I have question about Residues

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In summary, the equation at the bottom of the first page in the PDF is based on the Cauchy-Goursat Theorem, which states that the integral of a complex function around a closed path in the complex plane is zero if the function is analytic in the region inside the path. This theorem is used to calculate the integral of ∫c zn dz, where n<0, by deforming the path to encircle the origin and using a polar substitution to perform a real integration. The result of the integral is 2∏i c-1, which is the basis for the equation in question. To better understand this concept, it is recommended to refer to the chapters on complex variables in the book "Complex Variables with
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stgermaine
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Homework Statement


http://people.math.gatech.edu/~cain/winter99/ch10.pdf

In this PDF, I don't understand the equation at the bottom of the first page describint c sub (-1). On what basis is that equation correct? I don't know why but my diff eq teacher's giving out a test and he says there's going to be something related to the residue theorem. I haven't taken complex analysis and it's very confusing. I'd appreciate it if someone can help.

Thank you
 
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stgermaine said:
http://people.math.gatech.edu/~cain/winter99/ch10.pdf

In this PDF, I don't understand the equation at the bottom of the first page describint c sub (-1). On what basis is that equation correct?
The information you are missing is the Cauchy-Goursat Theorem.

It says if you take the integral of a complex function around a closed path in the complex plane and the complex function is everywhere analytic in the region inside the path, the result is zero (this means no singularities).

Usually what follows this is a demonstration of ∫c zn dz because the result is useful. If n>=0 then everywhere enclosed by the path is analytic and the integral will be zero. Here zn for n<0 has a singularity at the origin. If you integrate along a closed path that does not include the origin on its inside, then the result is still zero because zn is still analytic inside the closed curve. But if the closed curve encircles the origin (eg the unit circle is the path) the theorem cannot be used.

Instead a direct integration is done. Suppose the path is the unit circle enclosing zn where n<0. Then you can do a polar substitution z=r e and perform a real integration with respect to θ. If you do that, you will find the integral is still zero except when n=-1, for which you get the result:

c zn dz = 2∏i if n=-1 and 0 otherwise

Another theorem says paths can be deformed as long as they don't cross singularities. So this unit circle path can be deformed to any shape around the origin and you get the same result in the integral.... back to your page. You are supposing the complex function f(z) can be represented in a Laurent series. Now integrate that around a closed path enclosing zo. The only integral term of that series that is not zero is the term belonging to (z-zo)-1. And the result of the integral will be 2∏i c-1

If you are familiar with complex numbers, I can recommend chapters 3 and 4 of "Complex Variables with Applications 2nd Ed" by David Wunsch to get the ideas if this is going to be an important part of your course.
 

1. What are residues in chemistry?

Residues in chemistry refer to the remaining components or byproducts of a chemical reaction. They can also refer to the leftover components of a substance after it has been separated or purified.

2. How are residues formed?

Residues are formed when reactions occur between different chemicals, resulting in the creation of new substances. They can also form as a result of physical processes such as evaporation or filtration.

3. What are the effects of residues on the environment?

Residues can have both positive and negative effects on the environment. Some residues can be beneficial, such as those used in fertilizers, while others can be harmful and contribute to pollution or contamination of ecosystems.

4. How are residues monitored and controlled?

Residues are monitored and controlled through various methods, such as regular testing and analysis, setting limits and regulations, and implementing proper disposal and treatment methods. Many industries also have guidelines and protocols in place to minimize the production and release of residues.

5. What are some examples of residues in everyday life?

Residues can be found in many aspects of everyday life, such as in household cleaning products, food additives, and medications. They can also be present in industrial processes, such as in the production of plastics and fuels.

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