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 ℱ ={
(fα,Sα)|fα∈ Φ and Sα a ball in 𝒞n} with respect to a collection of maps, 𝒜 of
subsets of ℱ into ℱ. 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.
