From 14025d22ce3d66c9d235e57221ec4653e00f972c Mon Sep 17 00:00:00 2001 From: Jedidiah Barber Date: Fri, 26 Nov 2021 20:17:43 +1300 Subject: Switched to .xhtml extension, fixed some minor bugs --- project/templates/integral.xhtml | 241 +++++++++++++++++++++++++++++++++++++++ 1 file changed, 241 insertions(+) create mode 100644 project/templates/integral.xhtml (limited to 'project/templates/integral.xhtml') 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 %} + +{% 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