Article about area under the curve for complex valued integrals

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+{% extends "base.html" %}
+
+
+
+{% 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>
+            <msubsup>
+                <mo>&int;</mo>
+                <mn>0</mn>
+                <mn>1</mn>
+            </msubsup>
+            <msup>
+                <mfenced>
+                    <mn>-1</mn>
+                </mfenced>
+                <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>
+
+<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">
+    <figcaption>Figure 1 - Real values only</figcaption>
+</figure>
+
+<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/" target="_blank">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>
+
+<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">
+    <figcaption>Figure 2 - Full graph</figcaption>
+</figure>
+
+<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>
+            <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">
+                <figcaption>Figure 3 - Real component</figcaption>
+            </figure>
+        </td>
+        <td>
+            <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">
+                <figcaption>Figure 4 - Imaginary component</figcaption>
+            </figure>
+        </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>
+                        <msubsup>
+                            <mo>&int;</mo>
+                            <mn>0</mn>
+                            <mn>1</mn>
+                        </msubsup>
+                        <msup>
+                            <mfenced>
+                                <mn>-1</mn>
+                            </mfenced>
+                            <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>
+                        <msubsup>
+                            <mo>&int;</mo>
+                            <mn>0</mn>
+                            <mn>1</mn>
+                        </msubsup>
+                        <mi>sin</mi>
+                        <mfenced>
+                            <mrow>
+                                <mi>&pi;</mi>
+                                <mo>&it;</mo>
+                                <mi>x</mi>
+                            </mrow>
+                        </mfenced>
+                        <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>&pi;</mi>
+                        </mfrac>
+                        <msubsup>
+                            <menclose notation="right">
+                                <mrow>
+                                    <mi>cos</mi>
+                                    <mfenced>
+                                        <mrow>
+                                            <mi>&pi;</mi>
+                                            <mo>&it;</mo>
+                                            <mi>x</mi>
+                                        </mrow>
+                                    </mfenced>
+                                </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>&pi;</mi>
+                        </mfrac>
+                        <mfenced>
+                            <mrow>
+                                <mn>-1</mn>
+                                <mo>-</mo>
+                                <mn>1</mn>
+                            </mrow>
+                        </mfenced>
+                    </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</mn>
+                            </mrow>
+                            <mi>&pi;</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" target="_blank">right helicoid</a>.</p>
+
+{% endblock %}
+
+