Complex inequality

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  • Thread starter dyn
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  • #1
dyn
622
32
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
 

Answers and Replies

  • #2
35,248
11,502
Hint: Let a=sin(x) for real |x|<pi/2 and simplify.
 
  • #3
dyn
622
32
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
 
  • #4
35,248
11,502
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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