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      <text>25.4.1.7</text>
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      <text><![CDATA[<h4 class="HELM_example">Example</h4>
<p>Finally, use Fourier series techniques to deal with the final boundary condition \(u(x,0)=f(x)\):</p>
<h4 class="HELM_solution">Solution</h4>
<p>Superposing solutions of the form \((5)\) (with \(E = 0\)) gives
</p>

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            <td>\(\qquad u(x, y) = \sum_{n=0}^\infty F_n\cos(n\pi x)e^{-n\pi x}\)</td>
            <td>\((6)\)</td>
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so the boundary condition gives
<p>\(\qquad f(x) = \sum_{n=0}^\infty F_n\cos(n\pi x)\)</p>
We have here a half-range Fourier cosine series representation of a function \(f(x)\) defined over \(0 &lt; x &lt; 1\). Extending \(f(x)\) as an even periodic function with period 2 and using standard Fourier series theory gives

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            <td>\(\qquad F_n=2\int_0^{1}f(x)\cos(n\pi x) \;dx\)</td>
            <td>\[n = 1,2,...\]</td>
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with
<p>\(\qquad \dfrac{F_0}{2}=\int_0^{1}f(x) \;dx\) </p>
Hence \((6)\) is the solution of this given boundary value problem, the integrals giving us in principle the Fourier coefficients \(F_n\) for a given function \(f(x)\).
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