Real and Complex Analysis (Rudin)

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ANALYTIC CONTINUATION 331 PROOF Let Qo be the right half of Q. More precisely, Qo consists of all Z E II+ such that 0< Re z < I, 12z-11>1. (1) By Theorem 14.19 (and Remarks 14.20) there is a continuous function h on Qo which is one-to-one on Qo and holomorphic in Qo, such that h(Qo) = II + , h(0) = 0, h(l) = I, and h(oo) = 00. The reflection principle (Theorem 11.14) shows that the formula h( - x + iy) = h(x + iy) (2) extends h to a continuous function on the closure Q of Q which is a conformal mapping of the interior of Q onto the complex plane minus the nonnegative real axis. We also see that h is one-to-one on Q, that h(Q) is the region n described in (c), that and that h( -1 + iy) = h(l + iy) = h('t( -1 + iy» (0 < y < (0), (3) h( -! + !ei~ = h(! + !ei ("-9» = h(u( -! + !ei~) (0 < () < 1t). (4) Since h is real on the bo.dary of Q, (3) and (4) follow from (2) and the definitions of (1 and 'to I We now define the function A: (z E
332 REAL AND COMPLEX ANALYSIS value O. The latter theorem is actually true in a local situation: Iff has an isolated Singularity at a point Zo and iff omits two values in some neighborhood of zo, then Zo is a removable singularity or a pole of! This so-called "big Picard theorem" is a remarkable strengthening of the theorem of Weierstrass (Theorem 10.21) which merely asserts that the image of every neighborhood of Zo is dense in the plane iff has an essential singularity at Zo. We shall not prove it here. 16.22 Theorem Iff is an entire function and if there are two distinct complex numbers IX and fJ which are not in the range off, thenfis constant. PROOF Without loss of generality we assume that IX = 0 and fJ = 1; if not, replace f by (f - 1X)/(fJ - IX). Then f maps the plane into the region n described in Theorem 16.20. With each disc Dl en there is associated a region VI c rr+ (in fact, there are infinitely many such Vto one for each ((J E n such that A is one-toone on VI and A(V 1) = D 1 ; each such VI intersects at most two of the domains ((J(Q). Corresponding to each choice of VI there is a function "'1 E H(D1) such that "'1(A(Z» = z for all z E VI' If D2 is another disc in n and if Dl () D2 "# 0, we can choose a corresponding V2 so that VI () V2 "# 0. The function elements ("'to D 1) and ("'2, D2) will then be direct analytic continuations of each other. Note that "'i(DJ c rr+. Since the range off is in n, there is a disc Ao with center at 0 so that f(Ao) lies in a disc Do in n. Choose "'0 E H(Do), as above, put g(z) = "'o(f(z» for z E Ao, and let y be any curve in the plane which starts at O. The range of f 0 y is a compact subset of n. Hence y can be covered by a chain of discs, A o, ... , All' so that each f(AJ lies in a disc Di in n, and we can choose "'i E H(Di) so that ("'i' Di) is a direct analytic continuation of ("'i-I' Di- 1), for i = 1, ... n. This gives an analytic continuation of the function element (g, Ao) along the chain {Ao, ... , All}; note that "'" 0 f has positive imaginary part. Since (g, Ao) can be analytically continued along every curve in the plane and since the plane is simply connected, the monodromy theorem implies that g extends to an entire function. Also, the range of g is in rr +, hence (g - i)/(g + i) is bounded, hence constant, by Liouville's theorem. This shows that g is constant, and since "'0 was one-to-one onf(Ao) and Ao was a nonempty open set, we conclude thatfis constant. IIII Exercises 1 Suppose f(z) = :Ea. z·, a. ~ 0, and the radius of convergence of the series is 1. Prove that f has a singularity at z = 1. Hint: Expandfin powers of z - t. If 1 were a regular point off, the new series would converge at some x > 1. What would this imply about the original series? 2 Suppose (f, D) and (g, D) are function elements, P is a polynomial in two variables, and P(f, g) = 0 in D. Suppose f and g can be analytically continued along a curve y, to fl and gl' Prove that
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CONTENTS Preface xiii Prologue: The
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CONTENTS ix Chapter 10 Elementary P
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CHAPTER FOURTEEN CONFORMAL MAPPING
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CHAPTER TWENTY UNIFORM APROXIMATION
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BIBLIOGRAPHY 1. L. V. Ahlfors: "Com
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LIST OF SPECIAL SYMBOLS AND ABBREVI
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INDEX Absolute continuity, 120 of f
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INDEX 411 Double integral, 165 Dual
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INDEX 413 Lower limit, 14 Lower sem
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INDEX 415 Separable space, 92, 247
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Real and Complex Analysis (Rudin)

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