAD ANDINA DEL CUSCO P.A.P.INGENIERIA CIVIL
L0%"0",5( /%) "0)#& %! (0 %"0",5( %( /%&,90/0! *0&",0)%!=P&%9,0.%(+% 0) +&0?04# /% F#&,%&2 (# !% "#(#">0 !#)",5(0)(0 *0&0 )0 %"0",5( /% "0)#& %( -#&.0 %(%&0)2 0(8%!% "#(#">0( !#)",#(%! *0&+,")0&%! !, )0 -%(+% /% "0)#& !%"#.*#&+0?0 /% .0(%&0 !%(",))02 %( *0&+,")0&2 !, )0 -%(+%
%&0 (0 #(/0 /% !%(# # "#!%(#= E!+0! !#)",#(%! !,.*)%! 09%"%! !#( ))0.0/0! 90)#&%! *&#*,#!= L0 ,/%0 /% F#&,%& %&0.#/%)0& (0 -%(+% /% "0)#& "#.*)%40 "#( (0!*%&*#!,",5( ;# "#.?,(0",5( ),(%0)< /% !,.*)%! #(/0!!,(!#,/0)%! 3 *0&0 %!"&,?,& )0 !#)",5( "#.# (0!*%&*#!,",5( /% )#! "#&&%!*#(/,%(+%! 90)#&%! *&#*,#!= A )0!*%&*#!,",5( # "#.?,(0",5( ),(%0) !% )% ))0.0 S%&,% /%
F#&,%&=
T,%.*# /%!*@! -% #?9,# 8% !% *#/>0 !0& )0 .,!.0+@"(,"0 0 ( &0( "#(4(+# /% *&#?)%.0! ->!,"#! 3.0+%.6+,"#!2 %!*%",0).%(+% 08%))#! 8% ,(9#)"&0?0(%"0",#(%! /,-%&%(",0)%! ),(%0)%! "#( "#%1",%(+%!"#(!+0(+%!2 *0&0 )#! "0)%! !! !#)",#(%! '(,"0! %&0(
!,(!#,/0)%!=
CALCULO III Año-2015
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UNIVERSIDAD ANDINA DEL CUSCO P.A.P.INGENIERIA CIVIL
&%0! /% 0*),"0",5( ,(")3%(:
• A(6),!,! 9,?&0+#&,#
• A"'!+,"0
• Ó*+,"0
• P&#"%!0.,%(+# /% ,.6%(%! 3 !%0)%!
• C#.*&%!,5( /% /0+#!
E( ,(%(,%&>02 *0&0 %) "0!# /% )#! !,!+%.0! /%+%)%"#.(,"0",#(%!2 3 0 +&09@! /%) !# /% )#! "#.*#(%(+%!%!*%"+&0)%! /% -&%"%(",0 /% (0 !%0) /0/02 !% *%/%#*+,.,0& %) /,!%# /% ( !,!+%.0 *0&0 )0 !%0) *#&+0/#&0/%) .,!.#= R%1@&0!% 0) !# /% ( 0(0),0/#& /% %!*%"+&#!=
FUNCIONES PERIÓDICAS=$
U(0 función periódica f(t) ".*)% 8% *0&0 +#/# 90)#& /%t :
f(t) = f(t + T)=
CALCULO III Año-2015
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UNIVERSIDAD ANDINA DEL CUSCO P.A.P.INGENIERIA CIVIL
A) 90)#& .>(,.#2 .03#& 8% "% /% )0 "#(!+0(+% T 8%".*)% )# 0(+%&,#& !% )% ))0.0 %) periodo fundamental
;# !,.*)%.%(+% *%&,#/#< /% )0 -(",5(=
O?!%&90 8%:
f(t) = f(t + nT)2 /#(/% n = 0 ±1 ± 2 ±!...
E(+#("%!: E! -;+< "+%= (0 -(",5( *%&,5/,"0
Ejemplo: C6) %! %) *%&,#/# /% )0-(",5(
S, f(t) %! *%&,5/,"0 !% /%?% ".*),&:
C#.# "#!;+ *< "#!;+< *0&0 "0)8,%& %(+%&# 2%(+#("%!2 *0&0 8% !% ".*)0 )0 ,0)/0/2 !% &%8,%&% 8%:
T * 3 T *=
E! /%",&:
T * *
"#( 3 %(+%&#!=
E) valor mínimo de T !% #?+,%(% "#( 2 2 %! /%",&2T = 2"#=
CALCULO III Año-2015
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UNIVERSIDAD ANDINA DEL CUSCO P.A.P.INGENIERIA CIVIL
SERIE TRIGONOMÉTRICA DE FOURIER=$
...)3()2()(...
...)3cos()2cos()cos()(
030201
030201021
++++
++++=
t senbt senbt senb
t at at aat f
ω ω ω
ω ω ω
A)(0! -(",#(%! *%&,5/,"0! -;+< /% *%&,#/# T *%/%(%*&%!0&!% *#& )0 !,,%(+% !%&,%2 ))0.0/0 !%&,%
+&,#(#.@+&,"0 /% F#&,%&:
D#(/% $0 = 2#%T !% /%(#.,(0 -&%"%(",0 -(/0.%(+0):
])()cos([)(
1
00021 ∑
∞
=
++=
n
nn t n senbt naat f ω ω
CALCULO III Año-2015
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UNIVERSIDAD ANDINA DEL CUSCO P.A.P.INGENIERIA CIVIL
ORTOGONALIDAD=$S% /,"% 8% )0! -(",#(%! /%) "#(4(+# -;+< !#(#&+##(0)%! %( %) ,(+%&90)# 0 W + W ? !, /#! -(",#(%!"0)%!8,%&0 -.;+<2 -(;+< /% /,"7# "#(4(+# ".*)%(:
Ejemplo: L0! -(",#(%! + 3 + !#( #&+##(0)%! %( %),(+%&90)# X W + W 2 30 8%:
Ejemplo: L0!-(",#(%! !%( + 3 "#! + !#( #&+##(0)%! %( %) ,(+%&90)# X*W + W*2 30 8%:
FUNCIONES PARES E IMPARES.-
Un !"nc#$n (%&'#$#c o no) s& #c& !"nc#$n %' (o con s#&*'+ %') s# s",'!#c &s s#*'#c '&s%&c*o / && 1&'*#c/2 &s &c#'2 / !"nc#$n !(*) &s %' s#!(*) 4 !(-*)
CALCULO III Año-2015
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UNIVERSIDAD ANDINA DEL CUSCO P.A.P.INGENIERIA CIVIL
E( -#&.0 !,.,)0&2 (0 -(",5( -;+< !% /,"% -(",5( ,.*0& ;#"#( !,.%+&>0 ,.*0&<2 !, ! &61"0 %! !,.@+&,"0 &%!*%"+# 0)#&,%(2 %! /%",&2 !, ".*)% )# !,,%(+%: $-;+< -;$+<
CALCULO III Año-2015
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UNIVERSIDAD ANDINA DEL CUSCO P.A.P.INGENIERIA CIVIL
Ejemplo: L0! !,,%(+%! -(",#(%! !#( *0&%! # ,.*0&%!
f(t) = t + 1%t &(t) = 1%(t 2+1).
So'*n,
C#.# f(-t) = -t - 1%t = - f(t) *#& )# +0(+# f(t) %! -(",5(,.*0&=
C#.# &(-t) = 1%((-t)2+1) = 1%(t 2+1) = &(t) *#& )# +0(+#&(t) %! -(",5( *0&=
SIMETRIAS COEFICIENTES DE FOURIER
CALCULO III Año-2015
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UNIVERSIDAD ANDINA DEL CUSCO P.A.P.INGENIERIA CIVIL
PROPIEDADES DE LAS FUNCIONES PARES E IMPARES=$
• S# /s !"nc#on&s ! 5 ,R-6R
son !"nc#on&s %'&s2 &n*onc&s ! 7 , '&s"/* "n !"nc#$n %'.
• S# /s !"nc#on&s ! 5 , R-6R
son !"nc#on&s #%'&s2 &n*onc&s ! 7 , '&s"/* "n !"nc#$n %'.
• S# / !"nc#$n ! &s "n !"nc#$n %' 5 s# / !"nc#$n , &s "n !"nc#$n#%'2 &n*onc&s ! 7 , '&s"/*' s&' "n !"nc#$n #%'.
En*onc&s To !"nc#$n !(*) s& %"&& &7%'&s' coo / s" & os!"nc#on&s co%on&n*&s & /s c"/&s / "n &s %' 5 / o*' &s #%'
SIMETRÍA DE MEDIA ONDA=$
U(0 -(",5( *%&,5/,"0 /% *%&,#/# T !% /,"% !,.@+&,"0 /%.%/,0 #(/02 !, ".*)% )0*&#*,%/0/:
E! /%",&2 !, %( ! &61"0 )0! *0&+%! (%0+,90! !#( (&%%4# /% )0! *#!,+,90! *%&# /%!*)00/0! .%/,# *%&,#/#:
CALCULO III Año-2015
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UNIVERSIDAD ANDINA DEL CUSCO P.A.P.INGENIERIA CIVIL
SIMETRÍA DE CUARTO DE ONDA=$S# / !"nc#$n %&'#$#c !(*) *#&n& s#&*'+ & &# on 5 &s &s"n !"nc#$n %' o #%' &n*onc&s s& #c& 8 F(*) *#&n& "n s#&*'+ &c"'*o & on.
L s&'#& & Fo"'#&' & c"/8"#&' !"nc#$n %&'#o#c !(*) *#&n& s#&*'# &c"'*o & on %'2 cons* so/o &
'on#cos #%'&s & *&'#nos &/cos&no
L s&'#& & Fo"'#&' & c"/8"#&' !"nc#on %&'#o#c !(*) *#&n& s#&*'# & c"'*o& on #%'2 cons* & 'on#cos#%'&s & *'#nos s&no
CÓMO Y DÓNDE SE APLICAN LAS SERIES DE FOURIER
INTRODUCCION
M"c9s &c"c#on&s & /s c#&nc#s s& !o'"/n con &'#1s %'c#/&s 5 s&'&s"&/1&n2 &n ocs#on&s2 &sco%on#&no / #nc$,n#* &n s&'#&s (s"s
#n!#n#*s). Ls s&'#&s s #n*&'&sn*&s son /s & %o*&nc#s 5 %o' s"%"&s*o/s & Fo"'#&'. :o &/ c'c*&' %&'#$#co & */&s s"s2 /s s&'#&s &
CALCULO III Año-2015
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UNIVERSIDAD ANDINA DEL CUSCO P.A.P.INGENIERIA CIVIL
Fo"'#&' s& %/#cn2 %o' &&%/o2 on& s"',&n %'oc&sos osc#/n*&s2 coooc"''& &n /s s&'#&s *&%o'/&s & n*"'/&; &con$#c2 &n &/&c*'$n#c (s&
%/#cn %o' &&%/o &n *&o'+ & s&</&s )2 &n c=s*#c o &n $%*#c. Los %'o>/&s *&$'#cos '&/c#onos con / con1&',&nc# & /s s&'#&s &Fo"'#&' 9n #%"/so 1nc&s !"n&n*/&s &n #s*#n*os >#*os & /s*&*#cs 5 s#,"&n s#&no cons#&'os 9o5 coo %'o>/&s "5#!+c#/&s.
APLICACIÓN EN PROCESAMIENTO DIGITAL DE SEÑALES
Es #%o'*n*& cons#&'' / %/#cc#$n & /s s&'#&s & !o"'#&'2 5 8"& &s*ss#'1&n "c9o &n &/ %'oc&s#&n*o #,#*/ & s&</&s2 / c"/ &s "n '& &/s c#&nc#s & #n,&n#&'+ 8"& s& 9 &s''o//o '%#&n*& &n /os =/*#os30 <os.
Es*& '%#o &s''o//o &s '&s"/*o & 1nc&s *&cno/$,#cos *n*o &n /oso'&no'&s #,#*/&s coo &n / !>'#cc#$n & c#'c"#*os #n*&,'os. Es*osc#'c"#*os #,#*/&s >'*os 5 '&/*#1&n*& '%#os 9n 9&c9o %os#>/&cons*'"#' s#s*&s #,#*/&s /*&n*& so!#s*#cos2 c%c&s & '&/#;'!"nc#on&s 5 *'&s &/ %'oc&so & s&</&s 8"& con1&nc#on/&n*& s&'&/#;>n n/$,#c&n*&2 s& '&/#c&n 9o5 &#n*& 9'?'& #,#*/2 s >'*o 5 &n"o s !#>/&. Es '&/&1n*& #!&'&nc#& &n*'& "n s&</n/$,#c 5 #,#*/ %' co%'&n&' &o' &/ %'oc&s#&n*o & s&</&s2 &/no>'& & "n s&</ n/$,#c s& &'#1 &/ 9&c9o & 8"& &s "n s&</n/o, / s&</ !+s#c 8"& s& '&%'&s&n* .
CALCULO III Año-2015
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