Harmonic Mappings in the PlaneHarmonic mappings in the plane are univalent complex-valued harmonic functions of a complex variable. Conformal mappings are a special case where the real and imaginary parts are conjugate harmonic functions, satisfying the Cauchy-Riemann equations. Harmonic mappings were studied classically by differential geometers because they provide isothermal (or conformal) parameters for minimal surfaces. More recently they have been actively investigated by complex analysts as generalizations of univalent analytic functions, or conformal mappings. Many classical results of geometric function theory extend to harmonic mappings, but basic questions remain unresolved. This book is the first comprehensive account of the theory of planar harmonic mappings, treating both the generalizations of univalent analytic functions and the connections with minimal surfaces. Essentially self-contained, the book contains background material in complex analysis and a full development of the classical theory of minimal surfaces, including the Weierstrass-Enneper representation. It is designed to introduce non-specialists to a beautiful area of complex analysis and geometry. |
Contents
II | 1 |
III | 3 |
IV | 7 |
V | 11 |
VI | 13 |
VII | 16 |
VIII | 18 |
IX | 20 |
XXXIII | 95 |
XXXIV | 97 |
XXXV | 101 |
XXXVI | 103 |
XXXVII | 106 |
XXXVIII | 111 |
XXXIX | 112 |
XL | 115 |
X | 21 |
XI | 23 |
XII | 25 |
XIII | 27 |
XIV | 29 |
XV | 31 |
XVI | 34 |
XVII | 36 |
XVIII | 45 |
XIX | 48 |
XX | 54 |
XXI | 57 |
XXII | 59 |
XXIII | 62 |
XXIV | 66 |
XXV | 72 |
XXVI | 75 |
XXVII | 78 |
XXVIII | 79 |
XXIX | 82 |
XXX | 86 |
XXXI | 89 |
XXXII | 90 |
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Common terms and phrases
a₁ analytic function analytic univalent functions annulus argument principle b₁ Blaschke product boundary function bounded calculation Clunie and Sheil-Small coefficients conformal mapping constant convergence convex domain curve defined dilatation w(z Duren equation f(ei follows formula function f ƒ maps Gauss curvature h and g half-plane harmonic extension harmonic function harmonic Koebe function harmonic mapping Heinz Hengartner and Schober homeomorphism inequality isothermal parameters Jacobian Jordan domain Koebe function Let f locally univalent mapping ƒ maps the disk metric minimal graph minimal surface Möbius transformation modulus normal plane Proof of Theorem radial limits Radó-Kneser-Choquet theorem representation Riemann mapping theorem Schwarz lemma Schwarzian Schwarzian derivative Section sense-preserving harmonic function sense-preserving harmonic mapping sharp shows simply connected domain starlike typically real unit circle unit disk univalent functions univalent harmonic Weierstrass-Enneper zeros
Popular passages
Page 201 - AHLFORS, Conformal Invariants; Topics in Geometric Function Theory. McGraw-Hill, New York, 1973. 3.
Page 208 - MS Robertson, Analytic functions star-like in one direction, Amer. J. Math. 58 (1936), 465-472.
Page 208 - On the preservation of direction-convexity and the Goodman-Saff conjecture, Ann. Acad. Sci. Fenn. Ser. AI Math.
Page 201 - D. Bshouty, N. Hengartner, and W. Hengartner, A constructive method for starlike harmonic mappings, Numer. Math. 54 (1988), 167-178.
Page 209 - On the Fourier coefficients of homeomorphisms of the circle, Math. Res. Lett. 5 (1998), 383-390.
Page 209 - On univalent harmonic mappings and minimal surfaces, Pacific J. Math. 192 (2000), 191-200.
Page 209 - JL Ullman and CJ Titus [1] An integral inequality with applications to harmonic mappings, Michigan Math. J.