Complex Variables: An Introduction (Graduate Texts in Mathematics)

Complex Variables: An Introduction (Graduate Texts in Mathematics)

Language: English

Pages: 652

ISBN: 0387973494

Format: PDF / Kindle (mobi) / ePub

Textbooks, even excellent ones, are a reflection of their times. Form and content of books depend on what the students know already, what they are expected to learn, how the subject matter is regarded in relation to other divisions of mathematics, and even how fashionable the subject matter is. It is thus not surprising that we no longer use such masterpieces as Hurwitz and Courant's Funktionentheorie or Jordan's Cours d'Analyse in our courses. The last two decades have seen a significant change in the techniques used in the theory of functions of one complex variable. The important role played by the inhomogeneous Cauchy-Riemann equation in the current research has led to the reunification, at least in their spirit, of complex analysis in one and in several variables. We say reunification since we think that Weierstrass, Poincare, and others (in contrast to many of our students) did not consider them to be entirely separate subjects. Indeed, not only complex analysis in several variables, but also number theory, harmonic analysis, and other branches of mathematics, both pure and applied, have required a reconsidera­ tion of analytic continuation, ordinary differential equations in the complex domain, asymptotic analysis, iteration of holomorphic functions, and many other subjects from the classic theory of functions of one complex variable. This ongoing reconsideration led us to think that a textbook incorporating some of these new perspectives and techniques had to be written.

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bound is a consequence of the fact that (B(x m, !d(xm)))m~ 1 is a covering of IRft; the upper bound, from the bound on the number of balls of radius td(xm) intersecting at a single point. §3. Partitions of Unity 21 Let Mo = {m: B(x m , !d(x m )) (1 F by the formula qJ(X) := ( =1= L 0}. We define the required function qJ meMo t/!m (X»)/'P (x). It is easy to verify that qJ has all the properties stated in the proposition. o EXERCISES 1.3 1. Let cp, t/J be functions in ~(IR), with supp

x)'" = 2: n~O (_1)"1X(1X - 1) ... (IX n! - n + 1) xn. The power series obtained by replacing x with z converges in B(O, 1) and hence defines a hoI om orphic function there, also denoted (1 - z)"'. One can verify that (1 - z)'" coincides in B(O, 1) with e crLog(1-Z). In fact, these two holomorphic functions already coincide on ] -1,1[. 2. Analytic Properties of Holomorphic Functions 120 (3) Let f(z) = J: ~!~~2 = L (_I)"z2n 1+z n~O in B(O,I). The function F(z) f(Od( is a primitive of

[ -1, IJ consider the function 1.) =0 be? What ~ IR? (Hint: For the second part fez) f(z) + 1 12. Let B+ = {z E B: 1m z > O} and f E ff(B+) n 'G'(B+) such that If(x)1 = 1 for x E ] - 1, 1[. Show that f has a unique meromorphic extension to the whole disk B. Do poles actually occur? 13. For each example that follows define a hoi om orphic function in an annulus such that it has the following as its Laurent series development: 00 (a) )~oo (b) I z" Rf; 00 n=-oc r1nl(z - 1)". 146

function gin 1[:\ {O} such that f(z) = g(exp(2niz). Conclude that f can be represented in the form I 00 f(z) = k= ake hikz , -00 the series being convergent in ~(IC). It is called the Fourier series of f. Show that the Fourier series converges uniformly in every strip a < 1m z < b. Furthermore, the coefficients can be computed from the formula ak = II f(x + ib)e- 2 "ik(x+ib) dx, for any fixed b E IR. Generalize this expansion to the case when the periodic function f is not entire, but

J~"Z)jdX = -~. (8) One can apply the preceding method to Ioo R(x) log x dx when R has a simple pole at the point x = 1. In this case the vanishing of log x at x = 1 keeps the integral convergent. One needs to modify the previous contour as shown in Figure 2.6. If R is real-valued on the real axis one obtains the relation f 'Yj o R(x)logxdx = 1 n2Res(R, 1) - -Re 2 L Res(R(z)(logz)2,a). a", 1 An example of an application of this identity is given by: Ioo ~~~xTdx = ~2. One can

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