document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1185677>Question 1185677</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the Fourier sine and cosine series of: f(x) = x at 0<= x <= 1 , -1 at 1 < x <= 2, 2 at 2< x <= pi. ( piecewise function)  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849674 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849674\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to find the Fourier sine and cosine series for the given piecewise function:\r<BR>\n" );
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document.write( "**1. Define the function:**\r<BR>\n" );
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document.write( "The function f(x) is defined as:\r<BR>\n" );
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document.write( "* f(x) = x,  0 ≤ x ≤ 1<BR>\n" );
document.write( "* f(x) = -1, 1 < x ≤ 2<BR>\n" );
document.write( "* f(x) = 2,  2 < x ≤ π\r<BR>\n" );
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document.write( "**2. Fourier Sine Series:**\r<BR>\n" );
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document.write( "The Fourier sine series is given by:\r<BR>\n" );
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document.write( "f(x) = Σ (from n=1 to ∞)  bₙ sin(nx)\r<BR>\n" );
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document.write( "where the coefficients bₙ are calculated as:\r<BR>\n" );
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document.write( "bₙ = (2/π) ∫ (from 0 to π) f(x) sin(nx) dx\r<BR>\n" );
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document.write( "For the given piecewise function, this integral becomes:\r<BR>\n" );
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document.write( "bₙ = (2/π) [∫ (from 0 to 1) x sin(nx) dx + ∫ (from 1 to 2) -1 sin(nx) dx + ∫ (from 2 to π) 2 sin(nx) dx]\r<BR>\n" );
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document.write( "Solving these integrals (using integration by parts for the first integral) gives you the values for each bₙ.\r<BR>\n" );
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document.write( "**3. Fourier Cosine Series:**\r<BR>\n" );
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document.write( "The Fourier cosine series is given by:\r<BR>\n" );
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document.write( "f(x) = a₀/2 + Σ (from n=1 to ∞) aₙ cos(nx)\r<BR>\n" );
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document.write( "where the coefficients aₙ are calculated as:\r<BR>\n" );
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document.write( "a₀ = (2/π) ∫ (from 0 to π) f(x) dx<BR>\n" );
document.write( "aₙ = (2/π) ∫ (from 0 to π) f(x) cos(nx) dx\r<BR>\n" );
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document.write( "For the given piecewise function:\r<BR>\n" );
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document.write( "a₀ = (2/π) [∫ (from 0 to 1) x dx + ∫ (from 1 to 2) -1 dx + ∫ (from 2 to π) 2 dx]<BR>\n" );
document.write( "aₙ = (2/π) [∫ (from 0 to 1) x cos(nx) dx + ∫ (from 1 to 2) -1 cos(nx) dx + ∫ (from 2 to π) 2 cos(nx) dx]\r<BR>\n" );
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document.write( "Solving these integrals gives you the values for a₀ and each aₙ.  Again, integration by parts will be needed for the first integral in the aₙ calculation.\r<BR>\n" );
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document.write( "**4. Final Series:**\r<BR>\n" );
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document.write( "Once you've calculated the coefficients, substitute them back into the Fourier sine and cosine series formulas to get the final expressions for the series.\r<BR>\n" );
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document.write( "**Important Notes:**\r<BR>\n" );
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document.write( "* The integrals involved can be a bit tedious to solve by hand, but they are standard integrals that can be found in integral tables or solved using software.<BR>\n" );
document.write( "* The Fourier series will converge to the function f(x) at points of continuity. At the discontinuities (x=1 and x=2), the series will converge to the average of the left-hand and right-hand limits.<BR>\n" );
document.write( "* The sine series assumes an odd extension of the function, and the cosine series assumes an even extension.\r<BR>\n" );
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document.write( "Let me know if you'd like me to walk through the integration steps for a specific coefficient.<BR>\n" );
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