What is the Integral of a Function Around a Rectangle Oriented Clockwise?

In summary, a Rectangle Around the Pole is a geometric shape used to provide stable and secure support for a vertical pole. It is constructed using various materials and offers benefits such as increased stability and easy maintenance. However, it may require more space and may not be suitable for very tall or heavy poles.
  • #1
Dustinsfl
2,281
5
Find the integral
$$
\int_C\frac{dz}{z^2 - 3z + 5} = \int_C\frac{dz}{\left(z - \frac{3}{2}-i\frac{\sqrt{11}}{2}\right)\left(z-\frac{3}{2}+i\frac{\sqrt{11}}{2}\right)}
$$

Where the path is a rectangle oriented clockwise from (0,0) to (0,4) to (10,4) to (10,0) to (0,0).

So $z_1 = \frac{3}{2} + i \frac{ \sqrt{11} }{2}$ and $ z_2 = \frac{3}{2} - i \frac{ \sqrt{11} }{2}$

The $\int_C\frac{dz}{f(z)}=-2\pi i\text{Res}_{z_0}$

So the residue is
$$
\frac{1}{z_1-z_2} = \frac{1}{-i\sqrt{11}}
$$
Then
$$
\int_C\frac{dz}{z^2 - 3z + 5} = \frac{-2\pi i}{-i\sqrt{11}} =\frac{2\pi}{\sqrt{11}}
$$

Correct?
 
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  • #2
dwsmith said:
Find the integral
$$
\int_C\frac{dz}{z^2 - 3z + 5} = \int_C\frac{dz}{\left(z - \frac{3}{2}-i\frac{\sqrt{11}}{2}\right)\left(z-\frac{3}{2}+i\frac{\sqrt{11}}{2}\right)}
$$

Where the path is a rectangle oriented clockwise from (0,0) to (0,4) to (10,4) to (10,0) to (0,0).

So $z_1 = \frac{3}{2} + i \frac{ \sqrt{11} }{2}$ and $ z_2 = \frac{3}{2} - i \frac{ \sqrt{11} }{2}$

The $\int_C\frac{dz}{f(z)}=-2\pi i\text{Res}_{z_0}$

So the residue is
$$
\frac{1}{z_1-z_2} = \frac{1}{-i\sqrt{11}}
$$
Then
$$
\int_C\frac{dz}{z^2 - 3z + 5} = \frac{-2\pi i}{-i\sqrt{11}} =\frac{2\pi}{\sqrt{11}}
$$

Correct?

This is correct.
 
  • #3
Yes, that is correct! Your solution is well-explained and the final answer is correct. Good job!
 

FAQ: What is the Integral of a Function Around a Rectangle Oriented Clockwise?

What is a Rectangle Around the Pole?

A Rectangle Around the Pole is a geometric shape that is formed by drawing four straight lines around a vertical pole. The lines intersect at right angles to form a rectangular shape.

What is the purpose of a Rectangle Around the Pole?

The purpose of a Rectangle Around the Pole is to provide a stable and secure support for the pole. The rectangular shape evenly distributes the weight of the pole, making it less likely to lean or topple over.

How is a Rectangle Around the Pole constructed?

A Rectangle Around the Pole can be constructed using various materials such as wood, metal, or concrete. The four lines are typically connected at the corners to form a solid frame around the pole.

What are the benefits of using a Rectangle Around the Pole?

Using a Rectangle Around the Pole provides several benefits, including increased stability and durability for the supported pole. It also allows for easy access and maintenance of the pole, as well as a more aesthetically pleasing appearance.

Are there any limitations to using a Rectangle Around the Pole?

The main limitation of using a Rectangle Around the Pole is that it requires a larger area of land compared to other support structures, such as a single post. It also may not be suitable for very tall or heavy poles, as the weight may cause the rectangle to deform or collapse.

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