An Introduction to Classical Complex Analysis: Vol. 1 by R.B. Burckel

By R.B. Burckel

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This will mean one less distraction when the proofs of these later results, often lengthy enough in their own right, are carried out. The impatient reader may skip this material and return to it later as needed. 36 A subset A oflC is called simply-connectedifits complement qA has no bounded component, doubly-connected if qA has exactly one bounded component. In general we will say A has connectivity n E N U {ro} if qA has exactly n - 1 bounded components. 37 (i) Show that an open subset of IC is simply-connected if either (a) its boundary has no bounded component or (b) it is bounded and has a connected boundary.

0 Then there exists an R E [0, 00] such that: (i) The series (1) converges absolutely for all uniformly in D(zo, p)for any p < R. Z E C with Iz - zol < Rand 54 (ii) (iii) Power Series and the Exponential Function The terms of the series (I) are unbounded (and hence the series diverges) for every z E C with Iz - zol > R. The extended real number R is the reciprocal of the extended real number lim"~ and is called the radius of convergence of the series (I).

Log is continuous and (therefore) differentiable: (2) I '() r og x = II~" 1m log y - log x Y - X = I' 1m II-X I e - e 10.. " ) logy - log x (IOlrll I = elogx = x' since the exponential function is its own derivative. Since the real exponential is an isomorphism of the additive group IR onto the multiplicative group 65 § 3. The Complex Exponential Function (0, (0), its inverse log is an isomorphism of the multiplicative group (0, (0) onto the additive group IR, so + log y for all x, y E (0, (0).

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