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Bergman S. — The Kernel Function and Conformal Mapping
Bergman S. — The Kernel Function and Conformal Mapping



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Название: The Kernel Function and Conformal Mapping

Автор: Bergman S.

Аннотация:

The Kernel Function and Conformal Mapping by Stefan Bergman is a revised edition of The Kernel Function. The author has made extensive changes in the original volume. The present book will be of interest not only to mathematicians, but also to engineers, physicists, and computer scientists. The applications of orthogonal functions in solving boundary value problems and conformal mappings onto canonical domains are discussed; and publications are indicated where programs for carrying out numerical work using high-speed computers can be found. The unification of methods in the theory of functions of one and several complex variables is one of the purposes of introducing the kernel function and the domains with a distinguished boundary. This approach has been extensively developed during the last two decades. This second edition of Professor Bergman's book reviews this branch of the theory including recent developments not dealt with in the first edition. The presentation of the topics is simple and presupposes only knowledge of an elementary course in the theory of analytic functions of one variable.


Язык: en

Рубрика: Математика/Анализ/Комплексный анализ/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1950

Количество страниц: 161

Добавлена в каталог: 02.04.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$D{\varphi,\psi}$      44
$F_{\nu}(z)$      46
$G(z, \zeta)$      44
$k(z, \zeta)$      50
$k_0(z, \zeta)$      57
$l^2(B)$      47
$M_B^1(z,t)$      21
$N(z, \zeta)$      44
$N(z, \zeta, \zeta_0)$      51
$P(z, \zeta)$      46
$P_{\nu\mu}$, period of $F_{\nu}(z)$      46
$\hat{K}(z, \bar{\zeta})$      79
$\hat{K}(z, \zeta)$      79
$\hat{K}_n(z, \bar{\zeta})$      77
$\Lambda^2(B)$      45
$\Lambda_0^2(B)$      57
$\lambda_b^1(t)$      21
$\mathcal{L}^2(B)$      1
$\Omega^2$      118
$\omega_{\nu}(z)$      45
$\tilde{K}(z, \bar{\zeta})$      47 59
Affine transformations      142
Ahlfors method      150
Analytic continuation      18
Analytic hypersurface      145
Analytic surface      145
Analyticity in a variable      132
Approximation to analytic functions      13
Approximation to analytic functions, to solutions of elliptic differential equations      117
Average quadratic error      3
B-harmonic function      148
Basic space      33
Bessel inequality      3
Bicylinder      132
Boundary of four-dimensional domain      133
Boundary value problem of first and second kinds      112
Bounded functions      86
Canonical domains      61
Canonical mapping functions, in terms of kernel function      71
Cauchy formula      133 147
Cauchy sequence      41
Cauchy — Riemann equations      46 132
Circular domain      134
Class $l^2$      47
Class $\Lambda^2$      45
Class $\Lambda^2_0$      57
Class $\mathcal{L}^2$      1
Class $\Omega^2$      118
Classical domain functions      44
Closed orthogonal system for a simply-connected domain      12
Closed orthogonal system, existence of      10
Closed orthonormal system for $\Omega^2$      118
Closed system      3 135
Closed system of orthogonal harmonic functions      47
Closed system, construction of      10
Compact family      10 15
Complete space      41
Complex derivatives      35
Countable basis of H      41
Curvature, bounds for      36
Determinantal notation      25
Differential equations, of elliptic type      112
Differential equations, of elliptic type, non-definite coefficient      122
Differential geometry      31
Differential operators      35
Dirichlet integral      44 115
Dirichlet integral as metric      48
Dirichlet problem      52 112
Dirichlet problem, solved using kernel function      52
Distinguished boundary curve      147 150
Distinguished boundary surface      133 144
Distortion of length      140
Distortion of length, bounds for      142
Distortion theorems      67 71 96 144
Distortion theorems for two complex variables      138
domain functions      44
Domain functions, monoticity of      95
Domain, admitting a group of stability      142
Domain, admitting a group of stability for which polynomials are closed      13
Domain, admitting a group of stability, canonical      61
Domain, admitting a group of stability, of comparison      37 140
Doubly-harmonic functions      148
Doubly-harmonic measures      148
Doubly-orthogonal functions      14
Eigenf unctions      126
Eigenvalues      122
Elasticity equation      127
Elasticity equation, solution of boundary value problems for      130
Elliptic differential equations      40 112
Elliptic functions      10
Entire functions      150
Equation of elasticity      127
Existence of slit mappings      111
Expansion theorems      7 113 117 130 136
Extended class      149
Extremal curves      31
Extremal property of canonical domains      67
First boundary value problem      52
First fundamental form      31
Four-dimensional domains      132
Four-dimensional domains, visualization of      133
Fourier coefficients      2 119
Function of first kind      46 47
Function of second kind      47
Function of third kind      46 47
Functions of extended class      150
Functions of two complex variables      132
Functions of two variables      135
Fundamental solution S(Z, Z)      112
Gaussian      32
Gaussian of invariant metric      35
Gaussian of the metric      38
Gaussian, radius of      32
Gaussian, total      32
Geodesies      31
Gram — Schmidt orthogonalization process      3
Green’s formula      44 45 46
Green’s function      44 58 112 116 128 129 148
Green’s function, construction of      118
Green’s function, eigenfunctions      121
Green’s function, relation to Neumann’s function      50
Green’s function, representation      50 120
Green’s function, representation by kernel function      54
Green’s function, variation of      100
Group of stability      142
Hadamard variational method      90
Harmonic functions      40
Harmonic functions, closed systems of      44
Harmonic measure      45 144 150
Harmonic measure, conjugate harmonic function      46
Harmonic measure, representation in terms of kernel function      53
Hermitian form      18 27 28 138
Hermitian form, Jacobi reduction of      28
Hilbert space, convergence in      41
Hilbert space, convergence in, definitions      41
Hilbert space, convergence in, of functions      41
Hyperbolic measure, generalization of      144
Hypersphere      133 137 141
Identities connecting domain functions      54
Indicatrix      39
Integral operator of first kind      171
Integrals of first kind      47
Invariant metric, curvature of      35
Invariants of a domain      140
Jacobi reduction      28
Kernel function      41 58 77 106 115 116 120 126 127 129
Kernel function for circle      79
Kernel function for non-bounded, non-schiicht domains      37
Kernel function for simply-connected domains      79
Kernel function in two variables      136
Kernel function of $l^2$      47
Kernel function of an orthogonal system      9
Kernel function, boundary behavior of      38
Kernel function, boundary relation of      111
Kernel function, canonical mapping functions in      71
Kernel function, extremal property      87 136
Kernel function, harmonic      49
Kernel function, identities      57
Kernel function, independent of closed system      21 136
Kernel function, inequality in terms of      23
Kernel function, invariance of      137
Kernel function, minimum property of      42
Kernel function, monotonicity of      42
Kernel function, relative invariant      33
Kernel function, reproducing property of      22 42 83
Kernel function, uniqueness of      23
Kernel function, variation of      90
Kernel of Cauchy formula      148
Length, an absolute invariant      33
Length, an absolute invariant, of a curve      31
Linear independence      1
Linear space      41
Mapping function, for simply-connected domain      24
Mapping function, for simply-connected domain, existence of      70
Mapping theorems      61
Matrix notation      24
Meromorphic functions      150
Meromorphic functions of several complex variables      148
Method of orthogonal functions      135
Metric      113 122 138
Metric and invariant metric      32
Metric of Pick — Poincare      35
Metric, boundary behavior of      37
Metric, Gaussian curvature of      38
Metric, monotonicity of      34
Minimizing function      21
Minimum area      24
Minimum problem      24 30
Minimum problem and partial differentiation      113
Modulus of doubly-connected domain      102
Monotonic properties of minimum $\lambda$      140
Multiply-connected domains      61
Neumann problem      53 112
Neumann problem, solved with help of kernel function      53
Neumann’s function      44 112 116
Neumann’s function, construction of      118
Neumann’s function, relation to Green’s function      50
Neumann’s function, representation of      50 120
Nevanlinna’s theorems      150
Non-Euclidean length element      141
Norm, in Hilbert space      41
Normal family      17
Normal form      112
Normalization      2
Normed space      41
Orthogonal functions      2
Orthogonalization process      3
Orthonormal functions      2
Orthonormal systems      125
Orthonormal systems for elliptic differential equations      118
Orthonormal systems for equation of elasticity      128
Orthonormalization conditions      135
Parsevai identity      119
Parseval formula      3
Poincaire theorem      138
Polygonal domains      4
Positive definite Hermitian quadratic form      42
Positive definite Hermitian scalar product      41
Positive definite quadratic form      31
Principle, of hyperbolic measure      34
Principle, of hyperbolic measure, of minimum integral      39 144
Projection      121
Pseudo-conformal equivalence      150
Pseudo-conformal invariants      138
Pseudo-conformal transformation      132 138
Pseudo-conformal transformation of one circular domain      137
Pseudo-conformal transformation, invariant of      140
Pseudo-conformal transformation, non-existence of      137
Reinhardt circular domains      135
Representative domains      61
Riemann mapping theorem      34 138
Riemannian manifold      31 33
Riesz — Rischer theorem      5
Robin’s constant $\gamma (\zeta) = \zeta_B(\zeta)$      96
Robin’s problem      131
Runge theorem      13
Scalar product      1
Schiffer method      98
Schottky theory      60
Schwarz — Christoffel constants      75
Schwarz’s inequality integral      1 42
Schwarz’s lemma      42
Schwarz’s lemma, generalization      87
Second boundary value problem      53
Simply-connected domains      34
Simply-connected domains for two complex variables      138
Slit mappings      111
Smooth curves      44
Stieltjes integrals      99
Svmmetrv of G and N      44
Symmetry relation for periods      46
Triangle inequality in Hilbert space      41
Variation, of analytic kernel      94
Variation, of analytic kernel, of Green’s function      93 100
Variation, of analytic kernel, of harmonic measures      94
Variation, of analytic kernel, of kernel function      92
Variation, of analytic kernel, of period $p_{\nu\mu}$      95
Variational method, of Hadamard      90
Variational method, of Schiffer      98
Weierstrassian $\zeta$-function      10
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