CHAPTER Ⅰ．FUNDAMENTAL RESULTS 1
1．1．Terminology．Preliminaries 1
1．1．1．Point set terminology 1
1．1．2．Function-theoretic preliminaries 2
1．2．Gauss＇s Theorem 5
1．3．Lucas＇s Theorem 6
1．3．1．Statement and proof 6
1．3．2．Complements 8
1．4．Jensen＇s Theorem 9
1．4．1．Proof 10
1．4．2．Complements 11
1．5．Walsh＇s Theorem 13
1．5．1．Preliminaries 13
1．5．2．Proof 15
1．6．Lemniscates 17
1．6．1．Level curves 17
1．6．2．Lemniscates and their orthogonal trajectories 18
1．6．3．Behavior at infinity 21
CHAPTER Ⅱ．REAL POLYNOMIALS 24
2．1．Polynomials with real zeros 24
2．2．Jensen＇s Theorem，continued 27
2．2．1．Special cases 28
2．2．2．A region for non-real critical points 29
2．3．Number of critical points 31
2．4．Reality and non-reality of critical points 32
2．4．1．Sufficient conditions for non-real critical points 32
2．4．2．Sufficient conditions for real critical points 33
2．4．3．Real critical points，continued 36
2．5．Jensen configuration improved 38
2．6．W-curves 40
2．6．1．General results 41
2．6．2．Real polynomials 44
2．6．3．Zeros on horizontal segments 49
2．7．Non-real polynomials 52
2．7．1．Jensen circles 52
2．7．2．Infinite sectors 55
CHAPTER Ⅲ．POLYNOMIALS，CONTINUED 58
3．1．Infinite circular regions as loci 58
3．1．1．Exterior of a circle 58
3．1．2．Half-planes 60
3．2．A characteristic property of circular regions 62
3．3．Critical points of real polynomials as centers of circles 63
3．4．Lucas polygon improved 68
3．5．The Lucas polygons for a polynomial and its derivative 71
3．5．1．Relations of convex sets 71
3．5．2．Dependence of zeros of polynomial on those of its derivative 73
3．6．Symmetry in the origin 75
3．6．1．Ordinary symmetry 75
3．6．2．Multiple symmetry in O 78
3．7．Circular regions and symmetry 80
3．8．Higher derivatives 83
3．9．Further results 87
CHAPTER Ⅳ．RATIONAL FUNCTIONS 89
4．1．Field of force 89
4．1．1．Fundamental theorem 89
4．1．2．Transformations of the plane 92
4．1．3．Stereographic projection 94
4．2．Bocher＇s Theorem 97
4．2．1．Proof 98
4．2．2．Locus of critical points 99
4．2．3．Specializations 101
4．2．4．Circular regions as loci 103
4．2．5．Converse 105
4．3．Concentric circular regions as loci 106
4．4．The cross-ratio theorem 109
4．4．1．Geometric locus 109
4．4．2．Discussion of locus 113
4．4．3．Locus of critical points 115
4．4．4．Generalizations 120
4．5．Marden＇s Theorem 121
CHAPTER Ⅴ．RATIONAL FUNCTIONS WITH SYMMETRY 123
5．1．Real rational functions 123
5．1．1．Real zeros and poles 123
5．1．2．Non-real zeros 125
5．1．3．Regions as loci 128
5．1．4．Regions as loci，continued 131
5．1．5．W-curves 135
5．2．Zeros and poles concyclic 138
5．2．1．Zeros and poles interlaced 139
5．2．2．Extensions 141
5．2 3．Critical points near a given zero 145
5．2．4．Continuation 148
5．2．5．Zeros and poles on prescribed arcs 152
5．3．Hyperbolic plane 156
5．3．1．Analog of Lucas＇s Theorem 157
5．3．2．Extensions 159
5．3．3．Analog of Jensen＇s Theorem 163
5．3．4．Circular regions as loci 165
5．3．5．Analog of B？cher＇s Theorem 169
5．3．6．Continuation 170
5．3．7．NE half-planes as loci 172
5．4．Elliptic plane 176
5．4．1．Analog of Bocher＇s Theorem 176
5．4．2．Circular regions as loei 178
5．5．Symmetry in the origin 181
5．5．1．Regions bounded by concentric circles 181
5．5．2．Regions bounded by equilateral hyperbolas 183
5．5．3．Circular regions as loci of zeros and poles 186
5．5．4．Multiple symmetry in O 189
5．5．5．Multiple symmetry in O，continued 191
5．5．6．Polynomials and multiple symmetry in O 193
5．6．Skew symmetry in O 194
5．6．1．Sectors containing zeros and poles 194
5．6．2．W-curves 197
5．7．Symmetry in z and 1/z 202
5．7．1．Polynomials in z and 1/z 202
5．7．2．General rational functions 206
5．8．Miscellaneous results 208
5．8．1．Combinations of symmetries 208
5．8．2．Circular regions and symmetry in a line 209
5．8．3．Circular regions and real polynomials 212
CHAPTER Ⅵ．ANALYTIC FUNCTIONS 217
6．1．Entire and meromorphic functions 217
6．2．The hyperbolic plane 221
6．2．1．Analytic and meromorphic functions 221
6．2．2．Blaschke products 223
6．3．Functions with multiplicative periods 224
6．3．1．Uniform functions 224
6．3．2．Non-uniform functions 227
6．4．Simply periodic functions 231
6．4．1．General theorems 232
6．4．2．Functions with symmetry 234
6．5．Doubly periodic functions 235
6．6．General analytic functions 236
6．7．Analytic functions and harmonic functions 240
CHAPTER Ⅶ．GREEN＇S FUNCTIONS 241
7．1．Topology 241
7．1．1．Level curves 241
7．1．2．Variable regions 243
7．1．3．A formula for Green＇s function 245
7．1．4．Level curves and lemniscates 247
7．2．Geometry of level loci 249
7．2．1．Analog of Lucas＇s Theorem 249
7．2．2．Center of curvature 252
7．3．Symmetry in axis of reals 255
7．3．1．Analog of Jensen＇s Theorem 255
7．3．2．Number of critical points 257
7．4．Analog of Walsh＇s Theorem 259
7．4．1．Numbers of critical points 259
7．4．2．Numbers of critical points，alternate treatment 262
7．4．3．Symmetry 263
7．4．4．Inequalities on masses 264
7．4．5．Circles with collinear centers 266
7．5．Doubly-connected regions 266
CHAPTER Ⅷ．HARMONIC FUNCTIONS 269
8．1．Topology 269
8．1．1．Level loci 269
8．1．2．A formula for harmonic measure 271
8．1．3．Number of critical points 274
8．2．Analog of B？cher＇s Theorem 275
8．3．A cross-ratio theorem 278
8．4．Hyperbolic plane 279
8．5．Other symmetries 283
8．6．Periodic functions 286
8．7．Harmonic measure，Jordan region 288
8．7．1．Finite number of arcs 288
8．7．2．Infinite number of arcs 290
8．8．Hyperbolic geometry and conformal mapping 292
8．8．1．Doubly-connected region 292
8．8．2．Case p>2 293
8．9．Linear combinations of Green＇s functions 294
8．9．1．Simply-connected region 295
8．9．2．Regions of arbitrary connectivity 296
8．9．3．Multiply-connected regions 299
8．10．Harmonic measure of arcs 300
8．10．1．Doubly-connected regions 300
8．10．2．Arbitrary connectivity 301
8．10．3．Approximating circular regions 302
CHAPTER Ⅸ．FURTHER HARMONIC FUNCTIONS 304
9．1．A general field of force 304
9．2．Harmonic measure of arcs 307
9．2．1．A single arc 307
9．2．2．Several arcs 308
9．2．3．Linear combinations 310
9．3．Arcs of a circle 312
9．3．1．Disjoint arcs 313
9．3．2．Abutting arcs 315
9．3．3．Poisson＇s integral 317
9．3．4．Poisson＇s integral as potential of a double distribution 319
9．3．5．Harmonic functions and rational functions 322
9．4．A circle as partial boundary 324
9．4．1．Harmonic measure 324
9．4．2．Linear combinations 327
9．4．3．Harmonic measure，resumed 329
9．4．4．Symmetry 330
9．4．5．Symmetry，several arcs 333
9．4．6．Case p>1 337
9．5．Green＇s functions and harmonic measures 339
9．5．1．A numerical example 339
9．5．2．General linear combinations 341
9．6．Limits of boundary components 344
9．6．1．Hyperbolic geometry 344
9．6．2．Comparison of hyperbolic geometries 346
9．7．Methods of symmetry 348
9．7．1．Reflection in axis 349
9．7．2．Reflection in point 353
9．7．3．Reflection of an annulus，Green＇s function 356
9．7．4．Reflection of an annulus，harmonic measure 358
9．8．Topological methods 360
9．8．1．Simple-connectivity 360
9．8．2．Higher connectivity 362
9．8．3．Contours with boundary values zero 364
9．9．Assigned distributions of matter 365
9．9．1．Simple layers 365
9．9．2．Superharmonic functions 366
9．9．3．Double layers 370
BIBLIOGRAPHY 377
INDEX 381