By Wolfgang Ebeling
ISBN-10: 0821833197
ISBN-13: 9780821833193
The e-book offers an creation to the speculation of services of numerous complicated variables and their singularities, with detailed emphasis on topological points. the subjects contain Riemann surfaces, holomorphic capabilities of a number of variables, class and deformation of singularities, basics of differential topology, and the topology of singularities. the purpose of the e-book is to steer the reader from the basics to extra complicated themes of contemporary study. all of the valuable necessities are distinct and thoroughly defined. the overall concept is illustrated by way of a variety of examples and applications.
Readership: Graduate scholars and study mathematicians drawn to a number of advanced variables and complicated algebraic geometry.
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Extra resources for Functions of Several Complex Variables and Their Singularities
Sample text
Q(U, g), are disjoint neighborhoods of f , resp. g. For if Q(U, f) f1 Q(U, g) 0, we would then have px (f) = px(g) for some x E U. But it follows from the identity theorem that then f - g, and so f = g. Next we consider the analytic continuation of function germs. 4. 7. Analytic continuation along a path Definition. Let X be a Riemann surface, ry [0, 1] X a path with ,y(O) = xo and y(l) = xl, f E 0xo, g E oxl. One says that g arises from f by analytic continuation along y if and only if there are a subdivision of the interval [0, 1], regions Uj C X with ry([tj_1i tj]) C Uj, and holomorphic functions fj : Uj -> C (j = 1, ...
To passing to a suitable quotient space of their disjoint union with the quotient topology. The very graphic terminology of cutting and pasting was already introduced by B. Riemann, long before the development of set theoretic topology with whose help we can now define these concepts mathematically. The resulting surface is homeomorphic to a cylinder. If we are now to add in the two points lying over oo, we must 1. 8. The Riemann surface of the function A - z2 paste, respectively, a point to the upper and to the lower boundary of the cylinders, so obtaining the Riemann sphere C.
Let yo = (xo, 0), where 0 is the homotopy class of the constant paths with start point xo. Furthermore, let y = (x) a) E X. Let ry be a path from xo to x representing the homotopy class a. For T E I = [0, 1] we put ryT(t) = ry(Tt) and aT = ['y]. ) Then we consider the map ry:I t X (y(t),c). By construction '5'(0) = yo and =y(1) = y. We show that =y is continuous. Towards this, let r E I and V be an open neighborhood ofry(T). Without loss of generality V = U(aT), where U is an open neighborhood of ry(T).