By A. F. Beardon

ISBN-10: 0521271045

ISBN-13: 9780521271042

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**Extra resources for A Primer on Riemann Surfaces**

**Sample text**

3 coincide, each is locally a univalent holomorphic function of w. 3 z. Associated with f is a Riemann surface which, briefly, we con struct as follows. We cut the extended plane in some way so that after having removed the cuts, there remains a simply connected domain possible to choose the cut (in many ways) so that there are valued holomorphic functions w^ (z) in D, m D. It is single each satisfying P(z,Wj (z) ) = 0 throughout D. Now construct m distinct copies of D, say and regard distinct sides of the cut as distinct edges of Wj is regarded only on the closure of it is continuous on this closure.

Let but that there is no sequence in let Show that hq Q^, Q 2,... E be any sequence of is uncountable. 5 CURVES A curve in a topological space from a closed bounded real interval Y X [a,b] is a continuous function into X. y It is common to use to denote the corresponding point set as well although, for emphasis, we sometimes use Cyl = (y(t) The curve y final point : a < t < b}. is closed if the initial point y(b). The curve y is simple y(a) if y coincides with the is 1-1 on [a,b]: it is a simple closed curve if it is closed and if it is 1-1 on the interval Ea,b) .

1. The process above can be made rigorous: for example, by using (z,n) to denote the point to parametrize the copies of log z z z on (E^. It is more convenient though not by integers n but by the values of (the reader should be warned that this construction is not applicable to every many valued function). Thus we are led to consider R = { (z,w) : z = exp(w) } and, as a subset of the metric space £*£, this inherits, without further effort, a Hausdorff topology. Following the earlier intuitive ideas, it is natural to proceed as follows.

### A Primer on Riemann Surfaces by A. F. Beardon

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