14 CONTENTS

51.7.4 The Cauchy Integral Formula . . . . . . . . . . . . . . . . . . .163851.7.5 An Example Of A Cycle . . . . . . . . . . . . . . . . . . . . . .1646

51.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1649

52 The Open Mapping Theorem 165152.1 A Local Representation . . . . . . . . . . . . . . . . . . . . . . . . . . .165152.2 Branches Of The Logarithm . . . . . . . . . . . . . . . . . . . . . . . . .165352.3 Maximum Modulus Theorem . . . . . . . . . . . . . . . . . . . . . . . .165552.4 Extensions Of Maximum Modulus Theorem . . . . . . . . . . . . . . . .1656

52.4.1 Phragmên Lindelöf Theorem . . . . . . . . . . . . . . . . . . .165652.4.2 Hadamard Three Circles Theorem . . . . . . . . . . . . . . . . .165952.4.3 Schwarz’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . .166052.4.4 One To One Analytic Maps On The Unit Ball . . . . . . . . . .1660

52.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .166252.6 Counting Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .166352.7 An Application To Linear Algebra . . . . . . . . . . . . . . . . . . . . .166752.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1671

53 Residues 167353.1 Rouche’s Theorem And The Argument Principle . . . . . . . . . . . . . .1676

53.1.1 Argument Principle . . . . . . . . . . . . . . . . . . . . . . . .167653.1.2 Rouche’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . .167853.1.3 A Different Formulation . . . . . . . . . . . . . . . . . . . . . .1680

53.2 Singularities And The Laurent Series . . . . . . . . . . . . . . . . . . . .168153.2.1 What Is An Annulus? . . . . . . . . . . . . . . . . . . . . . . .168153.2.2 The Laurent Series . . . . . . . . . . . . . . . . . . . . . . . . .168353.2.3 Contour Integrals And Evaluation Of Integrals . . . . . . . . . .1687

53.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1696

54 Functional Analysis Applications 169954.1 The Spectral Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . .169954.2 Analytic Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . .170154.3 Sectorial Operators and Analytic Semigroups . . . . . . . . . . . . . . . .1701

54.3.1 The Numerical Range . . . . . . . . . . . . . . . . . . . . . . .171254.3.2 An Interesting Example . . . . . . . . . . . . . . . . . . . . . .171454.3.3 Fractional Powers Of Sectorial Operators . . . . . . . . . . . . .171754.3.4 A Scale Of Banach Spaces . . . . . . . . . . . . . . . . . . . . .1732

55 Complex Mappings 173755.1 Conformal Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .173755.2 Fractional Linear Transformations . . . . . . . . . . . . . . . . . . . . .1738

55.2.1 Circles And Lines . . . . . . . . . . . . . . . . . . . . . . . . .173855.2.2 Three Points To Three Points . . . . . . . . . . . . . . . . . . .1740

55.3 Riemann Mapping Theorem . . . . . . . . . . . . . . . . . . . . . . . . .174155.3.1 Montel’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . .174155.3.2 Regions With Square Root Property . . . . . . . . . . . . . . . .1744