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Unformatted text preview: The University of Texas at Austin Department of Electrical and Computer Engineering EE362K: Introduction to Automatic Control—Fall 2009 Solutions to Problem Set One C. Caramanis September 9, 2009. 1. Left to the students. 2. The Taylor expansions of cos( αt ), sin( αt ), e iαt and e iαt are: cos( αt ) = 1 α 2 t 2 2! + α 4 t 4 4! ... sin( αt ) = αt α 3 t 3 3! + α 5 t 5 5! ... e iαt = 1 α 2 t 2 2! + α 4 t 4 4! ... + i αt α 3 t 3 3! + α 5 t 5 5! ... e iαt = 1 α 2 t 2 2! + α 4 t 4 4! ... i αt α 3 t 3 3! + α 5 t 5 5! ... , from which one can see that we have the identities: cos( αt ) = e iαt + e iαt 2 sin( αt ) = e iαt e iαt 2 i , and also the desired identity: e iαt = cos( αt ) + i sin( αt ) . Now consider some a ∈ C , and write a = β + iα . Then we have: e at = e βt + iαt = e βt · e iαt = e βt (cos( αt ) + i sin( αt )) . It is clear from this expression that the magnitude of e at depends only on β = Re ( a )....
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