Complex Contour Integral Problem, meaning

In summary: In this case, the "force field" is a complex function, and the "work" done is the integral of the real part of the conjugate of the complex function.

Homework Statement

First, let's take a look at the complex line integral.

What is the geometry of the complex line integral?

If we look at the real line integral GIF:

[2]: http://en.wikipedia.org/wiki/File:Line_integral_of_scalar_field.gif

The real line integral is a path, but then you make a 3d figure, and it is the area under the 3d shape.

And

In the problem from Schaum's Outline:

[1]: http://i.stack.imgur.com/U65As.png

This is an interesting complex analysis problem; **The figure on the bottom left is what is being referred to,Fig7-10.**

How is in the solution:

$$\int_{BDEFG} = \int_{0}^{2\pi} \frac{(Re^{i\theta})^{p-1}iRe^{i\theta} d\theta}{1 + Re^{i\theta}}$$

How is the integral around the $$BDEFG$$, the same as the area from$$0 \to 2\pi$$

Thanks.

Above

The Attempt at a Solution

$$\int_{BDEFG} = \int_{0}^{2\pi} \frac{(Re^{i\theta})^{p-1}iRe^{i\theta} d\theta}{1 + Re^{i\theta}}$$

How is the integral around the $BDEFG$, the same as the area from 0 to $2\pi$

You can interpret the line integral of ##f(x,y) \geq 0## as an area. If ##f(x,y) \geq 0##, then ##\oint_C f(x,y) \space dr## is the area of one side of the "curtain" whose base is ##C## and whose height above any ##(x,y)## is ##f(x,y)##. Integrating along the whole path would give you the area of the curtain.

In the case where ##C## is located between two points on an axis, such as the line segment that joins ##(a,0)## to ##(b,0)##, then the line integral is actually just a regular single integral:

$$\int_C f(x,y) \space dr = \int_a^b f(x,0) \space dx$$

Which is simply the area under the function. A similar case is observed for ##C := (0, c) \rightarrow (0, d)##:

$$\int_C f(x,y) \space dr = \int_c^d f(0,y) \space dy$$

Complex numbers are really 2-D vectors, and so they are analogous to 2-D vector fields. The line integral of a 2-D vector field corresponds to the real part of the line integral of the conjugate of the complex function.

Usually the most intuitive way to view this is to think about the work done by a force field in moving a particle along a curve from one point to another.

1. What is a complex contour integral problem?

A complex contour integral problem involves calculating the integral of a complex-valued function along a given contour in the complex plane. This type of problem is often encountered in mathematics, physics, and engineering.

2. What is the importance of complex contour integrals?

Complex contour integrals are important in many areas of mathematics and science. They are used to solve problems in complex analysis, differential equations, and physics. They also have applications in signal processing, control theory, and image processing.

3. How is a complex contour integral problem solved?

The solution to a complex contour integral problem involves evaluating the integral using the Cauchy integral formula or the residue theorem. These techniques involve breaking down the complex function into simpler parts and then using known integration methods to solve the problem.

4. What are some common challenges in solving complex contour integral problems?

One of the main challenges in solving complex contour integral problems is determining the correct contour to use. This requires a good understanding of the function and its singularities. Another challenge is accurately evaluating the integral, which can be complicated and time-consuming for more complex functions.

5. What are some real-world applications of complex contour integrals?

Complex contour integrals have many real-world applications, including in physics, engineering, and signal processing. They are used to model and analyze systems with complex behavior, such as electric circuits, fluid flow, and quantum mechanics. They are also used in image processing for edge detection and reconstruction.

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