document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=366527>Question 366527</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find fourier series of cos^3 of x, and sin^3 of x ..<BR>\n" );
document.write( "any one could help ..pleez,,<BR>\n" );
document.write( "thanks </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #261277 by </B><A HREF=/tutors/aboutme.mpl?userid=Jk22><B>Jk22(389)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Jk22><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Jk22><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=261277\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given a function f(x) , we shall write the Fourier series of this.<BR>\n" );
document.write( "There are series in complex exponential, sine or cosine. Let f(x) a function which is 2Pi periodic<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+f%28x%29=sum%28+a%5Bn%5D+cos%28nx%29%2C+n=0%2C+%2Binfinity+%29+ ALIGN=MIDDLE  ALT=\"+f%28x%29=sum%28+a%5Bn%5D+cos%28nx%29%2C+n=0%2C+%2Binfinity+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the series in cosine, <BR>\n" );
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document.write( "with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a%5Bn%5D+=+%281%2Fpi%29+int%28+f%28x%29cos%28nx%29%2C+dx%2C+0%2C+2%2Api+%29 ALIGN=MIDDLE  ALT=\"+a%5Bn%5D+=+%281%2Fpi%29+int%28+f%28x%29cos%28nx%29%2C+dx%2C+0%2C+2%2Api+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "since cos(mx) and cos(nx) are orthogonal functions. <BR>\n" );
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document.write( "Consider an even function like cos(x)^3, only cosine (even) terms will appear.<BR>\n" );
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document.write( "However without doing the integration, we can write cos(x) with complex numbers : cos(x) = (exp(ix)+exp(-ix))/2<BR>\n" );
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document.write( "then cos(x)^3 = 1/8*(exp(i3x)+3exp(ix)+3exp(-ix)+exp(-i3x))<BR>\n" );
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document.write( "= cos(3x)/4 + 3/4*cos(x)<BR>\n" );
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document.write( " We can now show that this corresponds to the definition above :\r<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D=%282%2Fpi%29+int%28+cos%5E3%28x%29cos%28x%29%2C+dx%2C+0%2C+pi+%29=+%282%2Fpi%29%2A%28cos%28x%29%5E3sin%28x%29%2B3%2Aint%28+sin%28x%29%5E2cos%28x%29%5E2%2C+dx%2C+0%2C+pi%29%29+=+%286%2Fpi%29%2Aint%281-cos%28x%29%5E4%2C+dx%2C+0%2C+pi+%29++ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "hence <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+8%2Aa%5B1%5D+=+6%2Api%2Fpi+-%3E+a%5B1%5D+=+3%2F4+ ALIGN=MIDDLE  ALT=\"+8%2Aa%5B1%5D+=+6%2Api%2Fpi+-%3E+a%5B1%5D+=+3%2F4+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The same can be shown for : <BR>\n" );
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document.write( "the similar calculation can be done with sin(x) = (exp(ix) - exp(-ix))/(2i)<BR>\n" );
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document.write( "sin(x)^3 = (exp(3ix) - 3exp(ix) + 3exp(-ix) - exp(-i3x))/(2i)^3<BR>\n" );
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document.write( "= (- sin(3x) + 3sin(x))/4 </SPAN>\n" );
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