Residue Theory for Solving Complex Contour Integrals | Step-by-Step Guide

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Homework Help Overview

The problem involves evaluating a complex contour integral using residue theory, specifically focusing on the function \((z-i)^2 / \sin^2(z)\) over a rectangular contour defined by the vertices \(4+4i\), \(4-4i\), \(-4+4i\), and \(-4-4i\).

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to use Taylor series for solving the integral but expresses uncertainty about the effectiveness of this approach. Some participants suggest expanding \(1/\sin(z)\) in its Laurent series and question the order of the pole at \(z=0\). Others discuss the process of multiplying the Laurent series with \((z-i)^2\) and pulling residues from the \(z^{-1}\) term, while also considering the need to expand around other poles within the contour.

Discussion Status

The discussion is ongoing, with participants exploring different methods for expanding the series and calculating residues. There is a mix of approaches being considered, and while some guidance has been offered regarding the expansion of series, there is no explicit consensus on the best method yet.

Contextual Notes

Participants are navigating the complexities of residue theory and the specific behavior of the function \(\sin(z)\) within the defined contour. There is an acknowledgment of the need to consider multiple poles and the implications of their locations on the integration process.

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Homework Statement



\oint (z-i)^2 \div sin^2(z)
gamma : 4+4i, 4-4i, -4+4i, -4-4i rectangluar

Homework Equations



residue theory

The Attempt at a Solution



I don't know how to solve this problem.
I do my best using taylor series... however it doesn't solve the problem..
Please help me...
 
Last edited:
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Expand 1/sin z in its Laurent series and after multiplying the various terms together, pull the residues off the 1/z term.

Hint: what order pole does 1/sin z have? what's the residue of 1/sin z at 0?
 
thank you for reply :)

first, i expanded csc^2(z) in Laurent series

and (z-i)^2 = z^2 -2iz -1

after that, multiply Laurent series and z^2-2iz-1

and pull the residues off the z^-1 term.

but I wasn't sure this was a good idea.

Is this right?
 
Now you just need to do the same thing for the other poles inside your contour, except you have to expand the series around those points instead. (so the series will be in (z-pi) and (z+pi) instead of just z).
 

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