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.html | 241 ---------------------------------------- 1 file changed, 241 deletions(-) delete mode 100644 project/templates/integral.html (limited to 'project/templates/integral.html') diff --git a/project/templates/integral.html b/project/templates/integral.html deleted file mode 100644 index 364fe84..0000000 --- a/project/templates/integral.html +++ /dev/null @@ -1,241 +0,0 @@ - -{%- extends "base_math.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

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Scripts used to generate graphs: Link

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29/12/2018
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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.

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Consider the following example:

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- - - - ∫ - 0 - 1 - - - - ( - -1 - ) - - x - - dx - - -
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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.

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

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.

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Upon doing so, the following helical graph is obtained:

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

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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.

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

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- - - - - - - - - - - - - - - - - - - - -
- - - - ∫ - 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.

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

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