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525 lines
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525 lines
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<!DOCTYPE html>
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<h1 class="center">Fourier Series</h1>
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<!-- and INT{2, 3}x^2 and SIG{2, 3}x^2 -->
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<p>Sine and cosine waves can make other functions!</p>
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<p><span class="center">Here two different sine waves add together to make a new wave:</span></p>
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<p class="center"><img src="../physics/images/wave-superposition.svg" alt="wave superposition" height="307" width="333"><br>
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<i>Try "sin(x)+sin(2x)" at the <a href="../data/function-grapher.html">function grapher</a>.</i></p>
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<p>(You can also <b>hear it</b> at <a href="../physics/audio-spectrum-beats.html">Sound Beats</a>.)</p>
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<h2>Square Wave</h2>
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<p>Can we use sine waves to make a <b>square wave</b>?</p>
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<p>Our target is this square wave:</p>
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<p><img src="images/fourier-square.svg" alt="Square Wave -pi to pi" height="146" width="530"></p>
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<p>Start with <b>sin(x)</b>:</p>
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<p><img src="images/fourier-square-sine1.svg" alt="sin(x)" height="146" width="530"></p>
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<p>Then take <b>sin(3x)/3</b>:</p>
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<p><img src="images/fourier-square-sine3.svg" alt="sin(3x)/3:" height="146" width="530"></p>
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<p>And add it to make <b>sin(x)+sin(3x)/3</b>:</p>
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<p><img src="images/fourier-square-sine1-3.svg" alt="sin(x)+sin(3x)/3:" height="146" width="530"></p>
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<p>Can you see how it starts to look a little like a square wave?</p>
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<p>Now take <b>sin(5x)/5</b>:</p>
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<p><img src="images/fourier-square-sine5.svg" alt="sin(5x)/5:" height="146" width="530"></p>
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<p>Add it also, to make <b>sin(x)+sin(3x)/3+sin(5x)/5</b>:</p>
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<p><img src="images/fourier-square-sine1-5.svg" alt="sin(x)+sin(3x)/3+sin(5x)/5" height="146" width="530"></p>
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<p>Getting better! Let's add a lot more sine waves.</p>
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<p>Using 20 sine waves we get <b>sin(x)+sin(3x)/3+sin(5x)/5 + ... + sin(39x)/39</b>:</p>
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<p><img src="images/fourier-square-sine1-39.svg" alt="sin(x)+sin(3x)/3+sin(5x)/5" height="146" width="530"></p>
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<p>Using 100 sine waves we get <b>sin(x)+sin(3x)/3+sin(5x)/5 + ... + sin(199x)/199</b>:</p>
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<p><img src="images/fourier-square-sine1-199.svg" alt="sin(x)+sin(3x)/3+sin(5x)/5" height="146" width="530"></p>
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<p>And if we could add infinite sine waves in that pattern we would <b>have</b> a square wave!</p>
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<p>So we can say that:</p>
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<p class="center large">a square wave = sin(x) + sin(3x)/3 + sin(5x)/5 + ... (infinitely)</p>
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<p>That is the idea of a Fourier series.</p>
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<p>By adding infinite sine (and or cosine) waves we can make other functions, even if they are a bit weird.</p>
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<div class="fun">
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<p>You might like to have a little play with:</p>
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<p class="center"><a href="fourier-series-graph.html">The Fourier Series Grapher</a></p>
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<p>And it is also fun to use <a href="../geometry/spiral-artist.html">Spiral Artist</a> and see how circles make waves.</p>
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<p>They are designed to be experimented with, so play around and get a feel for the subject.</p>
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</div>
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<h2>Finding the Coefficients</h2>
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<p>How did we know to use sin(3x)/3, sin(5x)/5, etc?</p>
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<p>There are formulas!</p>
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<p>First let us write down a full series of sines and cosines, with a name for all coefficients:</p>
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<div class="center large" style="">f(x) = a<sub>0</sub> +
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<div class="sigma">
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<div class="to">∞</div>
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<div class="symb"></div>
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<div class="from">n=1</div>
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</div> a<sub>n</sub> cos(nx<span class="intbl"><em><span class="times">π</span></em><strong>L</strong></span>) + <div class="sigma">
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<div class="to">∞</div>
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<div class="symb"></div>
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<div class="from">n=1</div>
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</div> b<sub>n</sub> sin(nx<span class="intbl"><em><span class="times">π</span></em><strong>L</strong></span>)</div>
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<!-- f(x) = a_0 + SIG{n=1, INF} a_n cos(nx PI/L ) + SIG{n=1, INF} b_n sin(nx PI/L ) -->
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<p>Where:</p>
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<ul>
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<li>f(x) is the function we want (such as a square wave)</li>
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<li>L is <b>half of the period</b> of the function</li>
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<li>a<sub>0</sub>, a<sub>n</sub> and b<sub>n</sub> are <b>coefficients</b> that we need to calculate!</li>
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</ul>
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<div class="fun"><br>
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<div class="">What does
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<div class="sigma">
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<div class="to">∞</div>
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<div class="symb"></div>
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<div class="from">n=1</div>
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</div> a<sub>n</sub> cos(nx<span class="intbl"><em><span class="times">π</span></em><strong>L</strong></span>) mean?</div>
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<!-- What does SIG{n=1, INF} a_n cos(nx PI/L ) mean? -->
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<p>It uses <a href="../algebra/sigma-notation.html">Sigma Notation</a> to mean <b>sum</b> up the series of values starting at n=1:</p>
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<ul>
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<li>a<sub>1</sub> cos(1x <span class="times">π</span>/L)</li>
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<li>a<sub>2</sub> cos(2x <span class="times">π</span>/L)</li>
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<li>etc</li>
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</ul>
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<p>We do not (yet) know the values of a<sub>1</sub>, a<sub>2</sub> etc.</p>
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</div>
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<p>To find the coefficients a<sub>0</sub>, a<sub>n</sub> and b<sub>n</sub> we use these formulas:</p>
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<div class="center large" style="">a<sub>0</sub> = <span class="intbl"><em>1</em><strong>2L</strong></span>
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<div class="intgl">
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<div class="to">L</div>
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<div class="symb"></div>
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<div class="from">−L</div>
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</div> f(x) dx</div>
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<!-- a_0 = 1/2L INT{-L, L} f(x) dx -->
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<div class="center large" style="">a<sub>n</sub> = <span class="intbl"><em>1</em><strong>L</strong></span>
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<div class="intgl">
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<div class="to">L</div>
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<div class="symb"></div>
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<div class="from">−L</div>
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</div> f(x) cos(nx<span class="intbl"><em><span class="times">π</span></em><strong>L</strong></span>) dx</div>
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<!-- a_n = 1/L INT{-L, L} f(x) cos(nx PI/L ) dx -->
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<div class="center large" style="">b<sub>n</sub> = <span class="intbl"><em>1</em><strong>L</strong></span>
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<div class="intgl">
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<div class="to">L</div>
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<div class="symb"></div>
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<div class="from">−L</div>
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</div> f(x) sin(nx<span class="intbl"><em><span class="times">π</span></em><strong>L</strong></span>) dx</div>
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<!-- b_n = 1/L INT{-L, L} f(x) sin(nx PI/L ) dx -->
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<div class="fun"><br>
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<div style="">What does
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<div class="intgl">
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<div class="to">L</div>
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<div class="symb"></div>
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<div class="from">−L</div>
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</div>f(x) sin(nx<span class="intbl"><em><span class="times">π</span></em><strong>L</strong></span>) dx mean?</div>
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<!-- What does INT{-L, L} f(x) sin(nx PI/L ) dx mean? -->
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<p>It is an <a href="integration-introduction.html">integral</a>, but in practice it just means to find the <b>net area</b> of</p>
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<p class="center"><b>f(x) sin(nx<span class="intbl"><em><span class="times">π</span></em><strong>L</strong></span>)</b></p>
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<p>between −L and L</p>
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<p>We can often find that area just by sketching and using basic calculations, but other times we may need to use <a href="integration-rules.html">Integration Rules</a>.</p>
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</div>
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<p> </p>
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<p>So this is what we do:</p>
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<ul>
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<li>Take our <b>target function, multiply it by sine</b> (or cosine) and <b>integrate</b> (find the area)</li>
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<li>Do that for n=0, n=1, etc to calculate each coefficient</li>
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<li>And after we calculate all coefficients, we put them into the series formula above.</li>
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</ul>
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<p>Let us see how to do each step and then assemble the result at the end!</p>
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<div class="example">
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<h3>Example: This Square Wave:</h3>
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<p class="center"><img src="images/fourier-square.svg" alt="Square Wave -pi to pi" height="146" width="530"></p>
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<ul>
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<li><b>L = <span class="times">π</span></b> <i>(the Period is 2<span class="times">π</span>)</i></li>
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<li>The square wave is from <b>−h</b> to <b>+h</b></li>
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</ul>
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<p>Now our job is to calculate <b>a<sub>0</sub></b>, <b>a<sub>n</sub></b> and <b>b<sub>n</sub></b></p>
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<p> </p>
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<p><b>a<sub>0</sub></b> is the net area between −L and L, then divided by 2L. It is basically an <b>average</b> of f(x) in that range.</p>
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<p>Looking at this sketch:</p>
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<p class="center"><img src="images/fourier-square-area.svg" alt="Square Wave -pi to pi" height="146" width="530"><br>
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The net area of the square wave from −L to L is <b>zero</b>.</p>
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<p>So we know that:</p>
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<p class="center larger">a<sub>0</sub> = 0</p>
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<p class="center larger"> </p>
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<p>For <b>a<sub>1</sub></b> we know that n=1 and L=<span class="times">π</span>, so:</p>
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<div class="center large">a<sub>1</sub> = <span class="intbl"><em>1</em><strong><span class="times">π</span></strong></span>
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<div class="intgl">
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<div class="to"><span class="times">π</span></div>
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<div class="symb"></div>
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<div class="from">−<span class="times">π</span></div>
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</div> f(x) cos(1x<span class="intbl"><em><span class="times">π</span></em><strong><span class="times">π</span></strong></span>) dx</div>
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<!-- a_1 = 1/PI INT{-PI, PI} f(x) cos(1x PI/PI ) dx -->
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<p>Which simplifies to:</p>
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<div class="center large">a<sub>1</sub> = <span class="intbl"><em>1</em><strong><span class="times">π</span></strong></span>
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<div class="intgl">
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<div class="to"><span class="times">π</span></div>
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<div class="symb"></div>
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<div class="from">−<span class="times">π</span></div>
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</div> f(x) cos(x) dx</div>
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<!-- a_1 = 1/PI INT{-PI, PI} f(x) cos(x) dx -->
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<p>Now, because the square wave changes abruptly at x=0 we need to break the calculation into <b>−<span class=" times">π</span> to 0</b> and <b>0 to <span class=" times">π</span></b>,</p>
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<p><b>From −<span class=" times">π</span> to 0</b> we know f(x) is simply equal to <b>−h</b>:</p>
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<div class="center large"><span class="intbl"><em>1</em><strong><span class="times">π</span></strong></span>
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<div class="intgl">
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<div class="to">0</div>
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<div class="symb"></div>
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<div class="from">−<span class="times">π</span></div>
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</div> −h cos(x) dx</div>
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<!---− 1/PI INT{-PI, 0} -h cos(x) dx ---->
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<p>We can move the constant <b>−h</b> outside the integral:</p>
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<div class="center large"><span class="intbl"><em>−h</em><strong><span class="times">π</span></strong></span>
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<div class="intgl">
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<div class="to">0</div>
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<div class="symb"></div>
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<div class="from">−<span class="times">π</span></div>
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</div> cos(x) dx</div>
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<!-- -h/PI INT{-PI, 0} cos(x) dx -->
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<p>Let's sketch <b>cos(x)</b>:</p>
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<p class="center"><img src="images/fourier-square-cos1.svg" alt="Square Wave -pi to pi" height="146" width="530"><br>
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The net area of cos(x) from -<span class=" times">π</span> to 0 is <b>zero</b>.</p>
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<p>So the net area must be 0:</p>
|
||
<div class="center large"><span class="intbl"><em>−h</em><strong><span class="times">π</span></strong></span>
|
||
<div class="intgl">
|
||
<div class="to">0</div>
|
||
<div class="symb"></div>
|
||
<div class="from">−<span class="times">π</span></div>
|
||
</div> cos(x) dx = 0</div>
|
||
<!-- -h/PI INT{-PI, 0} cos(x) dx = 0 -->
|
||
<p> </p>
|
||
<p>The same idea applies from <b>0 to <span class="times">π</span></b>,</p>
|
||
<p class="center"><img src="images/fourier-square-cos1-pos.svg" alt="Square Wave -pi to pi" height="146" width="530"><br>
|
||
The net area of cos(x) from 0 to <span class="times">π</span> is <b>zero</b>.</p>
|
||
<p>and so we can conclude that:</p>
|
||
<p class="center larger">a<sub>1</sub> = 0</p>
|
||
<p>Now let us look at a<sub>2</sub></p>
|
||
<p>Aaaand ... the same thing happens!</p>
|
||
<p class="center"><img src="images/fourier-square-cos2-neg.svg" alt="Square Wave -pi to pi" height="146" width="530"><br>
|
||
The net area of cos(2x) from <b>-<span class="times">π</span> to 0</b> is <b>zero</b>.</p>
|
||
<p>And:</p>
|
||
<p class="center"><img src="images/fourier-square-cos2-pos.svg" alt="Square Wave -pi to pi" height="146" width="530"><br>
|
||
The net area of cos(2x) from <b>0 to <span class="times">π</span></b> is also <b>zero</b>.</p>
|
||
<p>So we know that:</p>
|
||
<p class="center larger">a<sub>2</sub> = 0</p>
|
||
<p>In fact we can extend this idea to every value of <b>a</b> and conclude that:</p>
|
||
<p class="center larger">a<sub>n</sub>= 0</p>
|
||
<p> </p>
|
||
<p><i>So far there has been no need for any major calculations! A few sketches and a little thought have been enough.</i></p>
|
||
<p><i>But now on to the <b>sine</b> function!</i></p>
|
||
<p> </p>
|
||
<p>For <b>b<sub>1</sub></b> we know that n=1 and L=<span class="pi times">π</span>, so:</p>
|
||
<div class="center large">b<sub>1</sub> = <span class="intbl"><em>1</em><strong><span class="times">π</span></strong></span>
|
||
<div class="intgl">
|
||
<div class="to"><span class="times">π</span></div>
|
||
<div class="symb"></div>
|
||
<div class="from">−<span class="times">π</span></div>
|
||
</div> sin(1x<span class="intbl"><em><span class="times">π</span></em><strong><span class="times">π</span></strong></span>) dx</div>
|
||
<!-- b_1 = 1/PI INT{-PI, PI} sin(1x PI/PI ) dx -->
|
||
<p>Which simplifies to:</p>
|
||
<div class="center large">b<sub>1</sub> = <span class="intbl"><em>1</em><strong><span class="times">π</span></strong></span>
|
||
<div class="intgl">
|
||
<div class="to"><span class="times">π</span></div>
|
||
<div class="symb"></div>
|
||
<div class="from">−<span class="times">π</span></div>
|
||
</div> sin(x) dx</div>
|
||
<!-- b_1 = 1/PI INT{-PI, PI} sin(x) dx -->
|
||
<p>and as before, because of the abrupt change at x=0, we need to break the calculation into <b>−<span class=" times">π</span> to 0</b> and <b>0 to <span class="times">π</span></b>,</p>
|
||
<p>So, just looking at the integral <b>from −<span class="pi times">π</span> to 0</b>, we know f(x) = −h:</p>
|
||
<p>We can move the constant <b>−h</b> outside the integral:</p>
|
||
<div class="center large"><span class="intbl"><em>−h</em><strong><span class="times">π</span></strong></span>
|
||
<div class="intgl">
|
||
<div class="to">0</div>
|
||
<div class="symb"></div>
|
||
<div class="from">−<span class="times">π</span></div>
|
||
</div> sin(x) dx</div>
|
||
<!-- -h/PI INT{-PI, 0} sin(x) dx -->
|
||
<p>And <b>sin(x)</b> looks like this:</p>
|
||
<p class="center"><img src="images/fourier-square-sine1-area.svg" alt="sin(x)" height="146" width="530"></p>
|
||
<p>How do we know the area is −2?</p>
|
||
<p>First we use <a href="integration-rules.html">Integration Rules</a> to find the integral of <b>sin(x)</b> is <span class="center"><b>−</b></span><b>cos(x)</b>:</p>
|
||
<p>Then we calculate the <a href="integration-definite.html">definite integral</a> between <span class="center">−</span><span class="times">π</span> and 0 by calculating the value of <span class="center">−</span>cos(x) for <b>0</b>, and for <b>−<span class="times">π</span></b>, and then subtracting:</p>
|
||
<p class="center">[−cos(0)] − [−cos(−<span class="times">π</span>)] = −1 − 1 = −2</p>
|
||
<p>So, between <span class="center">−</span><span class="times">π</span> and 0 we get</p>
|
||
<p class="center"><span class="intbl"><em>−h</em><strong><span class="times">π</span></strong></span>(−2)</p>
|
||
<p> </p>
|
||
<p>Next we look at the integral from <b>0 to <span class="times">π</span></b>:</p>
|
||
<div class="center large"><span class="intbl"><em>h</em><strong><span class="times">π</span></strong></span>
|
||
<div class="intgl">
|
||
<div class="to"><span class="times">π</span></div>
|
||
<div class="symb"></div>
|
||
<div class="from">0</div>
|
||
</div> sin(x) dx</div>
|
||
<!-- -h/PI INT{0, PI} sin(x) dx -->And its integral is:
|
||
<p class="center">[−cos(<span class="times">π</span>)] − [−cos(0<span class="times"></span>)] = 1 − [−1] = 2</p>
|
||
<p class="center"><img src="images/fourier-square-sine1-area-pos.svg" alt="sin(x)" height="146" width="530"></p>
|
||
<p>Now, combining both sides we get:</p>
|
||
<p class="center larger">b<sub>1</sub> = <span class="intbl"><em>1</em><strong><span class="times">π</span></strong></span>[ (−h) × (−2) + (h) × (2) ] = <span class="intbl"><em>4h</em><strong><span class="times">π</span></strong></span></p>
|
||
<p> </p>
|
||
<p>For <b>b<sub>2</sub></b> we have this integral:</p>
|
||
<div class="center large"><span class="intbl"><em>−h</em><strong><span class="times">π</span></strong></span>
|
||
<div class="intgl">
|
||
<div class="to"><span class="times">π</span></div>
|
||
<div class="symb"></div>
|
||
<div class="from">−<span class="times">π</span></div>
|
||
</div> sin(2x) dx</div>
|
||
<!-- -h/PI INT{-PI, PI} sin(2x) dx -->
|
||
<p>From −<span class="times">π </span>to 0 it looks like this:</p>
|
||
<p class="center"><img src="images/fourier-square-sin2-neg.svg" alt="Square Wave -pi to pi" height="146" width="530"><br>
|
||
The net area of sin(2x) from <b>−<span class="times">π</span> to 0</b> is <b>zero</b>.</p>
|
||
<p>And we have seen this kind of thing before, so we conclude that:</p>
|
||
<p class="center larger">b<sub>2</sub> = 0</p>
|
||
<p>For <b>b<sub>3</sub></b> we have this integral:</p>
|
||
<div class="center large"><span class="intbl"><em>−h</em><strong><span class="times">π</span></strong></span>
|
||
<div class="intgl">
|
||
<div class="to"><span class="times">π</span></div>
|
||
<div class="symb"></div>
|
||
<div class="from">−<span class="times">π</span></div>
|
||
</div> sin(3x) dx</div>
|
||
<!-- -h/PI INT{-PI, PI} sin(3x) dx -->
|
||
<p>From −<span class="times">π </span>to 0 we get this interesting situation:</p>
|
||
<p class="center"><img src="images/fourier-square-sin3-neg.svg" alt="Square Wave -pi to pi" height="146" width="530"><br>
|
||
Two areas cancel, but the third one is important!</p>
|
||
<p>So it is like the b<sub>1</sub> integral, but with only one-third of the area.</p>
|
||
<p>For <b>0 to <span class="times">π</span></b> we have:</p>
|
||
<p class="center"><img src="images/fourier-square-sin3-pos.svg" alt="Square Wave -pi to pi" height="146" width="530"><br>
|
||
Again two areas cancel, but not the third</p>
|
||
<p>And we can conclude:</p>
|
||
<p class="center larger">b<sub>3</sub> = <span class="intbl"><em>b<sub>1</sub></em><strong>3</strong></span><em> </em>= <span class="intbl"><em>4h</em><strong>3<span class="times">π</span></strong></span></p>
|
||
<p>The pattern continues:</p>
|
||
<p class="center"><img src="images/fourier-square-sin4.svg" alt="Square Wave -pi to pi" height="146" width="530"><br>
|
||
When n is even the areas cancel for a result of zero.</p>
|
||
<p> </p>
|
||
<p class="center"><img src="images/fourier-square-sin5.svg" alt="Square Wave -pi to pi" height="146" width="530"><br>
|
||
When n is odd, all except one area cancel for a result of 1/n.</p>
|
||
<p>So we can say</p>
|
||
<p class="center"><span class="larger">b<sub>n</sub> = <span class="intbl"><em>4h</em><strong>n<span class="times">π</span></strong></span></span> when n is odd, but <b>0</b> otherwise</p>
|
||
<p> </p>
|
||
<p>And we arrive at our last step: putting the coefficients into the master formula:</p>
|
||
<div class="center large">f(x) = a<sub>0</sub> +
|
||
|
||
<div class="sigma">
|
||
<div class="to">∞</div>
|
||
<div class="symb"></div>
|
||
<div class="from">n=1</div>
|
||
</div> a<sub>n</sub> cos(nx<span class="intbl"><em><span class="times">π</span></em><strong>L</strong></span>) + <div class="sigma">
|
||
<div class="to">∞</div>
|
||
<div class="symb"></div>
|
||
<div class="from">n=1</div>
|
||
</div> b<sub>n</sub> sin(nx<span class="intbl"><em><span class="times">π</span></em><strong>L</strong></span>)</div>
|
||
<!-- f(x) = a_0 + SIG{n=1, INF} a_n cos(nx PI/L ) + SIG{n=1, INF} b_n sin(nx PI/L ) -->
|
||
<p>And we know that:</p>
|
||
<ul>
|
||
<li>a<sub>0</sub> = 0</li>
|
||
<li>a<sub>n</sub> = 0 (all of them!),</li>
|
||
<li>b<sub>n</sub> = 0<strong><span class="times"></span></strong> when n is even</li>
|
||
<li>b<sub>n</sub> = <span class="intbl"><em>4h</em><strong>n<span class="times">π</span></strong></span> when n is odd</li>
|
||
</ul>
|
||
<p>So:</p>
|
||
<p class="center large">f(x) = <span class="intbl"><em>4h</em><strong><span class="times">π</span></strong></span> [ sin(x) + <span class="intbl"><em>sin(3x)</em><strong>3</strong></span> + <span class="intbl"><em>sin(5x)</em><strong>5</strong></span> + ... ]</p>
|
||
</div>
|
||
<p>In conclusion:</p>
|
||
<ul>
|
||
<li>Think about each coefficient, sketch the functions and see if you can find a pattern,</li>
|
||
<li>put it all together into the series formula at the end</li>
|
||
</ul>
|
||
<p> </p>
|
||
<div class="fun">
|
||
<p>And when you are done go over to:</p>
|
||
<p class="center"><a href="fourier-series-graph.html">The Fourier Series Grapher</a></p>
|
||
<p>and see if you got it right!</p>
|
||
<p>Why not try it with "sin((2n-1)*x)/(2n-1)", the 2n−1 neatly gives odd values, and see if you get a square wave.</p>
|
||
</div>
|
||
|
||
|
||
<h2>Other Functions</h2>
|
||
|
||
<p>Of course we can use this for many other functions!</p>
|
||
<p>But we must be able to work out all the coefficients, which in practice means that we work out the <b>area</b> of:</p>
|
||
<ul>
|
||
<li>the function</li>
|
||
<li>the function times sine</li>
|
||
<li>the function times cosine</li>
|
||
</ul>
|
||
<p>But as we saw above we can use tricks like breaking the function into pieces, using common sense, geometry and calculus to help us.</p>
|
||
<p>Here are a few well known ones:</p>
|
||
<div class="simple">
|
||
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<th>Wave</th>
|
||
<th>Series</th>
|
||
<th><a href="fourier-series-graph.html">Fourier Series Grapher</a></th>
|
||
</tr>
|
||
<tr>
|
||
<td>Square Wave</td>
|
||
<td>sin(x) + sin(3x)/3 + sin(5x)/5 + ...</td>
|
||
<td>sin((2n−1)*x)/(2n−1)</td>
|
||
</tr>
|
||
<tr>
|
||
<td>Sawtooth</td>
|
||
<td>sin(x) + sin(2x)/2 + sin(3x)/3 + ...</td>
|
||
<td>sin(n*x)/n</td>
|
||
</tr>
|
||
<tr>
|
||
<td>Pulse</td>
|
||
<td>sin(x) + sin(2x) + sin(3x) + ...</td>
|
||
<td>sin(n*x)*0.1</td>
|
||
</tr>
|
||
<tr>
|
||
<td>Triangle</td>
|
||
<td>sin(x) − sin(3x)/9 + sin(5x)/25 − ...</td>
|
||
<td>sin((2n−1)*x)*(−1)^n/(2n−1)^2</td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div>
|
||
<p> </p>
|
||
<p> </p>
|
||
<div class="fun">
|
||
<p><b>Footnote. Different versions of the formula!</b></p>
|
||
<p>On this page we used the general formula:</p>
|
||
<div class="center large">f(x) = a<sub>0</sub> +
|
||
|
||
<div class="sigma">
|
||
<div class="to">∞</div>
|
||
<div class="symb"></div>
|
||
<div class="from">n=1</div>
|
||
</div> a<sub>n</sub> cos(nx<span class="intbl"><em><span class="times">π</span></em><strong>L</strong></span>) + <div class="sigma">
|
||
<div class="to">∞</div>
|
||
<div class="symb"></div>
|
||
<div class="from">n=1</div>
|
||
</div> b<sub>n</sub> sin(nx<span class="intbl"><em><span class="times">π</span></em><strong>L</strong></span>)</div>
|
||
<!-- f(x) = a_0 + SIG{n=1, INF} a_n cos(nx PI/L ) + SIG{n=1, INF} b_n sin(nx PI/L ) -->
|
||
<p>But when the function f(x) has a period from -<span class="times">π</span> to <span class="times">π</span> we can use a simplified version:</p>
|
||
<div class="center large">f(x) = a<sub>0</sub> +
|
||
|
||
<div class="sigma">
|
||
<div class="to">∞</div>
|
||
<div class="symb"></div>
|
||
<div class="from">n=1</div>
|
||
</div> a<sub>n</sub> cos(nx) + <div class="sigma">
|
||
<div class="to">∞</div>
|
||
<div class="symb"></div>
|
||
<div class="from">n=1</div>
|
||
</div> b<sub>n</sub> sin(nx)</div>
|
||
<!-- f(x) = a_0 + SIG{n=1, INF} a_n cos(nx) + SIG{n=1, INF} b_n sin(nx) -->
|
||
<p>Or there is this one, where a<sub>0</sub> is rolled into the first sum (now n=<b>0</b> to ∞):</p>
|
||
<div class="center large">f(x) =
|
||
|
||
<div class="sigma">
|
||
<div class="to">∞</div>
|
||
<div class="symb"></div>
|
||
<div class="from">n=0</div>
|
||
</div> a<sub>n</sub> cos(nx) + <div class="sigma">
|
||
<div class="to">∞</div>
|
||
<div class="symb"></div>
|
||
<div class="from">n=1</div>
|
||
</div> b<sub>n</sub> sin(nx)</div>
|
||
<!-- f(x) = SIG{n=0, INF} a_n cos(nx) + SIG{n=1, INF} b_n sin(nx) -->
|
||
<p>But I prefer the one we use here, as it is more practical allowing for different periods.</p>
|
||
</div>
|
||
<p> </p>
|
||
|
||
<div class="related">
|
||
<a href="fourier-series-graph.html">Fourier Series Graph Tool</a>
|
||
<a href="integration-introduction.html">Integration</a>
|
||
<a href="../algebra/sigma-notation.html">Sigma Notation</a><a href="../algebra/small-angle-approximations.html">Small Angle Approximations</a>
|
||
<a href="index.html">Calculus Index</a>
|
||
</div>
|
||
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