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diff --git a/project/templates/integral.xhtml b/project/templates/integral.xhtml new file mode 100644 index 0000000..8f2f33a --- /dev/null +++ b/project/templates/integral.xhtml @@ -0,0 +1,241 @@ + +{%- extends "base_math.xhtml" -%} + + + +{%- block title -%}Area Under the Curve of a Complex Integral{%- endblock -%} + + + +{%- block style %} + <link href="/css/integral.css" rel="stylesheet" /> +{% endblock -%} + + + +{%- block content %} +<h4>Area Under the Curve of a Complex Integral</h4> + +<p>Scripts used to generate graphs: <a href="/dld/integral_scripts.zip">Link</a></p> + +<h5>29/12/2018</h5> + +<p>A definite integral can be represented on the xy-plane as the signed area bounded by the curve of +the function f(x), the x-axis, and the limits of integration a and b. But it's not immediately clear +how this definition applies for complex valued functions.</p> + +<p>Consider the following example:</p> + +<div class="precontain"><div class="mathblock"> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mrow> + <munderover> + <mo>∫</mo> + <mn>0</mn> + <mn>1</mn> + </munderover> + <msup> + <mrow> + <mo>(</mo> + <mn>-1</mn> + <mo>)</mo> + </mrow> + <mi>x</mi> + </msup> + <mi>dx</mi> + </mrow> + </math> +</div></div> + +<p>If the function is graphed on the xy-plane, the real valued outputs are sparse. Yet an elementary +antiderivative exists and the definite integral is well defined.</p> + +<div class="figure"> + <img src="/img/minus_one_exp_x_real_values_only.png" + alt="Real values only plot of minus one raised to the x power" + height="400" + width="520" /> + <div class="figcaption">Figure 1 - Real values only</div> +</div> + +<p>In order to plot a meaningful graph that can be used to potentially calculate the integral as a +signed area, some cues are taken from Philip Lloyd's work on +<a href="https://phantomgraphs.weebly.com/" class="external">Phantom Graphs</a>. In that work, an +additional z-axis is used to extend the x-axis into a complex xz-plane, allowing complex inputs to +be graphed. For the function considered here, the z-axis is instead used to extend the y-axis into a +complex yz-plane to allow graphing of complex outputs instead.</p> + +<p>Upon doing so, the following helical graph is obtained:</p> + +<div class="figure"> + <img src="/img/minus_one_exp_x_full_plot.png" + alt="Complete three dimensional graph of all values of minux one raised to the x power" + height="400" + width="520" /> + <div class="figcaption">Figure 2 - Full graph</div> +</div> + +<p>The curve is continuous and spirals around the x-axis, intersecting with the real xy-plane at the +points plotted in the initial graph of Figure 1. However it is still not clear how to represent the +area under the curve.</p> + +<p>Observing that complex numbers in cartesian form are composed of a real part and an imaginary +part, it is possible to decompose the function into real and imaginary components. These are easy to +obtain by rotating the graph above to view the real and imaginary parts as flat planes.</p> + +<table id="component"> + <tr> + <td> + <div class="figure"> + <img src="/img/cos_pi_x.png" + alt="Graph of the real component of minus one raised to the power of x" + height="400" + width="520" /> + <div class="figcaption">Figure 3 - Real component</div> + </div> + </td> + <td> + <div class="figure"> + <img src="/img/sin_pi_x.png" + alt="Graph of the imaginary component of minus one raised to the power of x" + height="400" + width="520" /> + <div class="figcaption">Figure 4 - Imaginary component</div> + </div> + </td> + </tr> +</table> + +<p>From this it can be seen that the function is a combination of a real valued cosine and an +imaginary valued sine. With the limits of integration under consideration, the real values disappear +and we are left with the following:</p> + +<div class="precontain"><div class="mathblock"> + <table> + <tr> + <td colspan="2"> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mrow> + <munderover> + <mo>∫</mo> + <mn>0</mn> + <mn>1</mn> + </munderover> + <msup> + <mrow> + <mo>(</mo> + <mn>-1</mn> + <mo>)</mo> + </mrow> + <mi>x</mi> + </msup> + <mi>dx</mi> + </mrow> + </math> + </td> + </tr> + <tr> + <td>=</td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mrow> + <mi>i</mi> + <munderover> + <mo>∫</mo> + <mn>0</mn> + <mn>1</mn> + </munderover> + <mi>sin</mi> + <mrow> + <mo>(</mo> + <mi>π</mi> + <mo>⁢</mo> + <mi>x</mi> + <mo>)</mo> + </mrow> + <mi>dx</mi> + </mrow> + </math> + </td> + </tr> + <tr> + <td>=</td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mrow> + <mo>-</mo> + <mfrac> + <mi>i</mi> + <mi>π</mi> + </mfrac> + <msubsup> + <menclose notation="right"> + <mrow> + <mi>cos</mi> + <mrow> + <mo>(</mo> + <mi>π</mi> + <mo>⁢</mo> + <mi>x</mi> + <mo>)</mo> + </mrow> + </mrow> + </menclose> + <mn>0</mn> + <mn>1</mn> + </msubsup> + </mrow> + </math> + </td> + </tr> + <tr> + <td>=</td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mrow> + <mo>-</mo> + <mfrac> + <mi>i</mi> + <mi>π</mi> + </mfrac> + <mrow> + <mo>(</mo> + <mn>-1</mn> + <mo>-</mo> + <mn>1</mn> + <mo>)</mo> + </mrow> + </mrow> + </math> + </td> + </tr> + <tr> + <td>=</td> + <td> + <math xmlns="http://www.w3.org/1998/Math/MathML"> + <mrow> + <mfrac> + <mrow> + <mn>2</mn> + <mi>i</mi> + </mrow> + <mi>π</mi> + </mfrac> + </mrow> + </math> + </td> + </tr> + </table> +</div></div> + +<p>This agrees with the answer obtained by ordinary evaluation of the integral without considering +the graph, so the informal area under the curve definition still works. Considering the area under +the curve using polar coordinates also works, but requires evaluating a less than pleasant infinite +sum and so won't be considered here.</p> + +<p>The next interesting question is how this relates to the surface area of a +<a href="https://www.mathcurve.com/surfaces.gb/helicoiddroit/helicoiddroit.shtml" class="external"> +right helicoid</a>.</p> +{% endblock -%} + + |