Article about area under the curve for complex valued integrals

14 files changed, 408 insertions, 0 deletions

diff --git a/project/assets/css/default.css b/project/assets/css/default.css
index fb1de72..83cd85a 100644
--- a/project/assets/css/default.css
+++ b/project/assets/css/default.css
@@ -21,6 +21,19 @@ div.precontain {
 }
 
 
+div.mathblock {
+    display: inline-block;
+    text-align: left;
+    background-color: #ffa5a5;
+    border-radius: 1.5em;
+    padding: 2em 2em 2em 2em;
+    margin: 1em 2em 1em 2em;
+    /* width should be equal to code and pre, but for some reason this needs 45em instead of 60em... bug? */
+    /* what are the default widths, margins, and paddings used in browsers anyway? */
+    width: 45em;
+}
+
+
 code {
     display: inline-block;
     text-align: left;
diff --git a/project/assets/css/integral.css b/project/assets/css/integral.css
new file mode 100644
index 0000000..94130ec
--- /dev/null
+++ b/project/assets/css/integral.css
@@ -0,0 +1,7 @@
+
+
+table#component {
+    margin: auto;
+}
+
+
diff --git a/project/assets/dld/integral_scripts.zip b/project/assets/dld/integral_scripts.zip
new file mode 100644
index 0000000..82ad952
--- /dev/null
+++ b/project/assets/dld/integral_scripts.zip
diff --git a/project/assets/img/cos_pi_x.png b/project/assets/img/cos_pi_x.png
new file mode 100644
index 0000000..7254624
--- /dev/null
+++ b/project/assets/img/cos_pi_x.png
diff --git a/project/assets/img/minus_one_exp_x_full_plot.png b/project/assets/img/minus_one_exp_x_full_plot.png
new file mode 100644
index 0000000..1927f52
--- /dev/null
+++ b/project/assets/img/minus_one_exp_x_full_plot.png
diff --git a/project/assets/img/minus_one_exp_x_real_values_only.png b/project/assets/img/minus_one_exp_x_real_values_only.png
new file mode 100644
index 0000000..f8cea87
--- /dev/null
+++ b/project/assets/img/minus_one_exp_x_real_values_only.png
diff --git a/project/assets/img/sin_pi_x.png b/project/assets/img/sin_pi_x.png
new file mode 100644
index 0000000..9996cd5
--- /dev/null
+++ b/project/assets/img/sin_pi_x.png
diff --git a/project/complexity.yml b/project/complexity.yml
index ef1bccc..8f680f4 100644
--- a/project/complexity.yml
+++ b/project/complexity.yml
@@ -20,4 +20,5 @@ unexpanded_templates:
  - "sokoban.html"
  - "links.html"
  - "steelman.html"
+ - "integral.html"
 
diff --git a/project/scripts/cos_pi_x.py b/project/scripts/cos_pi_x.py
new file mode 100755
index 0000000..551c142
--- /dev/null
+++ b/project/scripts/cos_pi_x.py
@@ -0,0 +1,36 @@
+#!/usr/bin/python
+
+
+import numpy as np
+import matplotlib.pyplot as plt
+from matplotlib.patches import Polygon
+
+
+plt.rcParams['legend.fontsize'] = 10
+
+fig = plt.figure ()
+ax = fig.gca ()
+
+x = np.linspace (-2, 2, 5000)
+y = np.cos (np.pi * x)
+
+ax.plot (x, y, label='$y=Re((-1)^{x})$')
+ax.legend ()
+
+a, b = 0, 1
+ix = np.linspace (a, b)
+iy = np.cos (np.pi * ix)
+verts = [(a,0)] + zip (ix, iy) + [(b,0)]
+poly = Polygon (verts, facecolor='0.9', edgecolor='0.5')
+ax.add_patch (poly)
+
+ax.set_xticks ([-2, -1, 1, 2])
+ax.spines['left'].set_position ('center')
+ax.spines['right'].set_position ('center')
+
+ax.set_yticks ([-1, -0.5, 0.5, 1])
+ax.spines['top'].set_position ('center')
+ax.spines['bottom'].set_position ('center')
+
+plt.show ()
+
diff --git a/project/scripts/minus_one_exp_x.py b/project/scripts/minus_one_exp_x.py
new file mode 100755
index 0000000..7073b47
--- /dev/null
+++ b/project/scripts/minus_one_exp_x.py
@@ -0,0 +1,36 @@
+#!/usr/bin/python
+
+
+from mpl_toolkits.mplot3d import Axes3D
+import numpy as np
+import matplotlib.pyplot as plt
+
+
+plt.rcParams['legend.fontsize'] = 10
+
+fig = plt.figure ()
+ax = fig.gca (projection='3d')
+
+# the z-axis is vertical in matplotlib
+x = np.linspace (-2, 2, 5000)
+y = np.sin (np.pi * x)
+z = np.cos (np.pi * x)
+
+ax.plot (x, y, z, label='$y=(-1)^{x}$')
+ax.legend ()
+
+ax.set_xlabel ('x', labelpad=7)
+ax.set_xticks ([-2, -1, 0, 1, 2])
+ax.xaxis.set_rotate_label (False)
+
+ax.set_ylabel ('$y_{im}$', labelpad=7)
+ax.set_yticks ([1, 0.5, 0, -0.5, -1])
+ax.invert_yaxis ()
+ax.yaxis.set_rotate_label (False)
+
+ax.set_zlabel ('$y_{re}$', labelpad=7)
+ax.set_zticks ([-1, -0.5, 0, 0.5, 1])
+ax.zaxis.set_rotate_label (False)
+
+plt.show ()
+
diff --git a/project/scripts/minus_one_scatter.py b/project/scripts/minus_one_scatter.py
new file mode 100755
index 0000000..4b26a4f
--- /dev/null
+++ b/project/scripts/minus_one_scatter.py
@@ -0,0 +1,34 @@
+#!/usr/bin/python
+
+
+import numpy as np
+import matplotlib.pyplot as plt
+
+
+plt.rcParams['legend.fontsize'] = 10
+
+fig = plt.figure ()
+ax = fig.gca ()
+
+data = np.array([
+    [-2, 1],
+    [-1,-1],
+    [ 0, 1],
+    [ 1,-1],
+    [ 2, 1]
+])
+x, y = data.T
+
+ax.set_xticks ([-2, -1, 1, 2])
+ax.spines['left'].set_position ('center')
+ax.spines['right'].set_position ('center')
+
+ax.set_yticks ([-1, -0.5, 0.5, 1])
+ax.spines['top'].set_position ('center')
+ax.spines['bottom'].set_position ('center')
+
+plt.scatter (x, y, label='$y=(-1)^{x}$')
+ax.legend ()
+
+plt.show ()
+
diff --git a/project/scripts/sin_pi_x.py b/project/scripts/sin_pi_x.py
new file mode 100755
index 0000000..8436726
--- /dev/null
+++ b/project/scripts/sin_pi_x.py
@@ -0,0 +1,36 @@
+#!/usr/bin/python
+
+
+import numpy as np
+import matplotlib.pyplot as plt
+from matplotlib.patches import Polygon
+
+
+plt.rcParams['legend.fontsize'] = 10
+
+fig = plt.figure ()
+ax = fig.gca ()
+
+x = np.linspace (-2, 2, 5000)
+y = np.sin (np.pi * x)
+
+ax.plot (x, y, label='$y=Im((-1)^{x})$')
+ax.legend ()
+
+a, b = 0, 1
+ix = np.linspace (a, b)
+iy = np.sin (np.pi * ix)
+verts = [(a,0)] + zip (ix, iy) + [(b,0)]
+poly = Polygon (verts, facecolor='0.9', edgecolor='0.5')
+ax.add_patch (poly)
+
+ax.set_xticks ([-2, -1, 1, 2])
+ax.spines['left'].set_position ('center')
+ax.spines['right'].set_position ('center')
+
+ax.set_yticks ([-1, -0.5, 0.5, 1])
+ax.spines['top'].set_position ('center')
+ax.spines['bottom'].set_position ('center')
+
+plt.show ()
+
diff --git a/project/templates/index.html b/project/templates/index.html
index 056231f..dfe396b 100644
--- a/project/templates/index.html
+++ b/project/templates/index.html
@@ -16,6 +16,9 @@
 {% block content %}
 
 <ul class="index">
+    <li><a href="/integral.html">Area Under the Curve of a Complex Integral</a><br>
+        <span class="post">(Posted 29/12/2018)</span></li>
+
     <li><a href="/steelman.html">D, Parasail, Pascal, and Rust vs The Steelman</a><br>
         <span class="post">(Posted 29/10/2017)</span></li>
 
diff --git a/project/templates/integral.html b/project/templates/integral.html
new file mode 100644
index 0000000..0e5307b
--- /dev/null
+++ b/project/templates/integral.html
@@ -0,0 +1,242 @@
+
+{% 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 %}
+
+