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Unformatted text preview: Analytic Continuation See Arfken & Weber pp 432-434 (in section 6.5 on Laurent expansions) for some of the material below. Our description here will closely follow . 1 Definition The intersection of two domains (regions in the complex plane) D 1 , D 2 , denoted D 1 ∩ D 2 , is the set of all points common to both D 1 and D 2 . The union of two domains D 1 , D 2 , denoted D 1 ∪ D 2 , is the set of all points in either D 1 or D 2 . Now, suppose you have two domains D 1 and D 2 , such that the intersection is nonempty and connected, and a function f 1 that is analytic over the domain D 1 . If there exists a function f 2 that is analytic over the domain D 2 and such that f 1 = f 2 on the intersection D 1 ∩ D 2 , then we say f 2 is an analytic continuation of f 1 into domain D 2 . Now, whenever an analytic continuation exists, it is unique. The reason for this is a basic mathematical result from the theory of complex variables: A function that is analytic in a domain D is uniquely determined over D by its values over a domain, or along an arc, interior to D ....
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This note was uploaded on 11/02/2011 for the course ECON 7125 taught by Professor Smith during the Spring '11 term at James Madison University.
- Spring '11