Understanding and Calculating Residues for 1/(1+z²+z⁴)

In summary, a residue in computing is the leftover remainder after a calculation or operation. It is important in analyzing functions and solving integrals and series. The process for computing the residue involves finding singularities and using a formula to calculate the residue at that point. A simple pole is a singularity where the function approaches infinity, while a higher-order pole is a singularity where the function approaches infinity at a faster rate. The residue can be negative, positive, or zero depending on the function and its singularities.
  • #1
oddiseas
73
0

Homework Statement



computing the residue;
1/(1+z²+z⁴)

Homework Equations



Can someone explain to me what a residue is and how to calculate it! Is it simply the discontinuities of the function?

The Attempt at a Solution

 
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  • #2


Its the co-efficient of the series expansion term at the singularity I think.
 
  • #3


Doesn't your complex analysis text define it? :confused:
 
  • #4


Gregg said:
Its the co-efficient of the series expansion term at the singularity I think.
The residue is the coefficient of (x- x0)-1 in a series expansion about the pole.
 

What is a residue in computing?

A residue in computing refers to the remainder left over after a calculation or operation. It is often used in complex analysis to solve integrals and series.

Why is computing the residue important?

Computing the residue can help in analyzing the behavior of functions, particularly in complex analysis. It can also be useful in solving integrals and series, and in finding solutions to differential equations.

What is the process for computing the residue?

The process for computing the residue involves finding the singularities of a function, determining their type (pole, removable singularity, etc.), and using the appropriate formula to calculate the residue at that singularity.

What is the difference between a simple pole and a higher-order pole?

A simple pole is a singularity of a function where the function approaches infinity, while a higher-order pole is a singularity where the function approaches infinity at a faster rate. The residue at a simple pole is determined by the coefficient of the term with the highest power, while the residue at a higher-order pole involves taking derivatives of the function.

Can the residue be negative?

Yes, the residue can be negative. It simply represents the remainder left over after a calculation, and can be positive, negative, or zero depending on the function and its singularities.

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