Real and Complex Analysis (Rudin)

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UNIFORM APPROXIMATION BY POLYNOMIALS 387 Some Lemmas 20.2 Lemma Suppose D is an open disc of radius r > 0, E c D, E is compact and connected, n = S2 - E is connected, and the diameter of E is at least r. Then there is afunction g E H(n) and a constant b, with the following property: If the inequalities Q((, z) = g(z) + (( - b)g2(Z), 100 IQ((, z)1
388 REAL AND COMPLEX ANALYSIS Since g is a conformal mapping of Q onto D(O; II I a I), (7) shows that 4 I g(z) I 2r). (11) Let robe a large circle with center at 0; (11) gives (by Cauchy's theorem) that A.2«() = 21 . r (z - ()g(z) dz = b - (. 1tl Jro Substitute this value of A.i() into (11). Then (1) shows that the function (12) is bounded as z--+ 00. Hence q> q>(z) = [ Q«(, z) - z ~ (}z - ()3 (13) z E Q (') D, then I z - (I < 2r, so (2) and (13) give has a removable singularity at 00. If I q>(z) I < 8r 3 I Q«(, z)1 + 4r2 < I,OOOr2. (14) By the maximum modulus theorem, (14) holds for all z E Q. This proves (3). IIII 20.3 Lemma Suppose f E C~(R2), the space of all continuously differentiable functions in the plane, with compact support. Put a =!(~+ i~). 2 ax ay Then the following" Cauchy formula" holds: f(z) = -! fT (afX() de d" n JR2 (- Z (1) (2)
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REAL AND COMPLEX ANALYSIS
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CONTENTS Preface xiii Prologue: The
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CONTENTS ix Chapter 10 Elementary P
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Page 424 and 425: INDEX Absolute continuity, 120 of f
Page 426 and 427: INDEX 411 Double integral, 165 Dual
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Page 430 and 431: INDEX 415 Separable space, 92, 247
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Real and Complex Analysis (Rudin)

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