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Stability Proof, continued

The same argument can be extended to the entire right-half plane as follows. Going back to

$\displaystyle \frac{\sin(\nu)}{\nu} = e^\tau,
$

since $ \left\vert\sin(\nu)/\nu\right\vert\leq 1$ for all real $ \nu$ , and since $ \left\vert e^\tau\right\vert>1$ for $ \tau>0$ , this equation clearly has no solutions in the right-half plane. Therefore,

$\displaystyle \zbox{\hbox{Any right-half-plane poles occur at $s=0$.}}
$


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``Stability Proof for a Cylindrical Bore with Conical Cap'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2014-03-24 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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