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...
View
Full
Document
This note was uploaded on 03/29/2010 for the course MATH 54257 taught by Professor Hald,oh during the Spring '10 term at Berkeley.
 Spring '10
 Hald,OH
 Math, Differential Equations, Linear Algebra, Algebra, Equations

Click to edit the document details