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Jens 'n' Frens |
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Consider a simply connected region of the complex plane with two open, disjoint subsets A and B whose boundaries intersect at the boundary of the region. (Conceptually, the boundary of the region can be approached from either side. E.g. an annulus about the origin with the negative real axis removed; A and B consist of those elements with positive and negative imaginary parts, respectively, then their boundaries intersect along the real line, and part of that intersection is on the boundary of the region.) Now consider an analytic function f on the region (e.g. log(z)); are there any good restrictions on how the limit of f|A compares to the limit of f|B as we approach [a connected piece of] the boundary of the region? (Conceptually, we approach the boundary from different sides. In the case of log, the two differ by an amount that is constant along the relevant stretch of boundary; this seems like too much to ask for in general. On the other hand, I can't construct an analytic function for which this isn't true.) |
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