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Real and complex analysis

Real and complex analysis

Real and complex

• Page 2: REAL AND COMPLEX ANALYSIS
• Page 5 and 6: REAL AND COMPLEX ANALYSISINTERNATIO
• Page 8 and 9: CONTENTSPrefacexiiiPrologue: The Ex
• Page 10: CONTENTS ixChapter 10Elementary Pro
• Page 16 and 17: PROLOGUETHE EXPONENTIAL FUNCTIONThi
• Page 18 and 19: PROLOGUE: THE EXPONENTIAL FUNCTION
• Page 20 and 21: CHAPTERONEABSTRACT INTEGRATIONTowar
• Page 22 and 23: ABSTRACT INTEGRA nON 7If no two mem
• Page 24 and 25: ABSTRACT INTEGRA nON 9It would perh
• Page 26 and 27: ABSTRACT INTEGRATION 111.8 Theorem
• Page 28 and 29: ABSTRACT INTEGRATION 131.12 Theorem
• Page 30 and 31: ABSTRACT INTEGRATION 15Corollaries(
• Page 32 and 33: ABSTRACT INTEGRA nON 17As the proof
• Page 34 and 35: ABSTRACT INTEGRATION 19The cancella
• Page 36 and 37: ABSTRACT INTEGRATION 21Next, let s
• Page 38 and 39: Corollary If aij :2= ° for i and j
• Page 40 and 41: ABSTRACT INTEGRATION 25each of thes
• Page 42 and 43: ABSTRACT INTEGRA nON 27PROOF Since
• Page 44 and 45: ABSTRACT INTEGRA TlON 291.37 The fa
• Page 46 and 47: ABSTRACT INTEGRATION 31PROOF Let a
• Page 48 and 49: CHAPTERTWOPOSITIVE BOREL MEASURESVe
• Page 50 and 51: POSITIVE BOREL MEASURES 35[The conv
• Page 52 and 53:

POSITIVE BOREL MEASURES 372.6 Theor

• Page 54 and 55:

POSITIVE BOREL MEASURES 39will mean

• Page 56 and 57:

POSITIVE BOREL MEASURES 41(b) I-'(K

• Page 58 and 59:

POSITIVE BOREL MEASURES 43Put V = U

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POSITIVE BOREL MEASURES 45PROOF If

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POSITIVE BOREL MEASURES 47Since hi

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POSITIVE BOREL MEASURES 49PROOF Put

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POSITIVE BOREL MEASURES 51(c) m is

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POSITIVE BOREL MEASURES 532.21 Rema

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POSITIVE BOREL MEASURES 55If T is o

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POSITIVE BOREL MEASURES 57Sincethe

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POSITIVE BOREL MEASURES S913 Is it

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CHAPTERTHREELP-SPACESConvex Functio

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If-SPACES 63(4) becomesexp U (Xl +

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I!'-SPACES 65Hence the left side of

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H-SPACES 67Suppose j, g, and h are

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I!-SPACES 69The simple functions pl

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l!'-SPACES 71Given / E Co(X) and

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If-SPACES 73(b) Prove that equality

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I!'-SPACES 75(b) If 1 :5 P < 00 and

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ELEMENTARY HILBERT SPACE THEORY 774

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ELEMENTARY HILBERT SPACE THEORY 79I

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ELEMENTARY HILBERT SPACE THEORY 81P

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ELEMENTARY HILBERT SPACE THEORY 834

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ELEMENTARY IDLBERT SPACE THEORY 8S4

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ELEMENTARY HILBERT SPACE THEORY 87(

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ELEMENTARY HILBERT SPACE THEORY 89W

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ELEMENTARY HILBERT SPACE THEORY 91T

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ELEMENTARY HILBERT SPACE THEORY 93M

• Page 110 and 111:

CHAPTERFIVEEXAMPLES OF BANACH SPACE

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EXAMPLES OF BANACH SPACE lECHNIQUES

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EXAMPLES OF BANACH SPACE TECHNIQUES

• Page 116 and 117:

EXAMPLES OF BANACH SPACE TECHNIQUES

• Page 118 and 119:

EXAMPLES OF BANACH SPACE TECHNIQUES

• Page 120 and 121:

EXAMPLES OF BANACH SPACE TECHNIQUES

• Page 122 and 123:

EXAMPLES OF BANACH SPACE TECHNIQUES

• Page 124 and 125:

EXAMPLES OF BANACH SPACE TECHNIQUES

• Page 126 and 127:

EXAMPLES OF BANACH SPACE TECHNIQUES

• Page 128 and 129:

EXAMPLFS OF BANACH SPACE TECHNIQUES

• Page 130 and 131:

EXAMPLES OF BANACH SPACE TECHNIQUES

• Page 132 and 133:

COMPLEX MEASURES 117This notation i

• Page 134 and 135:

COMPLEX MEASURES 119We have thus sp

• Page 136 and 137:

COMPLEX MEASURES 121(f) Since A2 .1

• Page 138 and 139:

COMPLEX MEASURES 123Hence g(x) E [0

• Page 140 and 141:

COMPLEX MEASURES 125By analogy with

• Page 142 and 143:

COMPLEX MEASURES 127then

• Page 144 and 145:

COMPLEX MEASURES 129The first part

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COMPLEX MEASURES 131Once we have th

• Page 148 and 149:

COMPLEX MEASURES 1334 Suppose 1 :5

• Page 150 and 151:

CHAPTERSEVENDIFFERENTIATIONIn eleme

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DIFFERENTIATION 1377.3 Lemma If W i

• Page 154 and 155:

DIFFERENTIATION 139PROOF Definefor

• Page 156 and 157:

DIFFERENTIA nON 141PROOF Let x be a

• Page 158 and 159:

DIFFERENTIATION 1437.14 Theorem Sup

• Page 160 and 161:

DIFFERENTIATION 1452- n - 1

• Page 162 and 163:

DIFFERENTIATION 147Assume next that

• Page 164 and 165:

DIFFERENTIATION 149The next theorem

• Page 166 and 167:

DIFFERENTIATION 151stronger hypothe

• Page 168 and 169:

DIFFERENTIATION, 153Since r" = m(B(

• Page 170 and 171:

DIFFERENTIATION 155Theorem 7.8 tell

• Page 172 and 173:

DIFFERENTIATION 1577 Construct a co

• Page 174 and 175:

DIFFERENTIATION 15921 Iffis a real

• Page 176 and 177:

INTEGRATION ON PRODUCT SPACES 161If

• Page 178 and 179:

INTEGRATION ON PRODUCT SPACES 163Pr

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INTEGRATION ON PRODUCT SPACES 165(c

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INTEGRATION ON PRODUCT SPACES 167To

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INTEGRATION ON PRODUCT SPACES 169Th

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INTEGRATION ON PRODUCT SPACES 171wh

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INTEGRATION ON PRODUCT SPACES 173Fo

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INTEGRATION ON PRODUCT SPACES 175S

• Page 192 and 193:

INTEGRATION ON PRODUCT SPACES 177Su

• Page 194 and 195:

FOURIER TRANSFORMS 179The formal pr

• Page 196 and 197:

FOURIER TRANSFORMS 181Let us see wh

• Page 198 and 199:

FOURIER TRANSFORMS 183The integrand

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FOURIER lRANSFORMS 185and Theorem 3

• Page 202 and 203:

FOURIER lRANSFORMS 187Theorem 9.2(d

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FOURIER TRANSFORMS 189orthogonal pr

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FOURIER TRANSFORMS 191follows: We k

• Page 208 and 209:

FOURIER TRANSFORMS 193Thenf'! (d (X

• Page 210 and 211:

FOURIER TRANSFORMS 195What does (*)

• Page 212 and 213:

ELEMENTARY PROPERTIES OF HOLOMORPHI

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ELEMENTARY PROPERTIES OF HOLOMORPHI

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ELEMENTARY PROPERTIES OF HOLOMORPHI

• Page 218 and 219:

ELEMENTARY PROPERTIES OF HOLOMORPHI

• Page 220 and 221:

ELEMENTARY PROPERTIES OF HOLOMORPlD

• Page 222 and 223:

ELEMENTARY PROPERTIES OF HOLOMORPHI

• Page 224 and 225:

ELEMENTARY PROPERTIES OF HOLOMORPHI

• Page 226 and 227:

ELEMENTARY PROPERTIES OF HOLOMORPHI

• Page 228 and 229:

ELEMENTARY PROPERTIES OF HOLOMORPHI

• Page 230 and 231:

ELEMENTARY PROPERTIES OF HOLOMORPHI

• Page 232 and 233:

ELEMENTARY PROPERTIES OF HOLOMORPHI

• Page 234 and 235:

ELEMENTARY PROPERTIES OF HOLOMORPHI

• Page 236 and 237:

ELEMENTARY PROPERTIES OF HOLOMORPHI

• Page 238 and 239:

ELEMENTARY PROPERTIES OF HOLOMORPHI

• Page 240 and 241:

ELEMENTARY PROPERTIES OF HOLOMORPlD

• Page 242 and 243:

ELEMENTARY PROPERTIES OF HOLOMORPHI

• Page 244 and 245:

ELEMENTARY PROPERTIES OF HOLOMORPHI

• Page 246 and 247:

CHAPTERELEVENHARMONIC FUNCTIONSThe

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HARMONIC FUNCTIONS 233The Poisson I

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HARMONIC FUNCTIONS 235Note: This th

• Page 252 and 253:

HARMONIC FUNCTIONS 237The Mean Valu

• Page 254 and 255:

HARMONIC FUNCTIONS 239Hence

• Page 256 and 257:

HARMONIC FUNCTIONS 241The regions n

• Page 258 and 259:

HARMONIC FUNCTIONS 243Hence, settin

• Page 260 and 261:

HARMONIC FUNCTIONS 245(a) If P. is

• Page 262 and 263:

HARMONIC FUNCTIONS 247PROOF To say

• Page 264 and 265:

HARMONIC FUNCTIONS 249As before, Lo

• Page 266 and 267:

HARMONIC FUNCTIONS 251If {u.} is a

• Page 268 and 269:

CHAPTERTWELVETHE MAXIMUM MODULUS PR

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THE MAXIMUM MODULUS PRINCIPLE 255Th

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THE MAXIMUM MODULUS PRINCIPLE 257Fi

• Page 274 and 275:

THE MAXIMUM MODULUS PRINCIPLE 259Fi

• Page 276 and 277:

THE MAXIMUM MODULUS PRINCIPLE 26112

• Page 278 and 279:

THE MAXIMUM MODULUS PRINCIPLE 263No

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THE MAXIMUM MODULUS PRINCIPLE 265Su

• Page 282 and 283:

APPROXIMATIONS BY RATIONAL FUNCTION

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APPROXIMATIONS BY RATIONAL FUNCTION

• Page 286 and 287:

APPROXIMA nONS BY RA nONAL FUNCTION

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APPROXIMA TIO~S BY RATIONAL FUNCTIO

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APPROXIMATIONS BY RATIONAL FUNCTION

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APPROXIMATIONS BY RATIONAL FUNCTION

• Page 294 and 295:

CONFORMAL MAPPING 279Here /'(zo) =

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CONFORMAL MAPPING 281Let us discuss

• Page 298 and 299:

CONFORMAL MAPPING 283Under these co

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CONFORMAL MAPPING 28514.9 Remarks T

• Page 302 and 303:

CONFORMAL MAPPING 287Divide Eqs. (5

• Page 304 and 305:

CONFORMAL MAPPING 289PROOF Iff= I/(

• Page 306 and 307:

CONFORMAL MAPPING 291Theorem 14.18(

• Page 308 and 309:

CONFORMAL MAPPING 293Put y(t) = .JR

• Page 310 and 311:

CONFORMAL MAPPING 29518 Suppose 0 i

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CONFORMAL MAPPING 297(d) Let ex be

• Page 314 and 315:

ZEROS OF HOLOMORPHIC FUNCTIONS 299I

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ZEROS OF HOLOMORPHIC FUNCTIONS 301c

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ZEROS OF HOLOMORPHIC FUNCTIONS 303I

• Page 320 and 321:

ZEROS OF HOLOMORPlDC FUNCTIONS 305a

• Page 322 and 323:

ZEROS OF HOWMORPHIC FUNCTIONS 307Je

• Page 324 and 325:

ZEROS OF HOLOMORPIDC FUNCTIONS 309a

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ZEROS OF HOWMORPHIC FUNCTIONS 311is

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ZEROS OF HOLOMORPHIC FUNCTIONS 313t

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ZEROS OF HOLOMORPHIC FUNCTIONS 315i

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ZEROS OF HOLOMORPHIC FUNCTIONS 317P

• Page 334 and 335:

CHAPTERSIXTEENANALYTIC CONTINUATION

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ANALYTIC CONTINUATION 321integers.

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• Page 340 and 341:

ANALYTIC CONTINUATION 325There are

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ANALYTIC CONTINUATION 32716.15 Tbeo

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ANALYTIC CONTINUATION 329We claim t

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ANALYTIC CONTINUATION 331PROOF Let

• Page 348 and 349:

ANALYTIC CONTINUATION 333P(f1' 01)

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CHAPTERSEVENTEENThis chapter is dev

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HP-SPACES 337some r > 0 we have D(z

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HP-SPACES 339PROOF Note first thatI

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For each z E U, 11 - e-ilz I and P(

• Page 358 and 359:

log I 9 I are 0 a.e. (Theorem 11.22

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Since Ilog+ u - log+ V I ~ I u - v

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X n , the one-dimensional spaces sp

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So there exists a qJ E Y such that

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HP-SPACES 351Let us recall that eve

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5 Suppose fe HP, qJ e H(U), and qJ(

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ffI' -SPACES 355exists (and is fini

• Page 372 and 373:

ELEMENTARY TIIEORY OF BANACH ALGEBR

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ELEMENTARY THEORY OF BANACH ALGEBRA

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ELEMENTARY THEORY OF BANACH ALGEBRA

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ELEMENTARY THEORY OF BANACH ALGEBRA

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ELEMENTARY THEORY OF BANACH ALGEBRA

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ELEMENTARY THEORY OF BANACH ALGEBRA

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ELEMENTARY THEORY OF BANACH ALGEBRA

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CHAPTERNINETEENHOLOMORPHIC FOURIER

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HOLOMORPHIC FOURIER TRANSFORMS 373a

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HOLOMORPHIC FOURIER TRANSFORMS 3751

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HOLOMORPHIC FOURIER TRANSFORMS 377i

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HOLOMORPHIC FOURIER TRANSFORMS 379C

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HOLOMORPHIC FOURIER TRANSFORMS 381T

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HOLOMORPHIC FOURIER TRANSFORMS 383F

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HOLOMORPHIC FOURIER TRANSFORMS 385h

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UNIFORM APPROXIMATION BY POLYNOMIAL

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UNIFORM APPROXIMATION BY POLYNOMIAL

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UNIFORM APPROXIMATION BY pOLYNOMIAL

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UNIFORM APPROXIMATION BY POLYNOMIAL

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APPENDIXHAUSDORFF'S MAXIMALITY THEO

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406 BIBLIOGRAPHY24. F. Riesz and B.

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408 LIST OF SPECIAL SYMBOLS AND ABB

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410 INDEXCartesian product, 7,160Ca

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412 INDEXHalmos, P. R., 398,403Hard

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414 INDEXOuter regular set, 47Overc

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416 INDEXUniform continuity, 51Unif

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