Análisis de Fourier en Espacios Complejos - Frecuencias "Negativas"
6 - Frecuencias "Negativas"
Tutorial creado por Michael Haag, Justin Romberg, Erika Jackson, Fara Meza. Extraido de: http://cnx.org/content/m12848/latest/
13 de Diciembre de 2006
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""<DIV class=para id=p1_ss3><FONT size=2>Si nos dan una frecuencia <SPAN class=mrow id=id5575592><SPAN class=mrow id=id5575596><SPAN class=mrow id=id5575602><SPAN class=mrow id=id5575604><SPAN class=mi1><EM>π</EM></SPAN><SPAN class=mo id=id5575608><</SPAN><SPAN class=mi1><EM>ω</EM></SPAN></SPAN><SPAN class=mo id=id5575613><</SPAN><SPAN class=mrow id=id5575617><SPAN class=mn>2</SPAN><SPAN class=mo id=id5575621></SPAN><SPAN class=mi1><EM>π</EM></SPAN></SPAN></SPAN></SPAN></SPAN>, entonces esta señal será representada en nuestro plano complejo como: </FONT></DIV>
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<DIV class=caption><SPAN class=caption-before><STRONG><FONT size=2>Subfigure 8.1</FONT></STRONG></SPAN></DIV></TD>
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<DIV class=caption><SPAN class=caption-before><STRONG><FONT size=2>Subfigure 8.2</FONT></STRONG></SPAN></DIV></TD></TR></TBODY></TABLE></DIV>
<DIV class=caption><FONT size=2><SPAN class=caption-before><STRONG>Figure 8: </STRONG></SPAN>Gráfica de nuestra senosoidal compleja con una frecuencia mayor que <SPAN class=mrow id=id5575849><SPAN class=mrow id=id5575753><SPAN class=mi1><EM>π</EM></SPAN></SPAN></SPAN>. </FONT></DIV></TD></TR></TBODY></TABLE>
<DIV class=para id=p2_ss3><FONT size=2>De nuestras imágenes mostradas arriba, el valor de nuestra senosoidal compleja en el plano complejo se puede interpretar como girar “hacia atrás” (en dirección de las manecillas del reloj) alrededor del círculo unitario con frecuencia <SPAN class=mrow id=id5575776><SPAN class=mrow id=id5575777><SPAN class=mrow id=id5575782><SPAN class=mrow id=id5575784><SPAN class=mn>2</SPAN><SPAN class=mo id=id5575787></SPAN><SPAN class=mi1><EM>π</EM></SPAN></SPAN><SPAN class=mo id=id5575790>−</SPAN><SPAN class=mi1><EM>ω</EM></SPAN></SPAN></SPAN></SPAN>. Girar en contra de las manecillas del reloj <SPAN class=mrow id=id3306861><SPAN class=mrow id=id3306865><SPAN class=mi1><EM>w</EM></SPAN></SPAN></SPAN> es lo mismo que girar en sentido de las manecillas del reloj <SPAN class=mrow id=id3306878><SPAN class=mrow id=id3306881><SPAN class=mrow id=id3306887><SPAN class=mrow id=id3306890><SPAN class=mn>2</SPAN><SPAN class=mo id=id3306894></SPAN><SPAN class=mi1><EM>π</EM></SPAN></SPAN><SPAN class=mo id=id3306897>−</SPAN><SPAN class=mi1><EM>ω</EM></SPAN></SPAN></SPAN></SPAN>. </FONT></DIV>
<DIV class=example id=eg5><FONT size=2><STRONG><SPAN class=example-before><FONT color=#336699>Example 5</FONT></SPAN><SPAN class=example-name> </SPAN></STRONG> </FONT>
<DIV class=para id=p1_eg5><FONT size=2>Graficaremos nuestro senosoidal compleja, <SPAN class=mrow id=id3306958><SPAN class=mrow id=id3306962><EM><SPAN><SPAN class=mi1>ⅇ</SPAN></SPAN><SUP><SPAN class=mrow id=id3306973><SPAN class=mi1>j</SPAN><SPAN class=mo id=id3306976></SPAN><SPAN class=mi1>ω</SPAN><SPAN class=mo id=id3306980></SPAN><SPAN class=mi1>n</SPAN></SPAN></SUP></EM></SPAN></SPAN>, donde tenemos </FONT><SPAN class=mrow id=id3307012><SPAN class=mrow id=id3307016><SPAN class=mrow id=id3307026><FONT size=2><SPAN class=mi1><EM>ω</EM></SPAN><SPAN class=mo id=id3307029>=</SPAN></FONT><SPAN class=mrow id=id3307031><FONT size=2> </FONT>
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<TD class=mfracaa><SPAN class=mrow id=id3307034><SPAN class=mrow id=id3307037><FONT size=2><SPAN class=mn>5</SPAN><SPAN class=mo id=id3307045></SPAN><SPAN class=mi1><EM>π</EM></SPAN></FONT></SPAN></SPAN></TD></TR>
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</SPAN></SPAN></SPAN></SPAN><FONT size=2>y <SPAN class=mrow id=id3307090><SPAN class=mrow id=id3307039><SPAN class=mrow id=id3307098><SPAN class=mi1><EM>n</EM></SPAN><SPAN class=mo id=id3307102>=</SPAN><SPAN class=mn>1</SPAN></SPAN></SPAN></SPAN>. </FONT></DIV>
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<DIV class=caption><FONT size=2><SPAN class=caption-before><STRONG>Figure 9: </STRONG></SPAN>La gráfica anterior de la frecuencia dada es idéntica a una donde </FONT><SPAN class=mrow id=id3307164><SPAN class=mrow id=id3307168><SPAN class=mrow id=id3307178><FONT size=2><SPAN class=mi1><EM>ω</EM></SPAN><SPAN class=mo id=id3307181>=</SPAN></FONT><SPAN class=mrow id=id3307184><FONT size=2><SPAN class=mo id=id3307185>−</SPAN><SPAN id=id3307187L>(</SPAN></FONT><SPAN id=id3307187M><SPAN class=mrow id=id3307192><FONT size=2> </FONT>
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<TD class=mfracaa><SPAN class=mrow id=id3307194><SPAN class=mrow id=id3307198><FONT size=2><SPAN class=mn>3</SPAN><SPAN class=mo id=id3307205></SPAN><SPAN class=mi1><EM>π</EM></SPAN></FONT></SPAN></SPAN></TD></TR>
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</FONT></SPAN></SPAN></SPAN></SPAN><FONT size=2>. </FONT></DIV></TD></TR></TBODY></TABLE>
<DIV class=para id=p2_eg5><FONT size=2>Esta gráfica es la misma que pondríamos con un senosoidal de frecuencia “negativa </FONT><SPAN class=mrow id=id3307266><SPAN class=mrow id=id3307268><SPAN class=mrow id=id3307273><SPAN class=mo id=id3307274><FONT size=2>−</FONT></SPAN><SPAN id=id3307276L><FONT size=2> </FONT>
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<TD><FONT size=2><FONT face=symbol>æ</FONT><BR><FONT face=symbol>è</FONT></FONT></TD></TR></TBODY></TABLE></SPAN><SPAN id=id3307276M><SPAN class=mrow id=id3307281>
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<TD class=mfracaa><SPAN class=mrow id=id3307283><SPAN class=mrow id=id3307286><FONT size=2><SPAN class=mn>3</SPAN><SPAN class=mo id=id3307293></SPAN><SPAN class=mi1><EM>π</EM></SPAN></FONT></SPAN></SPAN></TD></TR>
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<TD><SPAN class=mrow id=id3307296><SPAN class=mn><FONT size=2>4</FONT></SPAN></SPAN></TD></TR></TBODY></TABLE>
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<TD><FONT size=2><FONT face=symbol>ö</FONT><BR><FONT face=symbol>ø</FONT></FONT></TD></TR></TBODY></TABLE></SPAN>
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</SPAN></SPAN></SPAN><FONT size=2>. </FONT>
<DIV class=note><FONT size=2><SPAN class=note-before><STRONG><FONT color=#cc0000>point: </FONT></STRONG></SPAN>Tiene más sentido elegir un intervalo entre <SPAN class=mrow id=id3307349><SPAN class=mrow id=id3307350><SPAN style="COLOR: red"><apply></SPAN><SPAN id=id3307357L>[</SPAN><SPAN id=id3307357M><SPAN class=mrow id=id3307366><SPAN class=mo id=id3307367>−</SPAN><SPAN class=mi1><EM>π</EM></SPAN></SPAN><SPAN id=id3307366X1>,</SPAN><SPAN class=mi1><EM>π</EM></SPAN></SPAN><SPAN id=id3307357R>)</SPAN>
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<SPAN style="COLOR: red"></apply></SPAN></SPAN></SPAN> para <SPAN class=mrow id=id3307402><SPAN class=mrow id=id3307405><SPAN class=mi1><EM>ω</EM></SPAN></SPAN></SPAN>. </FONT></DIV></DIV></DIV>
<DIV class=para id=p3_ssub3><FONT size=2>Recuerde que <SPAN class=mrow id=id3307428><SPAN class=mrow id=id3307431><EM><SPAN><SPAN class=mi1>ⅇ</SPAN></SPAN><SUP><SPAN class=mrow id=id3307442><SPAN class=mi1>j</SPAN><SPAN class=mo id=id3307445></SPAN><SPAN class=mi1>ω</SPAN><SPAN class=mo id=id3307449></SPAN><SPAN class=mi1>n</SPAN></SPAN></SUP></EM></SPAN></SPAN> y <SPAN class=mrow id=id3307481><SPAN class=mrow id=id3307485><SPAN><SPAN class=mi1><EM>ⅇ</EM></SPAN></SPAN><SUP><SPAN class=mrow id=id3307496><SPAN class=mo id=id3307497>−</SPAN><SPAN id=id3307499L>(</SPAN><SPAN id=id3307499M><SPAN class=mrow id=id3307504><EM><SPAN class=mi1>j</SPAN><SPAN class=mo id=id3307507></SPAN><SPAN class=mi1>ω</SPAN><SPAN class=mo id=id3307511></SPAN><SPAN class=mi1>n</SPAN></EM></SPAN></SPAN><SPAN id=id3307499R>)</SPAN>
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</SPAN></SUP></SPAN></SPAN>son <SPAN class=term><STRONG>conjugados</STRONG></SPAN>. Esto nos da la siguiente notación y propiedad: </FONT>
<DIV class=equation id=eq6><FONT size=2><SPAN class=mrow id=id3307563><SPAN class=mrow id=id3307569><SPAN class=mrow id=id3307577><SPAN><EM><SPAN><SPAN class=mi1>ⅇ</SPAN></SPAN><SUP><SPAN class=mrow id=id3307585><SPAN class=mi1>j</SPAN><SPAN class=mo id=id3307589></SPAN><SPAN class=mi1>ω</SPAN><SPAN class=mo id=id3307592></SPAN><SPAN class=mi1>n</SPAN></SPAN></SUP></EM></SPAN><SUP><SPAN class=mo id=id3307596>*</SPAN></SUP></SPAN><SPAN class=mrow id=id3307599><SPAN class=mo id=id3307600>=</SPAN></SPAN><SPAN class=mrow id=id3307602><SPAN><SPAN class=mi1><EM>ⅇ</EM></SPAN></SPAN><SUP><SPAN class=mrow id=id3307610><SPAN class=mo id=id3307611>−</SPAN><SPAN id=id3307613L>(</SPAN><SPAN id=id3307613M><SPAN class=mrow id=id3307618><EM><SPAN class=mi1>j</SPAN><SPAN class=mo id=id3307621></SPAN><SPAN class=mi1>ω</SPAN><SPAN class=mo id=id3307624></SPAN><SPAN class=mi1>n</SPAN></EM></SPAN></SPAN><SPAN id=id3307613R>)</SPAN>
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</SPAN></SUP></SPAN></SPAN></SPAN><SPAN class=equation-number>(6)</SPAN> </FONT></DIV><FONT size=2>Las partes reales de ambas exponenciales de la ecuación de arriba son las mismas; la parte imaginaria son los negativos de una a la otra. Esta idea es la definición básica de un conjugado. </FONT></DIV>
<DIV class=para id=p4_sec2><FONT size=2>Ya que hemos visto los conceptos de senosoidales complejas, retomaremos la idea de encontrar una base para las señales periódicas en tiempo discreto. Después de observar las senosoidales complejas, tenemos que responder la pregunta sobre cuales senosoidales en tiempo discreto necesitamos para representar secuencias periódicas con periodo <SPAN class=mrow id=id3307720><SPAN class=mrow id=id3307724><SPAN class=mi1><EM>N</EM></SPAN></SPAN></SPAN>. </FONT>
<DIV class=note><FONT size=2><SPAN class=note-before><STRONG><FONT color=#cc0000>Equivalent Question: </FONT></STRONG></SPAN>Encuentre un conjunto de vectores <SPAN class=mrow id=id3307750><SPAN class=mrow id=id3307754><SPAN class=mrow id=id3307759><SPAN class=mrow id=id3307762><EM><SPAN><SPAN class=mrow id=id3307764><SPAN><SPAN class=mrow id=id3307768><SPAN class=mi1>b</SPAN></SPAN></SPAN></SPAN></SPAN><SUB><SPAN class=mrow id=id3307773><SPAN class=mi1>k</SPAN></SPAN></SUB></EM><SPAN class=mo id=id3307777>=</SPAN><EM><SPAN><SPAN class=mi1>ⅇ</SPAN></SPAN><SUP><SPAN class=mrow id=id3307786><SPAN class=mi1>j</SPAN><SPAN class=mo id=id3307789></SPAN><SPAN class=mrow id=id3307791><SPAN><SPAN class=mi1>ω</SPAN></SPAN><SUB><SPAN class=mi1>k</SPAN></SUB></SPAN><SPAN class=mo id=id3307803></SPAN><SPAN class=mi1>n</SPAN></SPAN></SUP></EM></SPAN><SPAN class=mo id=id3307806> </SPAN><SPAN class=mo id=id3307808>,</SPAN><SPAN class=mo id=id3307811> </SPAN><SPAN class=mrow id=id3307818><SPAN class=mi1><EM>n</EM></SPAN><SPAN class=mo id=id3307821>=</SPAN><SPAN class=mrow id=id3307823><SPAN id=id3307825L>{</SPAN><SPAN id=id3307825M><SPAN class=mn>0</SPAN><SPAN id=id3307827X1>,</SPAN><SPAN class=mi1><EM>…</EM></SPAN><SPAN id=id3307835X2>,</SPAN><SPAN class=mrow id=id3307840><SPAN class=mi1><EM>N</EM></SPAN><SPAN class=mo id=id3307843>−</SPAN><SPAN class=mn>1</SPAN></SPAN></SPAN><SPAN id=id3307825R>}</SPAN>
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</SPAN></SPAN><SPAN class=mo id=id3307849> </SPAN></SPAN></SPAN></SPAN> tal que <SPAN class=mrow id=id3307969><SPAN class=mrow id=id3307973><SPAN class=mrow id=id3307976><SPAN id=id3307979L>{</SPAN><SPAN id=id3307979M><EM><SPAN><SPAN class=mrow id=id3307989><SPAN><SPAN class=mrow id=id3307993><SPAN class=mi1>b</SPAN></SPAN></SPAN></SPAN></SPAN><SUB><SPAN class=mrow id=id3307998><SPAN class=mi1>k</SPAN></SPAN></SUB></EM></SPAN><SPAN id=id3307979R>}</SPAN>
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</SPAN></SPAN></SPAN>sea una <SPAN class=term><STRONG>base</STRONG></SPAN> para <SPAN class=mrow id=id3308034><SPAN class=mrow id=id3308037><SPAN class=mrow id=id3308042><EM><SPAN><SPAN class=mi1>ℂ</SPAN></SPAN><SUP><SPAN class=mi1>n</SPAN></SUP></EM></SPAN></SPAN></SPAN> </FONT></DIV><FONT size=2>Para resolver la pregunta de arriba, usaremos las senosoidales “Armónicos” con frecuencia fundamental de </FONT><SPAN class=mrow id=id3308074><SPAN class=mrow id=id3308078><SPAN class=mrow id=id3308085><FONT size=2><SPAN class=mrow id=id3308086><SPAN><SPAN class=mi1><EM>ω</EM></SPAN></SPAN><SUB><SPAN class=mn>0</SPAN></SUB></SPAN><SPAN class=mo id=id3308096>=</SPAN></FONT><SPAN class=mrow id=id3308100><FONT size=2> </FONT>
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<TD class=mfracaa><SPAN class=mrow id=id3308103><SPAN class=mrow id=id3308105><FONT size=2><SPAN class=mn>2</SPAN><SPAN class=mo id=id3308110></SPAN><SPAN class=mi1><EM>π</EM></SPAN></FONT></SPAN></SPAN></TD></TR>
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<TD><SPAN class=mrow id=id3308114><SPAN class=mi1><EM><FONT size=2>N</FONT></EM></SPAN></SPAN></TD></TR></TBODY></TABLE>
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<DIV class=equation id=eq7><FONT size=2><SPAN class=name>Senosoidales Armónicos</SPAN> </FONT><SPAN class=mrow id=id3308180><SPAN class=mrow id=id3308185><SPAN><SPAN class=mi1><EM><FONT size=2>ⅇ</FONT></EM></SPAN></SPAN><SUP><SPAN class=mrow id=id3308195><EM><FONT size=2><SPAN class=mi1>j</SPAN><SPAN class=mo id=id3308198></SPAN></FONT></EM><SPAN class=mrow id=id3308203><FONT size=2> </FONT>
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<TD class=mfracaa><SPAN class=mrow id=id3308205><SPAN class=mrow id=id3308209><FONT size=2><SPAN class=mn>2</SPAN><SPAN class=mo id=id3308216></SPAN><SPAN class=mi1><EM>π</EM></SPAN></FONT></SPAN></SPAN></TD></TR>
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<TD><SPAN class=mrow id=id3308219><SPAN class=mi1><EM><FONT size=2>N</FONT></EM></SPAN></SPAN></TD></TR></TBODY></TABLE>
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</SPAN><SPAN class=mo id=id3308223></SPAN><EM><FONT size=2><SPAN class=mi1>k</SPAN><SPAN class=mo id=id3308226></SPAN><SPAN class=mi1>n</SPAN></FONT></EM></SPAN></SUP></SPAN></SPAN><FONT size=2> <SPAN class=equation-number>(7)</SPAN> </FONT></DIV></DIV>
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<TD class=subfigure id=hs_subf1 vAlign=center><FONT size=2><IMG class=media src= http://www.wikilearning.com/imagescc/20402/hsin1.png" alt="image" /> ></FONT></TD>
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<DIV class=caption><FONT size=2><SPAN class=caption-before><STRONG>Subfigure 10.1: </STRONG></SPAN>Senosoidal armónico con <SPAN class=mrow id=id3308358><SPAN class=mrow id=id3308460><SPAN class=mrow id=id3308475><SPAN class=mi1><EM>k</EM></SPAN><SPAN class=mo id=id3308478>=</SPAN><SPAN class=mn>0</SPAN></SPAN></SPAN></SPAN> </FONT></DIV></TD>
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<DIV class=caption><FONT size=2><SPAN class=caption-before><STRONG>Subfigure 10.2: </STRONG></SPAN>Parte imaginaria del senosoidal, </FONT><SPAN class=mrow id=id3308473><SPAN class=mrow id=id3308400><SPAN class=mrow id=id3308405><FONT size=2><SPAN class=mi>Im</SPAN><SPAN class=mo id=id3308408></SPAN><SPAN id=id3308409L>(</SPAN></FONT><SPAN id=id3308409M><SPAN><SPAN class=mi1><EM><FONT size=2>ⅇ</FONT></EM></SPAN></SPAN><SUP><SPAN class=mrow id=id3308423><EM><FONT size=2><SPAN class=mi1>j</SPAN><SPAN class=mo id=id3308427></SPAN></FONT></EM><SPAN class=mrow id=id3308433><FONT size=2> </FONT>
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<TBODY>
<TR class=mfraca id=aid3308434 style="HEIGHT: 0px">
<TD class=mfracaa><SPAN class=mrow id=id3308435><SPAN class=mrow id=id3308439><FONT size=2><SPAN class=mn>2</SPAN><SPAN class=mo id=id3308446></SPAN><SPAN class=mi1><EM>π</EM></SPAN></FONT></SPAN></SPAN></TD></TR>
<TR class=mfracb id=bid3308434>
<TD><SPAN class=mrow id=id3308449><SPAN class=mi1><EM><FONT size=2>N</FONT></EM></SPAN></SPAN></TD></TR></TBODY></TABLE>
<SCRIPT type=text/javascript>
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else aid3308434.style.setExpression("height",bid3308434.offsetHeight );
</SCRIPT>
</SPAN><SPAN class=mo id=id7685758></SPAN><FONT size=2><SPAN class=mn>1</SPAN><SPAN class=mo id=id7685762></SPAN><SPAN class=mi1><EM>n</EM></SPAN></FONT></SPAN></SUP></SPAN><FONT size=2><SPAN id=id3308409R>)</SPAN>
<SCRIPT type=text/javascript>
var mrowH = id3308409M.offsetHeight;
mrowStretch(id3308409L,'æ','ç','ç','è');
mrowStretch(id3308409R,'ö','÷','÷','ø');</SCRIPT>
</FONT></SPAN></SPAN></SPAN><FONT size=2>, con <SPAN class=mrow id=id7685823><SPAN class=mrow id=id3308429><SPAN class=mrow id=id3308442><SPAN class=mi1><EM>k</EM></SPAN><SPAN class=mo id=id7685827>=</SPAN><SPAN class=mn>1</SPAN></SPAN></SPAN></SPAN> </FONT></DIV></TD>
<TD vAlign=top>
<DIV class=caption><FONT size=2><SPAN class=caption-before><STRONG>Subfigure 10.3: </STRONG></SPAN>Parte imaginaria del senosoidal, </FONT><SPAN class=mrow id=id7685862><SPAN class=mrow id=id7685866><SPAN class=mrow id=id7685873><FONT size=2><SPAN class=mi>Im</SPAN><SPAN class=mo id=id7685876></SPAN><SPAN id=id7685877L>(</SPAN></FONT><SPAN id=id7685877M><SPAN><SPAN class=mi1><EM><FONT size=2>ⅇ</FONT></EM></SPAN></SPAN><SUP><SPAN class=mrow id=id7685887><EM><FONT size=2><SPAN class=mi1>j</SPAN><SPAN class=mo id=id7685891></SPAN></FONT></EM><SPAN class=mrow id=id7685897><FONT size=2> </FONT>
<TABLE class=mfrac style="FONT-SIZE: 75%">
<TBODY>
<TR class=mfraca id=aid7685898 style="HEIGHT: 0px">
<TD class=mfracaa><SPAN class=mrow id=id7685899><SPAN class=mrow id=id7685903><FONT size=2><SPAN class=mn>2</SPAN><SPAN class=mo id=id7685910></SPAN><SPAN class=mi1><EM>π</EM></SPAN></FONT></SPAN></SPAN></TD></TR>
<TR class=mfracb id=bid7685898>
<TD><SPAN class=mrow id=id7685913><SPAN class=mi1><EM><FONT size=2>N</FONT></EM></SPAN></SPAN></TD></TR></TBODY></TABLE>
<SCRIPT type=text/javascript>
if ( aid7685898.offsetHeight > bid7685898.offsetHeight ) bid7685898.style.setExpression("height",aid7685898.offsetHeight );
else aid7685898.style.setExpression("height",bid7685898.offsetHeight );
</SCRIPT>
</SPAN><SPAN class=mo id=id7685917></SPAN><FONT size=2><SPAN class=mn>2</SPAN><SPAN class=mo id=id7685921></SPAN><SPAN class=mi1><EM>n</EM></SPAN></FONT></SPAN></SUP></SPAN><FONT size=2><SPAN id=id7685877R>)</SPAN>
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mrowStretch(id7685877L,'æ','ç','ç','è');
mrowStretch(id7685877R,'ö','÷','÷','ø');</SCRIPT>
</FONT></SPAN></SPAN></SPAN><FONT size=2>, con <SPAN class=mrow id=id7685988><SPAN class=mrow id=id7685901><SPAN class=mrow id=id7685989><SPAN class=mi1><EM>k</EM></SPAN><SPAN class=mo id=id7685994>=</SPAN><SPAN class=mn>2</SPAN></SPAN></SPAN></SPAN> </FONT></DIV></TD></TR></TBODY></TABLE></DIV>
<DIV class=caption><FONT size=2><SPAN class=caption-before><STRONG>Figure 10: </STRONG></SPAN>Ejemplos de nuestras armónicos senosoidales. </FONT></DIV></TD></TR></TBODY></TABLE>
<DIV class=para id=p5_sec2><SPAN class=mrow id=id7686035><SPAN class=mrow id=id7686038><SPAN><SPAN class=mi1><EM><FONT size=2>ⅇ</FONT></EM></SPAN></SPAN><SUP><SPAN class=mrow id=id7686048><EM><FONT size=2><SPAN class=mi1>j</SPAN><SPAN class=mo id=id7686051></SPAN></FONT></EM><SPAN class=mrow id=id7686054><FONT size=2> </FONT>
<TABLE class=mfrac style="FONT-SIZE: 75%">
<TBODY>
<TR class=mfraca id=aid7686055>
<TD class=mfracaa><SPAN class=mrow id=id7686056><SPAN class=mrow id=id7686058><FONT size=2><SPAN class=mn>2</SPAN><SPAN class=mo id=id7686064></SPAN><SPAN class=mi1><EM>π</EM></SPAN></FONT></SPAN></SPAN></TD></TR>
<TR class=mfracb id=bid7686055 style="HEIGHT: 18px">
<TD><SPAN class=mrow id=id7686067><SPAN class=mi1><EM><FONT size=2>N</FONT></EM></SPAN></SPAN></TD></TR></TBODY></TABLE>
<SCRIPT type=text/javascript>
if ( aid7686055.offsetHeight > bid7686055.offsetHeight ) bid7686055.style.setExpression("height",aid7686055.offsetHeight );
else aid7686055.style.setExpression("height",bid7686055.offsetHeight );
</SCRIPT>
</SPAN><SPAN class=mo id=id7686071></SPAN><EM><FONT size=2><SPAN class=mi1>k</SPAN><SPAN class=mo id=id7686074></SPAN><SPAN class=mi1>n</SPAN></FONT></EM></SPAN></SUP></SPAN></SPAN><FONT size=2> es periódica con periodo <SPAN class=mrow id=id7686135><SPAN class=mrow id=id7686137><SPAN class=mi1><EM>N</EM></SPAN></SPAN></SPAN> y tiene <SPAN class=mrow id=id7686152><SPAN class=mrow id=id7686156><SPAN class=mi1><EM>k</EM></SPAN></SPAN></SPAN> “ciclos” entre <SPAN class=mrow id=id7686167><SPAN class=mrow id=id7686171><SPAN class=mrow id=id7686178><SPAN class=mi1><EM>n</EM></SPAN><SPAN class=mo id=id7686181>=</SPAN><SPAN class=mn>0</SPAN></SPAN></SPAN></SPAN> y <SPAN class=mrow id=id7686204><SPAN class=mrow id=id7686208><SPAN class=mrow id=id7686214><SPAN class=mi1><EM>n</EM></SPAN><SPAN class=mo id=id7686217>=</SPAN><SPAN class=mrow id=id7686220><SPAN class=mi1><EM>N</EM></SPAN><SPAN class=mo id=id7686224>−</SPAN><SPAN class=mn>1</SPAN></SPAN></SPAN></SPAN></SPAN>. </FONT>
<DIV class=rule id=harmonic_sin><FONT size=2><STRONG><SPAN class=rule-before><FONT color=#336699>theorem 1</FONT></SPAN><SPAN class=rule-name> </SPAN></STRONG> </FONT>
<DIV class=statement>
<DIV class=para id=p1_state><FONT size=2>Si dejamos </FONT><SPAN class=mrow id=id7686297><SPAN class=mrow id=id7686300><SPAN class=mrow id=id7686304><SPAN class=mrow id=id7686314><FONT size=2><SPAN class=mrow id=id7686315><EM><SPAN class=mrow id=id7686316><SPAN><SPAN class=mi1>b</SPAN></SPAN><SUB><SPAN class=mi1>k</SPAN></SUB></SPAN><SPAN class=mo id=id7686327></SPAN></EM><SPAN id=id7686329L>[</SPAN><SPAN id=id7686329M><SPAN class=mi1><EM>n</EM></SPAN></SPAN><SPAN id=id7686329R>]</SPAN>
<SCRIPT type=text/javascript>
var mrowH = id7686329M.offsetHeight;
mrowStretch(id7686329L,'é','ê','ê','ë');
mrowStretch(id7686329R,'ù','ú','ú','û');</SCRIPT>
</SPAN><SPAN class=mo id=id7686336>=</SPAN></FONT><SPAN class=mrow id=id7686340><SPAN class=mrow id=id7686343><FONT size=2> </FONT>
<TABLE class=mfrac style="FONT-SIZE: 75%">
<TBODY>
<TR class=mfraca id=aid7686345>
<TD class=mfracaa><SPAN class=mrow id=id7686346><SPAN class=mn><FONT size=2>1</FONT></SPAN></SPAN></TD></TR>
<TR class=mfracb id=bid7686345 style="HEIGHT: 20px">
<TD><SPAN class=mrow id=id7686350><FONT size=2><SPAN class=msqrtx>\</SPAN><SPAN class=msqrt><SPAN class=mi1><EM>N</EM></SPAN></SPAN></FONT></SPAN></TD></TR></TBODY></TABLE>
<SCRIPT type=text/javascript>
if ( aid7686345.offsetHeight > bid7686345.offsetHeight ) bid7686345.style.setExpression("height",aid7686345.offsetHeight );
else aid7686345.style.setExpression("height",bid7686345.offsetHeight );
</SCRIPT>
</SPAN><SPAN class=mo id=id7686358></SPAN><SPAN><SPAN class=mi1><EM><FONT size=2>ⅇ</FONT></EM></SPAN></SPAN><SUP><SPAN class=mrow id=id7686365><EM><FONT size=2><SPAN class=mi1>j</SPAN><SPAN class=mo id=id7686369></SPAN></FONT></EM><SPAN class=mrow id=id7686376><FONT size=2> </FONT>
<TABLE class=mfrac style="FONT-SIZE: 75%">
<TBODY>
<TR class=mfraca id=aid7686377>
<TD class=mfracaa><SPAN class=mrow id=id7686378><SPAN class=mrow id=id7686382><FONT size=2><SPAN class=mn>2</SPAN><SPAN class=mo id=id7686389></SPAN><SPAN class=mi1><EM>π</EM></SPAN></FONT></SPAN></SPAN></TD></TR>
<TR class=mfracb id=bid7686377 style="HEIGHT: 18px">
<TD><SPAN class=mrow id=id7686392><SPAN class=mi1><EM><FONT size=2>N</FONT></EM></SPAN></SPAN></TD></TR></TBODY></TABLE>
<SCRIPT type=text/javascript>
if ( aid7686377.offsetHeight > bid7686377.offsetHeight ) bid7686377.style.setExpression("height",aid7686377.offsetHeight );
else aid7686377.style.setExpression("height",bid7686377.offsetHeight );
</SCRIPT>
</SPAN><SPAN class=mo id=id7686396></SPAN><EM><FONT size=2><SPAN class=mi1>k</SPAN><SPAN class=mo id=id7686399></SPAN><SPAN class=mi1>n</SPAN></FONT></EM></SPAN></SUP></SPAN></SPAN><FONT size=2><SPAN class=mo id=id7686403> </SPAN><SPAN class=mo id=id7686405>,</SPAN><SPAN class=mo id=id7686408> </SPAN><SPAN class=mrow id=id7686414><SPAN class=mi1><EM>n</EM></SPAN><SPAN class=mo id=id7686417>=</SPAN><SPAN class=mrow id=id7686419><SPAN id=id7686421L>{</SPAN><SPAN id=id7686421M><SPAN class=mn>0</SPAN><SPAN id=id7686306X1>,</SPAN><SPAN class=mi1><EM>…</EM></SPAN><SPAN id=id7686374X2>,</SPAN><SPAN class=mrow id=id7686426><SPAN class=mi1><EM>N</EM></SPAN><SPAN class=mo id=id7686429>−</SPAN><SPAN class=mn>1</SPAN></SPAN></SPAN><SPAN id=id7686421R>}</SPAN>
<SCRIPT type=text/javascript>
var mrowH = id7686421M.offsetHeight;
mrowStretch(id7686421L,'ì','ï','í','î');
mrowStretch(id7686421R,'ü','ú','ý','þ');</SCRIPT>
</SPAN></SPAN><SPAN class=mo id=id7686435> </SPAN></FONT></SPAN></SPAN></SPAN><FONT size=2> donde el término exponencial es un vector en <SPAN class=mrow id=id7686608><SPAN class=mrow id=id7686609><SPAN><SPAN class=mi>ℂ</SPAN></SPAN><SUP><SPAN class=mi1><EM>N</EM></SPAN></SUP></SPAN></SPAN>, entonces <SPAN class=mrow id=id7686638><SPAN class=mrow id=id7686642><SPAN class=mrow id=id7686649><SPAN style="COLOR: red"><apply></SPAN><SPAN class=mrow id=id7686653><SPAN id=id7686654L>{</SPAN><SPAN id=id7686654M><SPAN class=mrow id=id7686657><EM><SPAN><SPAN class=mi1>b</SPAN></SPAN><SUB><SPAN class=mi1>k</SPAN></SUB></EM></SPAN></SPAN><SPAN id=id7686654R>}</SPAN>
<SCRIPT type=text/javascript>
var mrowH = id7686654M.offsetHeight;
mrowStretch(id7686654L,'ì','ï','í','î');
mrowStretch(id7686654R,'ü','ú','ý','þ');</SCRIPT>
</SPAN><SPAN style="COLOR: red"></apply></SPAN><SPAN><SPAN class=mrow id=id7686675><SPAN class=mo id=id7686676>|</SPAN></SPAN>
<SCRIPT type=text/javascript>
var mrowH = id7686675.offsetHeight;
mrowStretch(id7686676,"ç","ç","ç","ç");</SCRIPT>
</SPAN><SUB><SPAN class=mrow id=id7686678><SPAN class=mrow id=id7686690><SPAN class=mi1><EM>k</EM></SPAN><SPAN class=mo id=id7686693>=</SPAN><SPAN class=mrow id=id7686695><SPAN id=id7686697L>{</SPAN><SPAN id=id7686697M><SPAN class=mn>0</SPAN><SPAN id=id7686702X1>,</SPAN><SPAN class=mi1><EM>…</EM></SPAN><SPAN id=id7686689X2>,</SPAN><SPAN class=mrow id=id7686710><SPAN class=mi1><EM>N</EM></SPAN><SPAN class=mo id=id7686714>−</SPAN><SPAN class=mn>1</SPAN></SPAN></SPAN><SPAN id=id7686697R>}</SPAN>
<SCRIPT type=text/javascript>
var mrowH = id7686697M.offsetHeight;
mrowStretch(id7686697L,'ì','ï','í','î');
mrowStretch(id7686697R,'ü','ú','ý','þ');</SCRIPT>
</SPAN></SPAN></SPAN></SUB></SPAN></SPAN></SPAN>es una <SPAN class=term><STRONG><FONT color=#996600>base ortonormal</FONT></STRONG></SPAN> para <SPAN class=mrow id=id7686808><SPAN class=mrow id=id7686809><SPAN><SPAN class=mi>ℂ</SPAN></SPAN><SUP><SPAN class=mi1><EM>N</EM></SPAN></SUP></SPAN></SPAN>. </FONT></DIV></DIV>
<DIV class=proof><FONT size=2><SPAN class=proof-before><STRONG><FONT color=#336699>Proof</FONT></STRONG></SPAN> </FONT>
<DIV class=para id=p1_prf><FONT size=2>Primero que todo, debemos demostrar que <SPAN class=mrow id=id7686858><SPAN class=mrow id=id7686861><SPAN style="COLOR: red"><apply></SPAN><SPAN class=mrow id=id7686869><SPAN id=id7686871L>{</SPAN><SPAN id=id7686871M><SPAN class=mrow id=id7686876><EM><SPAN><SPAN class=mi1>b</SPAN></SPAN><SUB><SPAN class=mi1>k</SPAN></SUB></EM></SPAN></SPAN><SPAN id=id7686871R>}</SPAN>
<SCRIPT type=text/javascript>
var mrowH = id7686871M.offsetHeight;
mrowStretch(id7686871L,'ì','ï','í','î');
mrowStretch(id7686871R,'ü','ú','ý','þ');</SCRIPT>
</SPAN><SPAN style="COLOR: red"></apply></SPAN></SPAN></SPAN> es ortonormal, <SPAN class=foreign><EM>por ejemplo </EM></SPAN><SPAN class=mrow id=id7686923><SPAN class=mrow id=id7686927><SPAN class=mrow id=id7686938><SPAN class=mrow id=id7686939><SPAN class=mo id=id7686941><</SPAN><SPAN class=mrow id=id7686943><EM><SPAN><SPAN class=mi1>b</SPAN></SPAN><SUB><SPAN class=mi1>k</SPAN></SUB></EM></SPAN><SPAN class=mo id=id7686955>,</SPAN><SPAN class=mrow id=id7686958><EM><SPAN><SPAN class=mi1>b</SPAN></SPAN><SUB><SPAN class=mi1>l</SPAN></SUB></EM></SPAN><SPAN class=mo id=id7686969>></SPAN></SPAN><SPAN class=mo id=id7686971>=</SPAN><SPAN class=mrow id=id7686975><EM><SPAN><SPAN class=mi1>δ</SPAN></SPAN><SUB><SPAN class=mrow id=id7686982><SPAN class=mi1>k</SPAN><SPAN class=mi1>l</SPAN></SPAN></SUB></EM></SPAN></SPAN></SPAN></SPAN> </FONT><SPAN class=mrow id=id7687060><SPAN class=mrow id=id7687061><SPAN class=mrow id=id7687070><FONT size=2><SPAN class=mrow id=id7687071><SPAN class=mo id=id7687073><</SPAN><SPAN class=mrow id=id7687075><EM><SPAN><SPAN class=mi1>b</SPAN></SPAN><SUB><SPAN class=mi1>k</SPAN></SUB></EM></SPAN><SPAN class=mo id=id7687087>,</SPAN><SPAN class=mrow id=id7687090><EM><SPAN><SPAN class=mi1>b</SPAN></SPAN><SUB><SPAN class=mi1>l</SPAN></SUB></EM></SPAN><SPAN class=mo id=id7687101>></SPAN></SPAN><SPAN class=mo id=id7687104>=</SPAN></FONT><SPAN class=mrow id=id7687108><FONT size=2> </FONT>
<TABLE class=munderover>
<TBODY>
<TR>
<TD><SPAN class=mrow id=id7687124><SPAN class=mrow id=id7687130><FONT size=2><SPAN class=mi1><EM>N</EM></SPAN><SPAN class=mo id=id7687133>−</SPAN><SPAN class=mn>1</SPAN></FONT></SPAN></SPAN></TD></TR>
<TR>
<TD><SPAN class=mo id=id7687111><FONT size=2>∑</FONT></SPAN></TD></TR>
<TR>
<TD><SPAN class=mrow id=id7687113><FONT size=2><SPAN class=mi1><EM>n</EM></SPAN><SPAN class=mo id=id7687118>=</SPAN><SPAN class=mn>0</SPAN></FONT></SPAN></TD></TR></TBODY></TABLE><FONT size=2><SPAN id=id7687139L>(</SPAN><SPAN id=id7687139M><SPAN class=mrow id=id7687143><SPAN class=mrow id=id7687147><EM><SPAN class=mrow id=id7687148><SPAN><SPAN class=mi1>b</SPAN></SPAN><SUB><SPAN class=mi1>k</SPAN></SUB></SPAN><SPAN class=mo id=id7687159></SPAN></EM><SPAN id=id7687163L>[</SPAN><SPAN id=id7687163M><SPAN class=mi1><EM>n</EM></SPAN></SPAN><SPAN id=id7687163R>]</SPAN>
<SCRIPT type=text/javascript>
var mrowH = id7687163M.offsetHeight;
mrowStretch(id7687163L,'é','ê','ê','ë');
mrowStretch(id7687163R,'ù','ú','ú','û');</SCRIPT>
</SPAN><SPAN class=mo id=id7687170></SPAN><SPAN><SPAN class=mrow id=id7687176><EM><SPAN class=mrow id=id7687177><SPAN><SPAN class=mi1>b</SPAN></SPAN><SUB><SPAN class=mi1>l</SPAN></SUB></SPAN><SPAN class=mo id=id7687188></SPAN></EM><SPAN id=id7687192L>[</SPAN><SPAN id=id7687192M><SPAN class=mi1><EM>n</EM></SPAN></SPAN><SPAN id=id7687192R>]</SPAN>
<SCRIPT type=text/javascript>
var mrowH = id7687192M.offsetHeight;
mrowStretch(id7687192L,'é','ê','ê','ë');
mrowStretch(id7687192R,'ù','ú','ú','û');</SCRIPT>
</SPAN></SPAN><SUP><SPAN class=mo id=id7687199>*</SPAN></SUP></SPAN></SPAN><SPAN id=id7687139R>)</SPAN>
<SCRIPT type=text/javascript>
var mrowH = id7687139M.offsetHeight;
mrowStretch(id7687139L,'æ','ç','ç','è');
mrowStretch(id7687139R,'ö','÷','÷','ø');</SCRIPT>
</FONT></SPAN><SPAN class=mo id=id7687201><FONT size=2>=</FONT></SPAN><SPAN class=mrow id=id7687204><SPAN class=mrow id=id7687206><FONT size=2> </FONT>
<TABLE class=mfrac style="FONT-SIZE: 75%">
<TBODY>
<TR class=mfraca id=aid7687207>
<TD class=mfracaa><SPAN class=mrow id=id7687208><SPAN class=mn><FONT size=2>1</FONT></SPAN></SPAN></TD></TR>
<TR class=mfracb id=bid7687207 style="HEIGHT: 20px">
<TD><SPAN class=mrow id=id7687211><SPAN class=mi1><EM><FONT size=2>N</FONT></EM></SPAN></SPAN></TD></TR></TBODY></TABLE>
<SCRIPT type=text/javascript>
if ( aid7687207.offsetHeight > bid7687207.offsetHeight ) bid7687207.style.setExpression("height",aid7687207.offsetHeight );
else aid7687207.style.setExpression("height",bid7687207.offsetHeight );
</SCRIPT>
</SPAN><SPAN class=mo id=id7687215></SPAN><SPAN class=mrow id=id7687218>
<TABLE class=munderover>
<TBODY>
<TR>
<TD><SPAN class=mrow id=id7687233><SPAN class=mrow id=id7687236><FONT size=2><SPAN class=mi1><EM>N</EM></SPAN><SPAN class=mo id=id7687239>−</SPAN><SPAN class=mn>1</SPAN></FONT></SPAN></SPAN></TD></TR>
<TR>
<TD><SPAN class=mo id=id7687220><FONT size=2>∑</FONT></SPAN></TD></TR>
<TR>
<TD><SPAN class=mrow id=id7687222><FONT size=2><SPAN class=mi1><EM>n</EM></SPAN><SPAN class=mo id=id7687228>=</SPAN><SPAN class=mn>0</SPAN></FONT></SPAN></TD></TR></TBODY></TABLE><SPAN id=id7687245L>
<TABLE class=lr>
<TBODY>
<TR>
<TD><FONT size=2><FONT face=symbol>æ</FONT><BR><FONT face=symbol>ç</FONT><BR><FONT face=symbol>è</FONT></FONT></TD></TR></TBODY></TABLE></SPAN><SPAN id=id7687245M><SPAN class=mrow id=id7687249><SPAN><SPAN class=mi1><EM><FONT size=2>ⅇ</FONT></EM></SPAN></SPAN><SUP><SPAN class=mrow id=id7687256><EM><FONT size=2><SPAN class=mi1>j</SPAN><SPAN class=mo id=id7687260></SPAN></FONT></EM><SPAN class=mrow id=id7687265><FONT size=2> </FONT>
<TABLE class=mfrac style="FONT-SIZE: 75%">
<TBODY>
<TR class=mfraca id=aid7687266>
<TD class=mfracaa><SPAN class=mrow id=id7687268><SPAN class=mrow id=id7687271><FONT size=2><SPAN class=mn>2</SPAN><SPAN class=mo id=id7767540></SPAN><SPAN class=mi1><EM>π</EM></SPAN></FONT></SPAN></SPAN></TD></TR>
<TR class=mfracb id=bid7687266 style="HEIGHT: 18px">
<TD><SPAN class=mrow id=id7687286><SPAN class=mi1><EM><FONT size=2>N</FONT></EM></SPAN></SPAN></TD></TR></TBODY></TABLE>
<SCRIPT type=text/javascript>
if ( aid7687266.offsetHeight > bid7687266.offsetHeight ) bid7687266.style.setExpression("height",aid7687266.offsetHeight );
else aid7687266.style.setExpression("height",bid7687266.offsetHeight );
</SCRIPT>
</SPAN><SPAN class=mo id=id7687290></SPAN><EM><FONT size=2><SPAN class=mi1>k</SPAN><SPAN class=mo id=id7687294></SPAN><SPAN class=mi1>n</SPAN></FONT></EM></SPAN></SUP><SPAN class=mo id=id7687298></SPAN><SPAN><SPAN class=mi1><EM><FONT size=2>ⅇ</FONT></EM></SPAN></SPAN><SUP><SPAN class=mrow id=id7687304><SPAN class=mo id=id7687305><FONT size=2>−</FONT></SPAN><SPAN id=id7687308L><FONT size=2> </FONT>
<TABLE class=lr>
<TBODY>
<TR>
<TD><FONT size=2><FONT face=symbol>æ</FONT><BR><FONT face=symbol>è</FONT></FONT></TD></TR></TBODY></TABLE></SPAN><SPAN id=id7687308M><SPAN class=mrow id=id7687312><EM><FONT size=2><SPAN class=mi1>j</SPAN><SPAN class=mo id=id7687315></SPAN></FONT></EM><SPAN class=mrow id=id7687318><FONT size=2> </FONT>
<TABLE class=mfrac style="FONT-SIZE: 75%">
<TBODY>
<TR class=mfraca id=aid7687319>
<TD class=mfracaa><SPAN class=mrow id=id7687320><SPAN class=mrow id=id7687322><FONT size=2><SPAN class=mn>2</SPAN><SPAN class=mo id=id7687328></SPAN><SPAN class=mi1><EM>π</EM></SPAN></FONT></SPAN></SPAN></TD></TR>
<TR class=mfracb id=bid7687319 style="HEIGHT: 18px">
<TD><SPAN class=mrow id=id7687331><SPAN class=mi1><EM><FONT size=2>N</FONT></EM></SPAN></SPAN></TD></TR></TBODY></TABLE>
<SCRIPT type=text/javascript>
if ( aid7687319.offsetHeight > bid7687319.offsetHeight ) bid7687319.style.setExpression("height",aid7687319.offsetHeight );
else aid7687319.style.setExpression("height",bid7687319.offsetHeight );
</SCRIPT>
</SPAN><SPAN class=mo id=id7687335></SPAN><EM><FONT size=2><SPAN class=mi1>l</SPAN><SPAN class=mo id=id7687339></SPAN><SPAN class=mi1>n</SPAN></FONT></EM></SPAN></SPAN><SPAN id=id7687308R><FONT size=2> </FONT>
<TABLE class=lr>
<TBODY>
<TR>
<TD><FONT size=2><FONT face=symbol>ö</FONT><BR><FONT face=symbol>ø</FONT></FONT></TD></TR></TBODY></TABLE></SPAN>
<SCRIPT type=text/javascript>
var mrowH = id7687308M.offsetHeight;
mrowStretch(id7687308L,'æ','ç','ç','è');
mrowStretch(id7687308R,'ö','÷','÷','ø');</SCRIPT>
</SPAN></SUP></SPAN></SPAN><SPAN id=id7687245R>
<TABLE class=lr>
<TBODY>
<TR>
<TD><FONT size=2><FONT face=symbol>ö</FONT><BR><FONT face=symbol>÷</FONT><BR><FONT face=symbol>ø</FONT></FONT></TD></TR></TBODY></TABLE></SPAN>
<SCRIPT type=text/javascript>
var mrowH = id7687245M.offsetHeight;
mrowStretch(id7687245L,'æ','ç','ç','è');
mrowStretch(id7687245R,'ö','÷','÷','ø');</SCRIPT>
</SPAN></SPAN></SPAN></SPAN></SPAN>
<DIV class=equation id=eq8><SPAN class=mrow id=id7687145><SPAN class=mrow id=id7687068><FONT size=2><SPAN class=mrow id=id7687324><SPAN class=mrow id=id7687067><SPAN class=mo id=id7687269><</SPAN><SPAN class=mrow id=id7981880><EM><SPAN><SPAN class=mi1>b</SPAN></SPAN><SUB><SPAN class=mi1>k</SPAN></SUB></EM></SPAN><SPAN class=mo id=id7981893>,</SPAN><SPAN class=mrow id=id7981896><EM><SPAN><SPAN class=mi1>b</SPAN></SPAN><SUB><SPAN class=mi1>l</SPAN></SUB></EM></SPAN><SPAN class=mo id=id7981908>></SPAN></SPAN></SPAN><SPAN class=mrow id=id7981910><SPAN class=mo id=id7981911>=</SPAN></SPAN></FONT><SPAN class=mrow id=id7981913><SPAN class=mrow id=id7981917><SPAN class=mrow id=id7981919><FONT size=2> </FONT>
<TABLE class=mfrac style="FONT-SIZE: 75%">
<TBODY>
<TR class=mfraca id=aid7981921>
<TD class=mfracaa><SPAN class=mrow id=id7981922><SPAN class=mn><FONT size=2>1</FONT></SPAN></SPAN></TD></TR>
<TR class=mfracb id=bid7981921 style="HEIGHT: 20px">
<TD><SPAN class=mrow id=id7981926><SPAN class=mi1><EM><FONT size=2>N</FONT></EM></SPAN></SPAN></TD></TR></TBODY></TABLE>
<SCRIPT type=text/javascript>
if ( aid7981921.offsetHeight > bid7981921.offsetHeight ) bid7981921.style.setExpression("height",aid7981921.offsetHeight );
else aid7981921.style.setExpression("height",bid7981921.offsetHeight );
</SCRIPT>
</SPAN><SPAN class=mo id=id7981930></SPAN><SPAN class=mrow id=id7981932>
<TABLE class=munderover>
<TBODY>
<TR>
<TD><SPAN class=mrow id=id7981948><SPAN class=mrow id=id7981955><FONT size=2><SPAN class=mi1><EM>N</EM></SPAN><SPAN class=mo id=id7981958>−</SPAN><SPAN class=mn>1</SPAN></FONT></SPAN></SPAN></TD></TR>
<TR>
<TD><SPAN class=mo id=id7981935><FONT size=2>∑</FONT></SPAN></TD></TR>
<TR>
<TD><SPAN class=mrow id=id7981937><FONT size=2><SPAN class=mi1><EM>n</EM></SPAN><SPAN class=mo id=id7981943>=</SPAN><SPAN class=mn>0</SPAN></FONT></SPAN></TD></TR></TBODY></TABLE><SPAN id=id7981964L>
<TABLE class=lr>
<TBODY>
<TR>
<TD><FONT size=2><FONT face=symbol>æ</FONT><BR><FONT face=symbol>è</FONT></FONT></TD></TR></TBODY></TABLE></SPAN><SPAN id=id7981964M><SPAN><SPAN class=mi1><EM><FONT size=2>ⅇ</FONT></EM></SPAN></SPAN><SUP><SPAN class=mrow id=id7981971><EM><FONT size=2><SPAN class=mi1>j</SPAN><SPAN class=mo id=id7981975></SPAN></FONT></EM><SPAN class=mrow id=id7981981><FONT size=2> </FONT>
<TABLE class=mfrac style="FONT-SIZE: 75%">
<TBODY>
<TR class=mfraca id=aid7981982>
<TD class=mfracaa><SPAN class=mrow id=id7981983><SPAN class=mrow id=id7981987><FONT size=2><SPAN class=mn>2</SPAN><SPAN class=mo id=id7981994></SPAN><SPAN class=mi1><EM>π</EM></SPAN></FONT></SPAN></SPAN></TD></TR>
<TR class=mfracb id=bid7981982 style="HEIGHT: 18px">
<TD><SPAN class=mrow id=id7981997><SPAN class=mi1><EM><FONT size=2>N</FONT></EM></SPAN></SPAN></TD></TR></TBODY></TABLE>
<SCRIPT type=text/javascript>
if ( aid7981982.offsetHeight > bid7981982.offsetHeight ) bid7981982.style.setExpression("height",aid7981982.offsetHeight );
else aid7981982.style.setExpression("height",bid7981982.offsetHeight );
</SCRIPT>
</SPAN><SPAN class=mo id=id7982001></SPAN><FONT size=2><SPAN id=id7982002L>(</SPAN><SPAN id=id7982002M><SPAN class=mrow id=id7982006><SPAN class=mi1><EM>l</EM></SPAN><SPAN class=mo id=id7982009>−</SPAN><SPAN class=mi1><EM>k</EM></SPAN></SPAN></SPAN><SPAN id=id7982002R>)</SPAN>
<SCRIPT type=text/javascript>
var mrowH = id7982002M.offsetHeight;
mrowStretch(id7982002L,'æ','ç','ç','è');
mrowStretch(id7982002R,'ö','÷','÷','ø');</SCRIPT>
<SPAN class=mo id=id7982014></SPAN><SPAN class=mi1><EM>n</EM></SPAN></FONT></SPAN></SUP></SPAN><SPAN id=id7981964R><FONT size=2> </FONT>
<TABLE class=lr>
<TBODY>
<TR>
<TD><FONT size=2><FONT face=symbol>ö</FONT><BR><FONT face=symbol>ø</FONT></FONT></TD></TR></TBODY></TABLE></SPAN>
<SCRIPT type=text/javascript>
var mrowH = id7981964M.offsetHeight;
mrowStretch(id7981964L,'æ','ç','ç','è');
mrowStretch(id7981964R,'ö','÷','÷','ø');</SCRIPT>
</SPAN></SPAN></SPAN></SPAN></SPAN><FONT size=2><SPAN class=equation-number>(8)</SPAN> </FONT></DIV><FONT size=2>Si <SPAN class=mrow id=id7981985><SPAN class=mrow id=id7981977><SPAN class=mrow id=id7982186><SPAN class=mi1><EM>l</EM></SPAN><SPAN class=mo id=id7982189>=</SPAN><SPAN class=mi1><EM>k</EM></SPAN></SPAN></SPAN></SPAN>, entonces </FONT>
<DIV class=equation id=eq9><SPAN class=mrow id=id7982219><SPAN class=mrow id=id7982224>
<TABLE class=mtable>
<TBODY>
<TR>
<TD><SPAN class=mrow id=id7982240><SPAN class=mrow id=id7982242><FONT size=2><SPAN class=mo id=id7982243><</SPAN><SPAN class=mrow id=id7982246><EM><SPAN><SPAN class=mi1>b</SPAN></SPAN><SUB><SPAN class=mi1>k</SPAN></SUB></EM></SPAN><SPAN class=mo id=id7982257>,</SPAN><SPAN class=mrow id=id7982260><EM><SPAN><SPAN class=mi1>b</SPAN></SPAN><SUB><SPAN class=mi1>l</SPAN></SUB></EM></SPAN><SPAN class=mo id=id7982272>></SPAN></FONT></SPAN></SPAN></TD>
<TD><SPAN class=mo id=id7982277><FONT size=2>=</FONT></SPAN></TD>
<TD><SPAN class=mrow id=id7982284><SPAN class=mrow id=id7982287><SPAN class=mrow id=id7982289>
<TABLE class=mfrac style="FONT-SIZE: 75%">
<TBODY>
<TR class=mfraca id=aid7982290 style="HEIGHT: 0px">
<TD class=mfracaa><SPAN class=mrow id=id7982291><SPAN class=mn><FONT size=2>1</FONT></SPAN></SPAN></TD></TR>
<TR class=mfracb id=bid7982290>
<TD><SPAN class=mrow id=id7982295><SPAN class=mi1><EM><FONT size=2>N</FONT></EM></SPAN></SPAN></TD></TR></TBODY></TABLE>
<SCRIPT type=text/javascript>
if ( aid7982290.offsetHeight > bid7982290.offsetHeight ) bid7982290.style.setExpression("height",aid7982290.offsetHeight );
else aid7982290.style.setExpression("height",bid7982290.offsetHeight );
</SCRIPT>
</SPAN><SPAN class=mo id=id7982299></SPAN><SPAN class=mrow id=id7982301>
<TABLE class=munderover>
<TBODY>
<TR>
<TD><SPAN class=mrow id=id7982317><SPAN class=mrow id=id7982324><FONT size=2><SPAN class=mi1><EM>N</EM></SPAN><SPAN class=mo id=id7982327>−</SPAN><SPAN class=mn>1</SPAN></FONT></SPAN></SPAN></TD></TR>
<TR>
<TD><SPAN class=mo id=id7982304><FONT size=2>∑</FONT></SPAN></TD></TR>
<TR>
<TD><SPAN class=mrow id=id7982306><FONT size=2><SPAN class=mi1><EM>n</EM></SPAN><SPAN class=mo id=id7982312>=</SPAN><SPAN class=mn>0</SPAN></FONT></SPAN></TD></TR></TBODY></TABLE><FONT size=2><SPAN id=id7982333L>(</SPAN><SPAN id=id7982333M><SPAN style="COLOR: red"><apply></SPAN><SPAN class=mn>1</SPAN><SPAN style="COLOR: red"></apply></SPAN></SPAN><SPAN id=id7982333R>)</SPAN>
<SCRIPT type=text/javascript>
var mrowH = id7982333M.offsetHeight;
mrowStretch(id7982333L,'æ','ç','ç','è');
mrowStretch(id7982333R,'ö','÷','÷','ø');</SCRIPT>
</FONT></SPAN></SPAN></SPAN></TD></TR>
<TR>
<TD><FONT size=2></FONT></TD>
<TD><SPAN class=mo id=id7982349><FONT size=2>=</FONT></SPAN></TD>
<TD><SPAN class=mrow id=id7982356><SPAN class=mn><FONT size=2>1</FONT></SPAN></SPAN></TD></TR></TBODY></TABLE>
<SCRIPT type=text/javascript></SCRIPT>
</SPAN></SPAN><FONT size=2><SPAN class=equation-number>(9)</SPAN> </FONT></DIV><FONT size=2>Si <SPAN class=mrow id=id7982322><SPAN class=mrow id=id7982485><SPAN class=mrow id=id7982489><SPAN class=mi1><EM>l</EM></SPAN><SPAN class=mo id=id7982493>≠</SPAN><SPAN class=mi1><EM>k</EM></SPAN></SPAN></SPAN></SPAN>, entonces tenemos que usar la “fórmula de sumatoria parcial” mostrada abajo: </FONT><SPAN class=mrow id=id7982518><SPAN class=mrow id=id7982522><SPAN class=mrow id=id7982529><SPAN class=mrow id=id7982530>
<TABLE class=munderover>
<TBODY>
<TR>
<TD><SPAN class=mrow id=id7982546><SPAN class=mrow id=id7982550><FONT size=2><SPAN class=mi1><EM>N</EM></SPAN><SPAN class=mo id=id7982553>−</SPAN><SPAN class=mn>1</SPAN></FONT></SPAN></SPAN></TD></TR>
<TR>
<TD><SPAN class=mo id=id7982533><FONT size=2>∑</FONT></SPAN></TD></TR>
<TR>
<TD><SPAN class=mrow id=id7982535><FONT size=2><SPAN class=mi1><EM>n</EM></SPAN><SPAN class=mo id=id7982540>=</SPAN><SPAN class=mn>0</SPAN></FONT></SPAN></TD></TR></TBODY></TABLE><FONT size=2><SPAN id=id7982559L>(</SPAN><SPAN id=id7982559M><EM><SPAN><SPAN class=mi1>α</SPAN></SPAN><SUP><SPAN class=mi1>n</SPAN></SUP></EM></SPAN><SPAN id=id7982559R>)</SPAN>
<SCRIPT type=text/javascript>
var mrowH = id7982559M.offsetHeight;
mrowStretch(id7982559L,'æ','ç','ç','è');
mrowStretch(id7982559R,'ö','÷','÷','ø');</SCRIPT>
</FONT></SPAN><SPAN class=mo id=id7982568><FONT size=2>=</FONT></SPAN><SPAN class=mrow id=id7982572><SPAN class=mrow id=id7982574><FONT size=2> </FONT>
<TABLE class=munderover>
<TBODY>
<TR>
<TD><SPAN class=mrow id=id7982588><SPAN class=mi1><EM><FONT size=2>∞</FONT></EM></SPAN></SPAN></TD></TR>
<TR>
<TD><SPAN class=mo id=id7982576><FONT size=2>∑</FONT></SPAN></TD></TR>
<TR>
<TD><SPAN class=mrow id=id7982578><FONT size=2><SPAN class=mi1><EM>n</EM></SPAN><SPAN class=mo id=id7982583>=</SPAN><SPAN class=mn>0</SPAN></FONT></SPAN></TD></TR></TBODY></TABLE><FONT size=2><SPAN id=id7982593L>(</SPAN><SPAN id=id7982593M><EM><SPAN><SPAN class=mi1>α</SPAN></SPAN><SUP><SPAN class=mi1>n</SPAN></SUP></EM></SPAN><SPAN id=id7982593R>)</SPAN>
<SCRIPT type=text/javascript>
var mrowH = id7982593M.offsetHeight;
mrowStretch(id7982593L,'æ','ç','ç','è');
mrowStretch(id7982593R,'ö','÷','÷','ø');</SCRIPT>
</FONT></SPAN><SPAN class=mo id=id7982603><FONT size=2>−</FONT></SPAN><SPAN class=mrow id=id7982607><FONT size=2> </FONT>
<TABLE class=munderover>
<TBODY>
<TR>
<TD><SPAN class=mrow id=id7982620><SPAN class=mi1><EM><FONT size=2>∞</FONT></EM></SPAN></SPAN></TD></TR>
<TR>
<TD><SPAN class=mo id=id7982609><FONT size=2>∑</FONT></SPAN></TD></TR>
<TR>
<TD><SPAN class=mrow id=id7982611><FONT size=2><SPAN class=mi1><EM>n</EM></SPAN><SPAN class=mo id=id7982615>=</SPAN><SPAN class=mi1><EM>N</EM></SPAN></FONT></SPAN></TD></TR></TBODY></TABLE><FONT size=2><SPAN id=id7982626L>(</SPAN><SPAN id=id7982626M><EM><SPAN><SPAN class=mi1>α</SPAN></SPAN><SUP><SPAN class=mi1>n</SPAN></SUP></EM></SPAN><SPAN id=id7982626R>)</SPAN>
<SCRIPT type=text/javascript>
var mrowH = id7982626M.offsetHeight;
mrowStretch(id7982626L,'æ','ç','ç','è');
mrowStretch(id7982626R,'ö','÷','÷','ø');</SCRIPT>
</FONT></SPAN></SPAN><SPAN class=mo id=id7982635><FONT size=2>=</FONT></SPAN><SPAN class=mrow id=id7982638><SPAN class=mrow id=id7982640><FONT size=2> </FONT>
<TABLE class=mfrac style="FONT-SIZE: 75%">
<TBODY>
<TR class=mfraca id=aid7982641>
<TD class=mfracaa><SPAN class=mrow id=id7982642><SPAN class=mn><FONT size=2>1</FONT></SPAN></SPAN></TD></TR>
<TR class=mfracb id=bid7982641 style="HEIGHT: 20px">
<TD><SPAN class=mrow id=id7982646><SPAN class=mrow id=id7982648><FONT size=2><SPAN class=mn>1</SPAN><SPAN class=mo id=id7982652>−</SPAN><SPAN class=mi1><EM>α</EM></SPAN></FONT></SPAN></SPAN></TD></TR></TBODY></TABLE>
<SCRIPT type=text/javascript>
if ( aid7982641.offsetHeight > bid7982641.offsetHeight ) bid7982641.style.setExpression("height",aid7982641.offsetHeight );
else aid7982641.style.setExpression("height",bid7982641.offsetHeight );
</SCRIPT>
</SPAN><SPAN class=mo id=id7982657><FONT size=2>−</FONT></SPAN><SPAN class=mrow id=id7982660><FONT size=2> </FONT>
<TABLE class=mfrac style="FONT-SIZE: 75%">
<TBODY>
<TR class=mfraca id=aid7982661>
<TD class=mfracaa><SPAN class=mrow id=id7982662><EM><FONT size=2><SPAN><SPAN class=mi1>α</SPAN></SPAN><SUP><SPAN class=mi1>N</SPAN></SUP></FONT></EM></SPAN></TD></TR>
<TR class=mfracb id=bid7982661 style="HEIGHT: 22px">
<TD><SPAN class=mrow id=id7982670><SPAN class=mrow id=id7982672><FONT size=2><SPAN class=mn>1</SPAN><SPAN class=mo id=id7982676>−</SPAN><SPAN class=mi1><EM>α</EM></SPAN></FONT></SPAN></SPAN></TD></TR></TBODY></TABLE>
<SCRIPT type=text/javascript>
if ( aid7982661.offsetHeight > bid7982661.offsetHeight ) bid7982661.style.setExpression("height",aid7982661.offsetHeight );
else aid7982661.style.setExpression("height",bid7982661.offsetHeight );
</SCRIPT>
</SPAN></SPAN><SPAN class=mo id=id7982681><FONT size=2>=</FONT></SPAN><SPAN class=mrow id=id7982684><FONT size=2> </FONT>
<TABLE class=mfrac style="FONT-SIZE: 75%">
<TBODY>
<TR class=mfraca id=aid7982685>
<TD class=mfracaa><SPAN class=mrow id=id7982686><SPAN class=mrow id=id7982688><FONT size=2><SPAN class=mn>1</SPAN><SPAN class=mo id=id7982692>−</SPAN><EM><SPAN><SPAN class=mi1>α</SPAN></SPAN><SUP><SPAN class=mi1>N</SPAN></SUP></EM></FONT></SPAN></SPAN></TD></TR>
<TR class=mfracb id=bid7982685 style="HEIGHT: 22px">
<TD><SPAN class=mrow id=id7982701><SPAN class=mrow id=id7982703><FONT size=2><SPAN class=mn>1</SPAN><SPAN class=mo id=id7982706>−</SPAN><SPAN class=mi1><EM>α</EM></SPAN></FONT></SPAN></SPAN></TD></TR></TBODY></TABLE>
<SCRIPT type=text/javascript>
if ( aid7982685.offsetHeight > bid7982685.offsetHeight ) bid7982685.style.setExpression("height",aid7982685.offsetHeight );
else aid7982685.style.setExpression("height",bid7982685.offsetHeight );
</SCRIPT>
</SPAN></SPAN></SPAN></SPAN><SPAN class=mrow id=id7982993><SPAN class=mrow id=id7982996><SPAN class=mrow id=id7983000><FONT size=2><SPAN class=mrow id=id7983001><SPAN class=mo id=id7983003><</SPAN><SPAN class=mrow id=id7983006><EM><SPAN><SPAN class=mi1>b</SPAN></SPAN><SUB><SPAN class=mi1>k</SPAN></SUB></EM></SPAN><SPAN class=mo id=id7983017>,</SPAN><SPAN class=mrow id=id7983020><EM><SPAN><SPAN class=mi1>b</SPAN></SPAN><SUB><SPAN class=mi1>l</SPAN></SUB></EM></SPAN><SPAN class=mo id=id7983032>></SPAN></SPAN><SPAN class=mo id=id7983034>=</SPAN></FONT><SPAN class=mrow id=id7983038><SPAN class=mrow id=id7983041><FONT size=2> </FONT>
<TABLE class=mfrac style="FONT-SIZE: 75%">
<TBODY>
<TR class=mfraca id=aid7983043>
<TD class=mfracaa><SPAN class=mrow id=id7983044><SPAN class=mn><FONT size=2>1</FONT></SPAN></SPAN></TD></TR>
<TR class=mfracb id=bid7983043 style="HEIGHT: 20px">
<TD><SPAN class=mrow id=id7983048><SPAN class=mi1><EM><FONT size=2>N</FONT></EM></SPAN></SPAN></TD></TR></TBODY></TABLE>
<SCRIPT type=text/javascript>
if ( aid7983043.offsetHeight > bid7983043.offsetHeight ) bid7983043.style.setExpression("height",aid7983043.offsetHeight );
else aid7983043.style.setExpression("height",bid7983043.offsetHeight );
</SCRIPT>
</SPAN><SPAN class=mo id=id7983052></SPAN><SPAN class=mrow id=id7983054>
<TABLE class=munderover>
<TBODY>
<TR>
<TD><SPAN class=mrow id=id7983070><SPAN class=mrow id=id7983077><FONT size=2><SPAN class=mi1><EM>N</EM></SPAN><SPAN class=mo id=id7983080>−</SPAN><SPAN class=mn>1</SPAN></FONT></SPAN></SPAN></TD></TR>
<TR>
<TD><SPAN class=mo id=id7983057><FONT size=2>∑</FONT></SPAN></TD></TR>
<TR>
<TD><SPAN class=mrow id=id7983059><FONT size=2><SPAN class=mi1><EM>n</EM></SPAN><SPAN class=mo id=id7983065>=</SPAN><SPAN class=mn>0</SPAN></FONT></SPAN></TD></TR></TBODY></TABLE><SPAN id=id7983086L>
<TABLE class=lr>
<TBODY>
<TR>
<TD><FONT size=2><FONT face=symbol>æ</FONT><BR><FONT face=symbol>è</FONT></FONT></TD></TR></TBODY></TABLE></SPAN><SPAN id=id7983086M><SPAN><SPAN class=mi1><EM><FONT size=2>ⅇ</FONT></EM></SPAN></SPAN><SUP><SPAN class=mrow id=id7983094><EM><FONT size=2><SPAN class=mi1>j</SPAN><SPAN class=mo id=id7983098></SPAN></FONT></EM><SPAN class=mrow id=id7983104><FONT size=2> </FONT>
<TABLE class=mfrac style="FONT-SIZE: 75%">
<TBODY>
<TR class=mfraca id=aid7983105>
<TD class=mfracaa><SPAN class=mrow id=id7983106><SPAN class=mrow id=id7983110><FONT size=2><SPAN class=mn>2</SPAN><SPAN class=mo id=id7983117></SPAN><SPAN class=mi1><EM>π</EM></SPAN></FONT></SPAN></SPAN></TD></TR>
<TR class=mfracb id=bid7983105 style="HEIGHT: 18px">
<TD><SPAN class=mrow id=id7983120><SPAN class=mi1><EM><FONT size=2>N</FONT></EM></SPAN></SPAN></TD></TR></TBODY></TABLE>
<SCRIPT type=text/javascript>
if ( aid7983105.offsetHeight > bid7983105.offsetHeight ) bid7983105.style.setExpression("height",aid7983105.offsetHeight );
else aid7983105.style.setExpression("height",bid7983105.offsetHeight );
</SCRIPT>
</SPAN><SPAN class=mo id=id7983124></SPAN><FONT size=2><SPAN id=id7983125L>(</SPAN><SPAN id=id7983125M><SPAN class=mrow id=id7983130><SPAN class=mi1><EM>l</EM></SPAN><SPAN class=mo id=id7983133>−</SPAN><SPAN class=mi1><EM>k</EM></SPAN></SPAN></SPAN><SPAN id=id7983125R>)</SPAN>
<SCRIPT type=text/javascript>
var mrowH = id7983125M.offsetHeight;
mrowStretch(id7983125L,'æ','ç','ç','è');
mrowStretch(id7983125R,'ö','÷','÷','ø');</SCRIPT>
<SPAN class=mo id=id7983138></SPAN><SPAN class=mi1><EM>n</EM></SPAN></FONT></SPAN></SUP></SPAN><SPAN id=id7983086R><FONT size=2> </FONT>
<TABLE class=lr>
<TBODY>
<TR>
<TD><FONT size=2><FONT face=symbol>ö</FONT><BR><FONT face=symbol>ø</FONT></FONT></TD></TR></TBODY></TABLE></SPAN>
<SCRIPT type=text/javascript>
var mrowH = id7983086M.offsetHeight;
mrowStretch(id7983086L,'æ','ç','ç','è');
mrowStretch(id7983086R,'ö','÷','÷','ø');</SCRIPT>
</SPAN></SPAN></SPAN></SPAN></SPAN><FONT size=2>en esta ecuación podemos decir que </FONT><SPAN class=mrow id=id7983307><SPAN class=mrow id=id7983102><SPAN class=mrow id=id7983111><FONT size=2><SPAN class=mi1><EM>α</EM></SPAN><SPAN class=mo id=id7983074>=</SPAN><SPAN><SPAN class=mi1><EM>ⅇ</EM></SPAN></SPAN></FONT><SUP><SPAN class=mrow id=id7983315><EM><FONT size=2><SPAN class=mi1>j</SPAN><SPAN class=mo id=id7983319></SPAN></FONT></EM><SPAN class=mrow id=id7983324><FONT size=2> </FONT>
<TABLE class=mfrac style="FONT-SIZE: 75%">
<TBODY>
<TR class=mfraca id=aid7983326>
<TD class=mfracaa><SPAN class=mrow id=id7983327><SPAN class=mrow id=id7983330><FONT size=2><SPAN class=mn>2</SPAN><SPAN class=mo id=id7983337></SPAN><SPAN class=mi1><EM>π</EM></SPAN></FONT></SPAN></SPAN></TD></TR>
<TR class=mfracb id=bid7983326 style="HEIGHT: 18px">
<TD><SPAN class=mrow id=id7983340><SPAN class=mi1><EM><FONT size=2>N</FONT></EM></SPAN></SPAN></TD></TR></TBODY></TABLE>
<SCRIPT type=text/javascript>
if ( aid7983326.offsetHeight > bid7983326.offsetHeight ) bid7983326.style.setExpression("height",aid7983326.offsetHeight );
else aid7983326.style.setExpression("height",bid7983326.offsetHeight );
</SCRIPT>
</SPAN><SPAN class=mo id=id7983344></SPAN><FONT size=2><SPAN id=id7983346L>(</SPAN><SPAN id=id7983346M><SPAN class=mrow id=id7983349><SPAN class=mi1><EM>l</EM></SPAN><SPAN class=mo id=id7983352>−</SPAN><SPAN class=mi1><EM>k</EM></SPAN></SPAN></SPAN><SPAN id=id7983346R>)</SPAN>
<SCRIPT type=text/javascript>
var mrowH = id7983346M.offsetHeight;
mrowStretch(id7983346L,'æ','ç','ç','è');
mrowStretch(id7983346R,'ö','÷','÷','ø');</SCRIPT>
</FONT></SPAN></SUP></SPAN></SPAN></SPAN><FONT size=2>, así podemos ver como esta expresión se encuentra en la forma que necesitamos utilizar para nuestra fórmula de sumatoria parcial. </FONT><SPAN class=mrow id=id7983428><SPAN class=mrow id=id7983321><SPAN class=mrow id=id7983430><FONT size=2><SPAN class=mrow id=id7983431><SPAN class=mo id=id7983434><</SPAN><SPAN class=mrow id=id7983437><EM><SPAN><SPAN class=mi1>b</SPAN></SPAN><SUB><SPAN class=mi1>k</SPAN></SUB></EM></SPAN><SPAN class=mo id=id7983448>,</SPAN><SPAN class=mrow id=id7983451><EM><SPAN><SPAN class=mi1>b</SPAN></SPAN><SUB><SPAN class=mi1>l</SPAN></SUB></EM></SPAN><SPAN class=mo id=id7983462>></SPAN></SPAN><SPAN class=mo id=id7983465>=</SPAN></FONT><SPAN class=mrow id=id7983469><SPAN class=mrow id=id7983474><FONT size=2> </FONT>
<TABLE class=mfrac style="FONT-SIZE: 75%">
<TBODY>
<TR class=mfraca id=aid7983476>
<TD class=mfracaa><SPAN class=mrow id=id7983477><SPAN class=mn><FONT size=2>1</FONT></SPAN></SPAN></TD></TR>
<TR class=mfracb id=bid7983476 style="HEIGHT: 20px">
<TD><SPAN class=mrow id=id7983481><SPAN class=mi1><EM><FONT size=2>N</FONT></EM></SPAN></SPAN></TD></TR></TBODY></TABLE>
<SCRIPT type=text/javascript>
if ( aid7983476.offsetHeight > bid7983476.offsetHeight ) bid7983476.style.setExpression("height",aid7983476.offsetHeight );
else aid7983476.style.setExpression("height",bid7983476.offsetHeight );
</SCRIPT>
</SPAN><SPAN class=mo id=id7983485></SPAN><SPAN id=id7983494L>
<TABLE class=lr>
<TBODY>
<TR>
<TD><FONT size=2><FONT face=symbol>æ</FONT><BR><FONT face=symbol>ç</FONT><BR><FONT face=symbol>è</FONT></FONT></TD></TR></TBODY></TABLE></SPAN><SPAN id=id7983494M><SPAN class=mrow id=id7983497>
<TABLE class=mfrac style="FONT-SIZE: 75%">
<TBODY>
<TR class=mfraca id=aid7983498>
<TD class=mfracaa><SPAN class=mrow id=id7983500><SPAN class=mrow id=id7983502><FONT size=2><SPAN class=mn>1</SPAN><SPAN class=mo id=id7785911>−</SPAN><SPAN><SPAN class=mi1><EM>ⅇ</EM></SPAN></SPAN></FONT><SUP><SPAN class=mrow id=id7785921><EM><FONT size=2><SPAN class=mi1>j</SPAN><SPAN class=mo id=id7785925></SPAN></FONT></EM><SPAN class=mrow id=id7785931><FONT size=2> </FONT>
<TABLE class=mfrac style="FONT-SIZE: 100%">
<TBODY>
<TR class=mfraca id=aid7785932>
<TD class=mfracaa><SPAN class=mrow id=id7785933><SPAN class=mrow id=id7785937><FONT size=2><SPAN class=mn>2</SPAN><SPAN class=mo id=id7785944></SPAN><SPAN class=mi1><EM>π</EM></SPAN></FONT></SPAN></SPAN></TD></TR>
<TR class=mfracb id=bid7785932>
<TD><SPAN class=mrow id=id7785947><SPAN class=mi1><EM><FONT size=2>N</FONT></EM></SPAN></SPAN></TD></TR></TBODY></TABLE></SPAN><SPAN class=mo id=id7785951></SPAN><FONT size=2><SPAN id=id7785952L>(</SPAN><SPAN id=id7785952M><SPAN class=mrow id=id7785956><SPAN class=mi1><EM>l</EM></SPAN><SPAN class=mo id=id7785959>−</SPAN><SPAN class=mi1><EM>k</EM></SPAN></SPAN></SPAN><SPAN id=id7785952R>)</SPAN>
<SCRIPT type=text/javascript>
var mrowH = id7785952M.offsetHeight;
mrowStretch(id7785952L,'æ','ç','ç','è');
mrowStretch(id7785952R,'ö','÷','÷','ø');</SCRIPT>
<SPAN class=mo id=id7785964></SPAN><SPAN class=mi1><EM>N</EM></SPAN></FONT></SPAN></SUP></SPAN></SPAN></TD></TR>
<TR class=mfracb id=bid7983498 style="HEIGHT: 40px">
<TD><SPAN class=mrow id=id7785967><SPAN class=mrow id=id7785969><FONT size=2><SPAN class=mn>1</SPAN><SPAN class=mo id=id7785972>−</SPAN><SPAN><SPAN class=mi1><EM>ⅇ</EM></SPAN></SPAN></FONT><SUP><SPAN class=mrow id=id7785980><EM><FONT size=2><SPAN class=mi1>j</SPAN><SPAN class=mo id=id7785984></SPAN></FONT></EM><SPAN class=mrow id=id7785986><FONT size=2> </FONT>
<TABLE class=mfrac style="FONT-SIZE: 100%">
<TBODY>
<TR class=mfraca id=aid7785987>
<TD class=mfracaa><SPAN class=mrow id=id7785988><SPAN class=mrow id=id7785991><FONT size=2><SPAN class=mn>2</SPAN><SPAN class=mo id=id7785994></SPAN><SPAN class=mi1><EM>π</EM></SPAN></FONT></SPAN></SPAN></TD></TR>
<TR class=mfracb id=bid7785987>
<TD><SPAN class=mrow id=id7785998><SPAN class=mi1><EM><FONT size=2>N</FONT></EM></SPAN></SPAN></TD></TR></TBODY></TABLE></SPAN><SPAN class=mo id=id7786001></SPAN><FONT size=2><SPAN id=id7786003L>(</SPAN><SPAN id=id7786003M><SPAN class=mrow id=id7786007><SPAN class=mi1><EM>l</EM></SPAN><SPAN class=mo id=id7786010>−</SPAN><SPAN class=mi1><EM>k</EM></SPAN></SPAN></SPAN><SPAN id=id7786003R>)</SPAN>
<SCRIPT type=text/javascript>
var mrowH = id7786003M.offsetHeight;
mrowStretch(id7786003L,'æ','ç','ç','è');
mrowStretch(id7786003R,'ö','÷','÷','ø');</SCRIPT>
</FONT></SPAN></SUP></SPAN></SPAN></TD></TR></TBODY></TABLE>
<SCRIPT type=text/javascript>
if ( aid7983498.offsetHeight > bid7983498.offsetHeight ) bid7983498.style.setExpression("height",aid7983498.offsetHeight );
else aid7983498.style.setExpression("height",bid7983498.offsetHeight );
</SCRIPT>
</SPAN></SPAN><SPAN id=id7983494R>
<TABLE class=lr>
<TBODY>
<TR>
<TD><FONT size=2><FONT face=symbol>ö</FONT><BR><FONT face=symbol>÷</FONT><BR><FONT face=symbol>ø</FONT></FONT></TD></TR></TBODY></TABLE></SPAN>
<SCRIPT type=text/javascript>
var mrowH = id7983494M.offsetHeight;
mrowStretch(id7983494L,'æ','ç','ç','è');
mrowStretch(id7983494R,'ö','÷','÷','ø');</SCRIPT>
</SPAN><SPAN class=mo id=id7786016><FONT size=2>=</FONT></SPAN><SPAN class=mrow id=id7786018><SPAN class=mrow id=id7786019><FONT size=2> </FONT>
<TABLE class=mfrac style="FONT-SIZE: 75%">
<TBODY>
<TR class=mfraca id=aid7786021>
<TD class=mfracaa><SPAN class=mrow id=id7786022><SPAN class=mn><FONT size=2>1</FONT></SPAN></SPAN></TD></TR>
<TR class=mfracb id=bid7786021 style="HEIGHT: 20px">
<TD><SPAN class=mrow id=id7786026><SPAN class=mi1><EM><FONT size=2>N</FONT></EM></SPAN></SPAN></TD></TR></TBODY></TABLE>
<SCRIPT type=text/javascript>
if ( aid7786021.offsetHeight > bid7786021.offsetHeight ) bid7786021.style.setExpression("height",aid7786021.offsetHeight );
else aid7786021.style.setExpression("height",bid7786021.offsetHeight );
</SCRIPT>
</SPAN><SPAN class=mo id=id7786030></SPAN><SPAN id=id7786033L>
<TABLE class=lr>
<TBODY>
<TR>
<TD><FONT size=2><FONT face=symbol>æ</FONT><BR><FONT face=symbol>ç</FONT><BR><FONT face=symbol>è</FONT></FONT></TD></TR></TBODY></TABLE></SPAN><SPAN id=id7786033M><SPAN class=mrow id=id7786036>
<TABLE class=mfrac style="FONT-SIZE: 75%">
<TBODY>
<TR class=mfraca id=aid7786038 style="HEIGHT: 39px">
<TD class=mfracaa><SPAN class=mrow id=id7786039><SPAN class=mrow id=id7786041><FONT size=2><SPAN class=mn>1</SPAN><SPAN class=mo id=id7786044>−</SPAN><SPAN class=mn>1</SPAN></FONT></SPAN></SPAN></TD></TR>
<TR class=mfracb id=bid7786038>
<TD><SPAN class=mrow id=id7786050><SPAN class=mrow id=id7786052><FONT size=2><SPAN class=mn>1</SPAN><SPAN class=mo id=id7786055>−</SPAN><SPAN><SPAN class=mi1><EM>ⅇ</EM></SPAN></SPAN></FONT><SUP><SPAN class=mrow id=id7786063><EM><FONT size=2><SPAN class=mi1>j</SPAN><SPAN class=mo id=id7786066></SPAN></FONT></EM><SPAN class=mrow id=id7786069><FONT size=2> </FONT>
<TABLE class=mfrac style="FONT-SIZE: 100%">
<TBODY>
<TR class=mfraca id=aid7786070>
<TD class=mfracaa><SPAN class=mrow id=id7786071><SPAN class=mrow id=id7786073><FONT size=2><SPAN class=mn>2</SPAN><SPAN class=mo id=id7786077></SPAN><SPAN class=mi1><EM>π</EM></SPAN></FONT></SPAN></SPAN></TD></TR>
<TR class=mfracb id=bid7786070>
<TD><SPAN class=mrow id=id7786080><SPAN class=mi1><EM><FONT size=2>N</FONT></EM></SPAN></SPAN></TD></TR></TBODY></TABLE></SPAN><SPAN class=mo id=id778
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Autor y licencia de 'Análisis de Fourier en Espacios Complejos'
Tutorial de Michael Haag, Justin Romberg, Erika Jackson, Fara Meza. Extraido de: http://cnx.org/content/m12848/latest/
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