From 23b53a46d263a30a466ee0d48543b6ed64f29141 Mon Sep 17 00:00:00 2001 From: Jed Barber Date: Sat, 29 Dec 2018 12:45:47 +1100 Subject: Article about area under the curve for complex valued integrals --- project/templates/integral.html | 242 ++++++++++++++++++++++++++++++++++++++++ 1 file changed, 242 insertions(+) create mode 100644 project/templates/integral.html (limited to 'project/templates/integral.html') 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 %} + +{% endblock %} + + + +{% block content %} + +

Area Under the Curve of a Complex Integral

+ +

Scripts used to generate graphs: Link

+ +
29/12/2018
+ +

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.

+ +

Consider the following example:

+ +
+ + + + ∫ + 0 + 1 + + + + -1 + + x + + dx + + +
+ +

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.

+ +
+ Real values only plot of minus one raised to the x power +
Figure 1 - Real values only
+
+ +

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 +Phantom Graphs. +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.

+ +

Upon doing so, the following helical graph is obtained:

+ +
+ Complete three dimensional graph of all values of minux one raised to the x power +
Figure 2 - Full graph
+
+ +

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.

+ +

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.

+ + + + + + +
+
+ Graph of the real component of minus one raised to the power of x +
Figure 3 - Real component
+
+ 		+
+ Graph of the imaginary component of minus one raised to the power of x +
Figure 4 - Imaginary component
+
+
+ +

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:

+ +
+ + + + + + + + + + + + + + + + + + + + +
+ + + + ∫ + 0 + 1 + + + + -1 + + x + + dx + + +
= + + + i + + ∫ + 0 + 1 + + sin + + + π + ⁢ + x + + + dx + + +
= + + + - + + i + π + + + + + cos + + + π + ⁢ + x + + + + + 0 + 1 + + + +
= + + + - + + i + π + + + + -1 + - + 1 + + + + +
= + + + + + 2 + i + + π + + + +
+
+ +

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.

+ +

The next interesting question is how this relates to the surface area of a +right helicoid.

+ +{% endblock %} + + -- cgit