Solving Trig Integrals with Residue Theorem

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    Complex Inequality
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Discussion Overview

The discussion revolves around solving trigonometric integrals using the residue theorem, specifically focusing on the conditions under which certain singularities lie within the unit circle. Participants explore the implications of the variable 'a' and its relationship to the integral.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions the clarity of why the singularity z1 = (-1+(1-a2)1/2)/a lies inside the unit circle for |a| < 1, expressing difficulty in understanding this condition.
  • Another participant suggests substituting a = sin(x) for real |x| < π/2 as a potential simplification method.
  • A different participant expresses confusion regarding the results obtained from the substitution, specifically mentioning the expression -cosec x + cot x and questioning the restriction |x| < π/2.
  • One participant reflects on the assumption that 'a' is real, considering the possibility that it may be complex, which could affect the analysis of the range of x.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus, as there are varying interpretations of the variable 'a' and its implications for the problem. Multiple competing views remain regarding the conditions and simplifications involved.

Contextual Notes

There are limitations related to the assumptions about the nature of 'a' (real vs. complex) and the implications for the range of x, which remain unresolved in the discussion.

dyn
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Hi.
I have looked through an example of working out a trig integral using the residue theorem. The integral is converted into an integral over the unit circle centred at the origin. The singularities are found.
One of them is z1 = (-1+(1-a2)1/2)/a
It then states that for |a| < 1 , z1 lies inside the unit circle.
Should this be obvious just by looking at z1 ? Because I can't see it. I have tried a few values and it seems to be true but that doesn't prove it. If its not obvious how do I go about trying to show it ?
Thanks
 
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Hint: Let a=sin(x) for real |x|<pi/2 and simplify.
 
I must be missing something as the answer I get doesn't seem to help : -cosec x + cot x
Also I'm not sure why I can specify | x | < π/2
 
Hmm... I was assuming a is real but it is probably complex? It might still work but the range of x will need more thought.
 

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