Download Complex Analysis by Serge Lang (auth.) PDF

By Serge Lang (auth.)

ISBN-10: 1475718713

ISBN-13: 9781475718713

ISBN-10: 147571873X

ISBN-13: 9781475718737

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It is an open set. Let S be a subset of the plane. A boundary point of S is a point oc such that every disc D(oc, r) centered at oc and of radius r > 0 contains both points of S and points not in S. In the closed first quadrant of Fig. 9(b), the points on the x-axis with x ~ 0 and on the y-axis with y ~ 0 are boundary points of the quadrant. A point oc is said to be adherent to S if every disc D(oc, r) with r > 0 contains some element of S. A point oc is said to be an interior point of S if there exists a disc D(oc, r) which is contained in S.

Between the three possible definitions to the reader. 1. A set of complex numbers is compact if and only if it is closed and bounded. Proof. Assume that S is compact. If S is not bounded, for each positive integer n there exists z" E S such that Then the sequence {z,,} does not have a point of accumulation. Indeed, if v is a point of accumulation, pick m > 21 v I, and note that Iv I > O. Then This contradicts the fact that for infinitely many m we must have Zm close to v. Hence S is bounded. To show S is closed, let v be in its closure.

The general formula for a quotient is obtained from this by writing fig = f·1/g, and using the rules for the derivative of a product, and the derivative of I/g. Examples. As in ordinary calculus, from the formula for a product and induction, we see that for any positive integer n, The rule for a quotient also shows that this formula remains valid when n is a negative integer. The derivative of z2/(2z - I) is (2z - 1)2z - 2Z2 (2z - 1)2 This formula is valid for any complex number z such that 2z - 1 =1= More generally, let fez) o.

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