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Unformatted text preview: Math 185. Sample Answers to Second Midterm 1. (15 points) Carefully define the following. (In each definition you may use without defining them any terms or symbols that were used in the text prior to that definition.) (a). Analytic continuation Answer: If f 1 is analytic on a domain D 1 , if f 2 is analytic on a domain D 2 , if D 1 ∩ D 2 6 = ∅ , and if f 1 ( z ) = f 2 ( z ) for all z ∈ D 1 ∩ D 2 , then we say that f 2 is an analytic continuation of f 1 to D 2 . (Usually we’ll have D 2 ⊆ D 1 , but this not required in the definition.) (b). Residue Answer: Assume that z is an isolated singularity for an analytic function f ( z ), and that f has a Laurent series ∑ ∞ n =-∞ c n ( z- z ) n in some annular neighborhood < | z- z | < (where > 0). Then the residue Res z = z ( f ) of f at z is the coefficient c- 1 . (c). Pole of order m Answer: A function f has a pole of order m at z if its principal part at that point is of the form b 1 z- z + b 2 ( z- z ) 2 + ··· + b m ( z- z ) m with b m 6 = 0. (Alternatively, one could use the principal part in part (b), or the Laurent series on the annular domain 0 < | z- z | < in part (c).) 2. (20 points) Find Res z =0 tan z- z (1- cos z ) 2 ....
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This note was uploaded on 09/01/2010 for the course MATH 185 taught by Professor Lim during the Fall '07 term at Berkeley.
- Fall '07