Vol. 42, No. 3, 1972

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Generalized continuation

Alan Seymour Cover

Vol. 42 (1972), No. 3, 589–601

In this paper the operation of analytic continuation is generalized by relaxing the condition that a direct continuation of a function must have the same values as the original on the intersection of their domains of definition. Thus the generalized continuations of a function can have some other property in common with the original function such as being preimages of a single function under a local integral operator. This generalization is accomplished by developing 𝒜-continuation of ={ (fα,Sα)|fα Φ and Sα a ball in 𝒞n} with respect to a collection of maps, 𝒜 of subsets of into . 𝒜 must satisfy some compatibility conditions. Many of the proofs in this development parallel those for analytic continuation and lead to the introduction of a manifold on which the generalized continuation is single valued. A generalized continuation of function elements (fα,Sα) is achieved when all the fα’s are complex valued functions defined on Sα and some examples are given.

Mathematical Subject Classification 2000
Primary: 32D15
Secondary: 32A30
Received: 22 December 1970
Published: 1 September 1972
Alan Seymour Cover