Undergrad Solving Trig Integrals with Residue Theorem

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SUMMARY

The discussion focuses on solving trigonometric integrals using the residue theorem, specifically addressing the integral's conversion into a contour integral over the unit circle. The singularity identified is z1 = (-1+(1-a²)¹/²)/a, which lies within the unit circle for |a| < 1. The participant seeks clarification on the conditions under which this is evident and how to demonstrate it rigorously. Additionally, they explore the implications of setting a = sin(x) for real |x| < π/2 and express confusion regarding the nature of 'a' as real or complex.

PREREQUISITES
  • Understanding of complex analysis, specifically the residue theorem
  • Familiarity with contour integration techniques
  • Knowledge of trigonometric identities and their applications in integrals
  • Basic concepts of singularities in complex functions
NEXT STEPS
  • Study the application of the residue theorem in evaluating integrals over the unit circle
  • Learn how to identify and classify singularities in complex functions
  • Explore the relationship between trigonometric functions and complex exponentials
  • Investigate the implications of variable substitutions in complex integrals
USEFUL FOR

Mathematicians, physics students, and anyone interested in advanced calculus, particularly those working with complex analysis and integral evaluation techniques.

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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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