Question about computing residues after substitution

In summary, the speaker asks a question and receives an answer regarding finding the residuum of a function. They are instructed to transform the function and calculate the residuum of the original and substituted functions. The speaker's error is explained and the solution is provided.
  • #1
Belgium 12
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Hi members,
See attacged PDF file for my question

Thank you
 

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  • #2
The first option is correct. Your substitution function gives a residuum of 1/2.
Now, you need to transform it to the residuum of the original function.

Your function (around the pol z=1) can be written as a Laurent series: ##f(z)=\sum_{n=-1}^\infty a_n (z-1)^n ##. Note that all terms with n<-1 vanish because z=1 is a pol of first order. The residuum is defined as the first coefficient of the Laurent-series with negative index, i.e. a-1.

Now, you found the residuum of the original function to be 1, i.e. a-1=1.
If you do your substitution in the Laurentseries, namely (z-1)=2u, you will be left with a new Laurent series of a different function: ##g(u)=\sum_{n=-1}^\infty b_n (u-0)^n##, where ##b_n=a_n*2^n##. If you calculate the residuum of g in the usual way, you will get b-1=1/2 (which you got).
Now you just have to calculate a-1 from that.

To explain your error: The residuum of the substituted function is not exactly the same as the residuum of the original function.
 
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Likes jim mcnamara and Belgium 12
  • #3
Hi,
thank you very much for the answer
Belgium 12
 

Related to Question about computing residues after substitution

1. What is the purpose of computing residues after substitution?

Computing residues after substitution is a method used in mathematics and computer science to simplify complex equations and expressions. It can also help in finding the roots or solutions of a given equation.


2. How do you compute residues after substitution?

To compute residues after substitution, you first need to substitute a given variable or expression with a specific value. Then, you can use techniques such as expansion, simplification, or evaluation to find the residue or solution of the equation.


3. What is the difference between computing residues after substitution and solving equations?

The main difference between computing residues after substitution and solving equations is the approach. Computing residues after substitution focuses on simplifying and evaluating a given expression, while solving equations aims to find the unknown variables or values that satisfy the equation.


4. When is computing residues after substitution useful?

Computing residues after substitution is useful in various fields such as mathematics, physics, and engineering. It can be used to solve differential equations, evaluate complex integrals, and simplify complex expressions in physics and engineering problems.


5. Are there any limitations or drawbacks to computing residues after substitution?

One limitation of computing residues after substitution is that it may not always give an exact solution or residue. In some cases, it may only provide an approximate solution due to the complexity of the equation or the chosen substitution value. Additionally, it may not be applicable to all types of equations or expressions.

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