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Additional resources for Calculus - Single Variable

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The Divisor of a Meromorphic Function. This is a convenient means for cataloguing the zeros and poles of a meromorphic function together with their multiplicities. The notion of a divisor permeates the theory of algebraic functions (in both its concrete and its abstract algebraic forms) and number theory. Given f meromorphic on a region Ω, by the divisor ∂f of f is meant the function with domain Ω given by Algebraic Structure. Given f and g meromorphic on Ω, let E denote the union of the sets of poles of f and g.

This observation now assures us that if f possesses a derivative at each point of , then f′ is continuous in . We conclude this sequence of exercises with a more serious application of Fourier methods, namely, an application to an elementary problem concerning the boundary values of analytic functions. Suppose that f is a finite complex-valued function with domain C(0;1), continuous on its domain. Under what circumstances does there exist a function F with domain , continuous on its domain, analytic in Δ(0;1), and satisfying Clearly, there is at most one such function since the kth coefficient ak of the power-series expansion of F in Δ(0;1) satisfies This assertion may also be established with the aid of the maximum principle.

Math. , 54: 1948), which shows that if Ω1 and Ω2 are two regions K and if φ is an isomorphism of the ring A(Ω1) of analytic functions on Ω1 onto the ring A(Ω2), of analytic functions on Ω2 then either there exists a univalent analytic function ψ1 mapping Ω2 onto Ω1 such that or else there exists a univalent function ψ2 mapping Ω2 onto Ω1 whose conjugate is analytic and which satisfies The function-theoretic requirements of these two papers are modest. There are open questions in this field which are worthy of investigation.