Unformatted text preview: âˆ’ s ) t a + ib âˆ’ s Â± Â± Â± N Â· = 1 a + ib âˆ’ s lim N â†’âˆž ( e ( a âˆ’ s + ib ) N âˆ’ 1 ) . (7.1) Since e ( a âˆ’ s + ib ) x = e ( a âˆ’ s ) x e ibx , where the Frst factor vanishes at âˆž if a âˆ’ s < 0 while the second factor is a bounded ( Â± Â± e ibx Â± Â± â‰¡ 1) and periodic function, the limit in (7.1) exists if and only if a âˆ’ s < 0. Assuming that s > a , we get 1 a + ib âˆ’ s lim N â†’âˆž ( e ( a âˆ’ s + ib ) N âˆ’ 1 ) = 1 a + ib âˆ’ s (0 âˆ’ 1) = 1 s âˆ’ ( a + ib ) . 394...
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 Spring '10
 Hald,OH
 Math, Differential Equations, Calculus, Linear Algebra, Algebra, Equations, Limit, lim, triangle, exponential order, ib âˆ’ s

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