《THE LOCATION OF CRITICAL POINTS OF ANALYTIC AND HARMONIC FUNCTIONS》求取 ⇩

CHAPTER Ⅰ．FUNDAMENTAL RESULTS1

1．1．Terminology．Preliminaries1

1．1．1．Point set terminology1

1．1．2．Function-theoretic preliminaries2

1．2．Gauss＇s Theorem5

1．3．Lucas＇s Theorem6

1．3．1．Statement and proof6

1．3．2．Complements8

1．4．Jensen＇s Theorem9

1．4．1．Proof10

1．4．2．Complements11

1．5．Walsh＇s Theorem13

1．5．1．Preliminaries13

1．5．2．Proof15

1．6．Lemniscates17

1．6．1．Level curves17

1．6．2．Lemniscates and their orthogonal trajectories18

1．6．3．Behavior at infinity21

CHAPTER Ⅱ．REAL POLYNOMIALS24

2．1．Polynomials with real zeros24

2．2．Jensen＇s Theorem，continued27

2．2．1．Special cases28

2．2．2．A region for non-real critical points29

2．3．Number of critical points31

2．4．Reality and non-reality of critical points32

2．4．1．Sufficient conditions for non-real critical points32

2．4．2．Sufficient conditions for real critical points33

2．4．3．Real critical points，continued36

2．5．Jensen configuration improved38

2．6．W-curves40

2．6．1．General results41

2．6．2．Real polynomials44

2．6．3．Zeros on horizontal segments49

2．7．Non-real polynomials52

2．7．1．Jensen circles52

2．7．2．Infinite sectors55

CHAPTER Ⅲ．POLYNOMIALS，CONTINUED58

3．1．Infinite circular regions as loci58

3．1．1．Exterior of a circle58

3．1．2．Half-planes60

3．2．A characteristic property of circular regions62

3．3．Critical points of real polynomials as centers of circles63

3．4．Lucas polygon improved68

3．5．The Lucas polygons for a polynomial and its derivative71

3．5．1．Relations of convex sets71

3．5．2．Dependence of zeros of polynomial on those of its derivative73

3．6．Symmetry in the origin75

3．6．1．Ordinary symmetry75

3．6．2．Multiple symmetry in O78

3．7．Circular regions and symmetry80

3．8．Higher derivatives83

3．9．Further results87

CHAPTER Ⅳ．RATIONAL FUNCTIONS89

4．1．Field of force89

4．1．1．Fundamental theorem89

4．1．2．Transformations of the plane92

4．1．3．Stereographic projection94

4．2．Bocher＇s Theorem97

4．2．1．Proof98

4．2．2．Locus of critical points99

4．2．3．Specializations101

4．2．4．Circular regions as loci103

4．2．5．Converse105

4．3．Concentric circular regions as loci106

4．4．The cross-ratio theorem109

4．4．1．Geometric locus109

4．4．2．Discussion of locus113

4．4．3．Locus of critical points115

4．4．4．Generalizations120

4．5．Marden＇s Theorem121

CHAPTER Ⅴ．RATIONAL FUNCTIONS WITH SYMMETRY123

5．1．Real rational functions123

5．1．1．Real zeros and poles123

5．1．2．Non-real zeros125

5．1．3．Regions as loci128

5．1．4．Regions as loci，continued131

5．1．5．W-curves135

5．2．Zeros and poles concyclic138

5．2．1．Zeros and poles interlaced139

5．2．2．Extensions141

5．2 3．Critical points near a given zero145

5．2．4．Continuation148

5．2．5．Zeros and poles on prescribed arcs152

5．3．Hyperbolic plane156

5．3．1．Analog of Lucas＇s Theorem157

5．3．2．Extensions159

5．3．3．Analog of Jensen＇s Theorem163

5．3．4．Circular regions as loci165

5．3．5．Analog of B？cher＇s Theorem169

5．3．6．Continuation170

5．3．7．NE half-planes as loci172

5．4．Elliptic plane176

5．4．1．Analog of Bocher＇s Theorem176

5．4．2．Circular regions as loei178

5．5．Symmetry in the origin181

5．5．1．Regions bounded by concentric circles181

5．5．2．Regions bounded by equilateral hyperbolas183

5．5．3．Circular regions as loci of zeros and poles186

5．5．4．Multiple symmetry in O189

5．5．5．Multiple symmetry in O，continued191

5．5．6．Polynomials and multiple symmetry in O193

5．6．Skew symmetry in O194

5．6．1．Sectors containing zeros and poles194

5．6．2．W-curves197

5．7．Symmetry in z and 1/z202

5．7．1．Polynomials in z and 1/z202

5．7．2．General rational functions206

5．8．Miscellaneous results208

5．8．1．Combinations of symmetries208

5．8．2．Circular regions and symmetry in a line209

5．8．3．Circular regions and real polynomials212

CHAPTER Ⅵ．ANALYTIC FUNCTIONS217

6．1．Entire and meromorphic functions217

6．2．The hyperbolic plane221

6．2．1．Analytic and meromorphic functions221

6．2．2．Blaschke products223

6．3．Functions with multiplicative periods224

6．3．1．Uniform functions224

6．3．2．Non-uniform functions227

6．4．Simply periodic functions231

6．4．1．General theorems232

6．4．2．Functions with symmetry234

6．5．Doubly periodic functions235

6．6．General analytic functions236

6．7．Analytic functions and harmonic functions240

CHAPTER Ⅶ．GREEN＇S FUNCTIONS241

7．1．Topology241

7．1．1．Level curves241

7．1．2．Variable regions243

7．1．3．A formula for Green＇s function245

7．1．4．Level curves and lemniscates247

7．2．Geometry of level loci249

7．2．1．Analog of Lucas＇s Theorem249

7．2．2．Center of curvature252

7．3．Symmetry in axis of reals255

7．3．1．Analog of Jensen＇s Theorem255

7．3．2．Number of critical points257

7．4．Analog of Walsh＇s Theorem259

7．4．1．Numbers of critical points259

7．4．2．Numbers of critical points，alternate treatment262

7．4．3．Symmetry263

7．4．4．Inequalities on masses264

7．4．5．Circles with collinear centers266

7．5．Doubly-connected regions266

CHAPTER Ⅷ．HARMONIC FUNCTIONS269

8．1．Topology269

8．1．1．Level loci269

8．1．2．A formula for harmonic measure271

8．1．3．Number of critical points274

8．2．Analog of B？cher＇s Theorem275

8．3．A cross-ratio theorem278

8．4．Hyperbolic plane279

8．5．Other symmetries283

8．6．Periodic functions286

8．7．Harmonic measure，Jordan region288

8．7．1．Finite number of arcs288

8．7．2．Infinite number of arcs290

8．8．Hyperbolic geometry and conformal mapping292

8．8．1．Doubly-connected region292

8．8．2．Case p>2293

8．9．Linear combinations of Green＇s functions294

8．9．1．Simply-connected region295

8．9．2．Regions of arbitrary connectivity296

8．9．3．Multiply-connected regions299

8．10．Harmonic measure of arcs300

8．10．1．Doubly-connected regions300

8．10．2．Arbitrary connectivity301

8．10．3．Approximating circular regions302

CHAPTER Ⅸ．FURTHER HARMONIC FUNCTIONS304

9．1．A general field of force304

9．2．Harmonic measure of arcs307

9．2．1．A single arc307

9．2．2．Several arcs308

9．2．3．Linear combinations310

9．3．Arcs of a circle312

9．3．1．Disjoint arcs313

9．3．2．Abutting arcs315

9．3．3．Poisson＇s integral317

9．3．4．Poisson＇s integral as potential of a double distribution319

9．3．5．Harmonic functions and rational functions322

9．4．A circle as partial boundary324

9．4．1．Harmonic measure324

9．4．2．Linear combinations327

9．4．3．Harmonic measure，resumed329

9．4．4．Symmetry330

9．4．5．Symmetry，several arcs333

9．4．6．Case p>1337

9．5．Green＇s functions and harmonic measures339

9．5．1．A numerical example339

9．5．2．General linear combinations341

9．6．Limits of boundary components344

9．6．1．Hyperbolic geometry344

9．6．2．Comparison of hyperbolic geometries346

9．7．Methods of symmetry348

9．7．1．Reflection in axis349

9．7．2．Reflection in point353

9．7．3．Reflection of an annulus，Green＇s function356

9．7．4．Reflection of an annulus，harmonic measure358

9．8．Topological methods360

9．8．1．Simple-connectivity360

9．8．2．Higher connectivity362

9．8．3．Contours with boundary values zero364

9．9．Assigned distributions of matter365

9．9．1．Simple layers365

9．9．2．Superharmonic functions366

9．9．3．Double layers370

BIBLIOGRAPHY377

INDEX381