By A. F. Beardon

ISBN-10: 0521271045

ISBN-13: 9780521271042

**Read or Download A Primer on Riemann Surfaces PDF**

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

**Example text**

Have a common point, then there are points within a distance We insist that V (say f(U) are disjoint. 1 1. Show that if a sequence lies in a compact subset of a surface, then it has a convergent subsequence. 2. Let E be a subset of a surface S and suppose that E does not contain any convergent sequence of distinct points. Show that S - E is a surface. 2 RIEMANN SURFACES A surface is a Riemann surface if the change from one coordinate system to another is holomorphic. As this is our major concern, we give a formal definition.

As f is continuous, subset of W h is continuous whenever and note that g is. Let A be any open 28 h -1 (A) = f_ 1 (g_ 1 (A)). If Tf so g h g_ 1 (A) is continuous then is in the quotient topology is continuous: this proves (2). Because f is surjective, if B e y we have hf_ 1 (B) = gff_ 1 (B) = g(B). By taking B f are open and g open, we see that ifhis open then so is h then so is g. Finally, = gf. Exercise 2,7 1, Let generated by x that the map g G (x) b- exp(ix) G x + 2irand let of X/G into W be the group of translations of the real line f : X -► X/G be the natural map.

And let G be generated by g:z 2z. 7 y = {y = e X } Show that the G-images of D/G. ALGEBRAIC FUNCTIONS The following ideas are necessarily intuitive and will be justified in Chapter 7. Let say, in w. P(z,w) be a polynomial in Now consider the m-valued function takes as values the solutions yields m distinct values w_. w. of z and f P(z,w) = 0. w, of degree which, for each m, z, In general, this and apart from certain values of z where 52 the w. 3 coincide, each is locally a univalent holomorphic function of w.

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

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