Residue Calculus: Evaluating Poles and Contours

In summary, the conversation discusses the evaluation of poles at specific points and whether or not they lie within a given contour. The participants also mention drawing a graph to determine the location of the poles and confirm that for (i), there are no poles within the contour, and for (iii), the poles at z=-1 and z=2 are inside the contour.
  • #1
eeuler
25
0

Homework Statement


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The Attempt at a Solution


So there are poles at: z=[tex]\pm2[/tex] and at z= -1 of order 4. Right?

My query is, when evaluating these poles (using the residue theorem), is it right that for (i) Z = 1/2, no residues lie in that contour?
for (iii), do all residues lie in the contour?
 
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  • #2
Right about the poles, but for the rest, are you just guessing? Why don't you draw a graph? No, I don't think the triangle in iii) includes all poles.
 
  • #3
^^I wasn't sure how to determine whether a pole is inside a specified contour but my tutor suggested drawing a graph, which i have been doing, in which case shouldn't (i) have no poles in the contour? As for (iii) then, would Z=+2, and Z=-1 lie in the contour?
 
  • #4
Your tutor is wise. Yes, i) contains no poles and iii) contains z=(-1) and z=2.
 
  • #5
^^Thanks :)
 

What is residue calculus?

Residue calculus is a branch of mathematics that deals with the calculation of residues of complex functions. It is used to evaluate complex integrals, particularly those that cannot be solved using traditional methods.

What are residues of a complex function?

Residues are the values of a complex function at its singular points. They are calculated by taking the limit of the function as it approaches the singular point.

How is residue calculus used to evaluate complex integrals?

Residue calculus uses the residues of a complex function to calculate complex integrals. This is done by using the residue theorem, which states that the integral of a function around a closed contour is equal to the sum of the residues of the function inside the contour.

What are some real-world applications of residue calculus?

Residue calculus has many applications in engineering, physics, and other scientific fields. It is used in the analysis of electrical circuits, fluid dynamics, and quantum mechanics, among others.

What are the prerequisites for learning residue calculus?

A strong understanding of complex numbers, complex functions, and basic calculus is necessary for learning residue calculus. Knowledge of vector calculus and multivariable calculus is also helpful in understanding more advanced concepts.

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