Constructive function theory, - interpolation and by I. P. Natanson

By I. P. Natanson

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Example text

If n − 1 of the connected components of the complement C \ Ω are bounded, we also say that Ω is n-connected. A typical example is, when Ω is the open unit disc with the exception of n−1 different points inside the unit circle. This set Ω is n-connected. Again we refer to residue calculus for important applications. We have noted that simple closed curves have started to appear. ), and their two endpoints will always coincide. e. the way it is traversed according to its parametric description. We shall adopt the following convention that if a simply or multiply connected domain Ω is bounded by closed curves as above, then Ω always lies to the left on any of these curves, seen in their sense of orientation.

A formal differentiation gives P (z) = 3z 2 − (2 + 4i)z − (1 − 2i), and it is not hard by division to find that P (z) = 1 + 2i 1 z− 3 9 P (z) + 4 4 iz + = 9 9 1 + 2i 1 z− 3 9 P (z) + 4 i(z − i). 9 The only possible multiple root (in fact of multiplicity 2) is z = i. Instead of continuing the procedure as described above we simple check it by dividing P (z) by (z − i)2 = z − 2iz − 1. This gives P (z) = z 3 − (1 + 2i)z 2 − (1 − 2i)z + 1 = (z − 1)(z − i)2 , and z = i is indeed a root of multiplicity 2.

E. A = { A}◦ . It is easy to prove that A¯ = A◦ ∪ ∂A, that A◦ , ∂A and { A}◦ form a disjoint splitting of C = A◦ ∪ ∂A ∪ { A}◦ for every subset A, and that an isolated point is always a boundary point. In Complex Functions Theory we shall mostly deal with (nonempty) open connected sets. Such sets are also called open domains. A domain is a set, such that its interior is an open domain. If a domain also is closed, we call it a closed domain. 6 In an open domain A, any two points z0 and z ∈ A can always be joined by a continuous curve, which lies entirely in A.

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