Definite Integral using Residue Thm

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Homework Help Overview

The discussion revolves around calculating the integral of the function [ z^4/(1 + z^8) ] over the interval from negative infinity to positive infinity, utilizing the Residue Theorem. The problem is situated within the context of complex analysis, specifically focusing on integrals involving rational functions and their poles.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to identify the poles of the function and calculate the residues in the upper half-plane. They express confusion regarding the steps needed to arrive at the answer provided in their textbook, particularly in relation to expressing the result in terms of arcsin.

Discussion Status

Some participants offer guidance on calculating residues at poles, suggesting a method involving the derivative of the denominator. Others question the form of the answer, with a participant correcting an earlier assertion about the presence of arcsin in the final result. The discussion reflects a mix of interpretations and approaches without reaching a consensus.

Contextual Notes

There is mention of previous attempts to seek help, indicating that the original poster has not fully resolved their confusion despite prior guidance. The complexity of the expressions involved and the specific requirement for the answer format are also noted as constraints in the discussion.

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Homework Statement



Calculate the integral [ z^4/(1 + z^8) ] over negative infinity to positive infinity.

Homework Equations



Residue Theorem. Specifically for real-valued rational functions (on the real axis) where the denominator exceeds the degree of the numerator by at least two or more. The denominator has only complex poles, and the integral is given by 2πi(sum of all residues in the upper half plane U).

The Attempt at a Solution


I found the eight poles, only four of which are in the upper plane. So the residues in U we are looking for are for poles exp(πi/8), exp(3πi/8), exp(5πi/8), exp(7πi/8).

I posted this question before but unfortunately the help I was given didn't clear my confusion. The answer in the back of the book is π/4 * arcsin(3π/8) and I can't trace the steps to get this. I would greatly appreciate if somebody could walk me through finding the residue of one of these poles so that I can complete the rest of the question myself. I need to be able to get the answer in terms of arcsin which is part of my problem. When I multiplied the denominator all out for a pole then I get a huge expression of various exponents of e.

Thank you!
 
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Huge protip: If you have f(x)/g(x), and you have a simple pole, the residue at the pole is just f(x)/g'(x) evaluated at the point. This will save you hours of anguish trying to sum up different combinations of roots of unity
 
I think the answer shouldn't contain arcsin. It must be 1/sin.
 
Ok, yeah I foolishly wrote the answer is arcsine but it is actually π/4 * 1/sin(3π/8).
 

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