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Tables of Contents for Integral Theorems for Functions and Differential Forms in Cm
Chapter/Section Title
Page #
Page Count
Introduction
1
8
Differential forms
9
10
Usual notation
9
1
Complex differential forms
10
1
Operations on complex differential forms
11
3
Integration with respect to a part of variables
14
1
The differential form |F|
15
1
More spaces of differential forms
16
3
Differential forms with coefficients in 2 x 2-matrices
19
42
Classes Gp(Ω), Gp(Omega;)
19
1
Matrix-valued differential forms
19
2
The hyperholomorphic Cauchy-Riemann operators on B1 and B1
21
3
Formula for d (F * G)
24
1
Differential matrix forms of the unit normal
24
4
Formula for d (F * σ * G)
28
4
Exterior differentiation and the hyperholomorphic Cauchy-Riemann operators
32
1
Stokes formula compatible with the hyperholomorphic Cauchy-Riemann operators
32
2
The Cauchy kernel for the null-sets of the hyperholomorphic Cauchy-Riemann operators
34
1
Structure of the product KD * σ
35
4
Borel-Pompeiu (or Cauchy-Green) formula for smooth differential matrix-forms
39
22
Structure of the Borel-Pompeiu formula
44
3
The case m = 1
47
1
The case m = 2
48
3
Notations for some integrals in C2
51
3
Formulas of the Borel-Pompeiu type in C2
54
1
Complements to the Borel-Pompeiu-type formulas in C2
55
1
The case m > 2
55
2
Notations for some integrals in Cm
57
1
Formulas of the Borel-Pompeiu type in Cm
58
1
Complements to the Borel-Pompeiu-type formulas in Cm
58
3
Hyperholomorphic functions and differential forms in Cm
61
14
Hyperholomorphy in Cm
61
1
Hyperholomorphy in one variable
62
1
Hyperholomorphy in two variables
63
2
Hyperholomorphy in three variables
65
5
Hyperholomorphy for any number of variables
70
3
Observation about right-hand-side hyperholomorphy
73
2
Hyperholomorphic Cauchy's integral theorems
75
6
The Cauchy integral theorem for left-hyperholo-morphic matrix-valued differential forms
75
1
The Cauchy integral theorem for right-G-hyper-holomorphic m.v.d.f.
75
1
Some auxiliary computations
76
1
More auxiliary computations
77
1
The Cauchy integral theorem for holomorphic functions of several complex variables
78
1
The Cauchy integral theorem for antiholomorphic functions of several complex variables
78
1
The Cauchy integral theorem for functions holomorphic in some variables and antiholomorphic in the rest of variables
79
1
Concluding remarks
80
1
Hyperholomorphic Morera's theorems
81
8
Left-hyperholomorphic Morera theorem
81
1
Version of a right-hyperholomorphic Morera theorem
82
2
Morera's theorem for holomorphic functions of several complex variables
84
1
Morera's theorem for antiholomorphic functions of several complex variables
85
1
The Morera theorem for functions holomorphic in some variables and antiholomorphic in the rest of variables
86
3
Hyperholomorphic Cauchy's integral representations
89
6
Cauchy's integral representation for left-hyperholomorphic matrix-valued differential forms
89
1
A consequence for holomorphic functions
90
1
A consequence for antiholomorphic functions
90
1
A consequence for holomorphic-like functions
91
1
Bochner-Martinelli integral representation for holomorphic functions of several complex variables, and hyperholomorphic function theory
92
1
Bochner-Martinelli integral representation for antiholomorphic functions of several complex variables, and hyperholomorphic function theory
92
1
Bochner-Martinelli integral representation for functions holomorphic in some variables and antiholomorphic in the rest, and hyperholomorphic function theory
93
2
Hyperholomorphic D-problem
95
22
Some reasonings from one variable theory
95
2
Right inverse operators to the hyperholomorphic Cauchy-Riemann operators
97
13
Structure of the formula of Theorem 7.2
99
2
Case m = 1
101
1
Case m = 2
102
4
Case m > 2
106
3
Analogs of (7.17)
109
1
Commutativity relations for T-type operators
110
1
Solution of the hyperholomorphic D-problem
110
1
Structure of the general solution of the hyperholomorphic D-problem
111
3
D-type problem for the Hodge-Dirac operator
114
3
Complex Hodge-Dolbeault system, the ∂-problem and the Koppelman formula
117
50
Definition of the complex Hodge-Dolbeault system
117
1
Relation with hyperholomorphic case
118
1
The Cauchy integral theorem for solutions of degree p for the complex Hodge-Dolbeault system
119
2
The Cauchy integral theorem for arbitrary solutions of the complex Hodge-Dolbeault system
121
1
Morera's theorem for solutions of degree p for the complex Hodge-Dolbeault system
122
1
Morera's theorem for arbitrary solutions of the complex Hodge-Dolbeault system
123
1
Solutions of a fixed degree
124
1
Arbitrary solutions
124
1
Bochner-Martinelli-type integral representation for solutions of degree s of the complex Hodge-Dolbeault system
125
1
Bochner-Martinelli-type integral representation for arbitrary solutions of the complex Hodge-Dolbeault system
126
1
Solution of the ∂-type problem for the complex Hodge-Dolbeault system in a bounded domain in Cm
127
1
Complex ∂-problem and the ∂-type problem for the complex Hodge-Dolbeault system
128
2
∂-problem for differential forms
130
1
∂-problem for functions of several complex variables
131
1
General situation of the Borel-Pompeiu representation
132
6
Partial derivatives of integrals with a weak singularity
138
2
Theorem 8.15 in C2
140
1
Formula (8.14.3) in C2
141
2
Integral representation (8.14.3) for a (0, 1)-differential form in C2, in terms of its coefficients
143
1
Koppelman's formula in C2
143
1
Koppelman's formula in C2 for a (0, 1) - differential form, in terms of its coefficients
144
1
Comparison of Propositions 8.18 and 8.20
145
2
Koppelman's formula in C2 and hyperholomorphic theory
147
1
Definition of H,K
147
1
A reformulation of the Borel-Pompeiu formula
148
3
Identity (8.14.4) for a d.f. of a fixed degree
151
2
About the Koppelman formula
153
6
Auxiliary computations
159
3
The Koppelman formula for solutions of the complex Hodge-Dolbeault system
162
1
Appendix: properties of H,K
163
4
Hyperholomorphic theory and Clifford analysis
167
28
One way to introduce a complex Clifford algebra
167
3
Classical definition of a complex Clifford algebra
168
2
Some differential operators on Wm-valued functions
170
3
Factorization of the Laplace operator
171
2
Relation of the operators ∂ and ∂ with the Dirac operator of Clifford analysis
173
1
Matrix algebra with entries from Wm
174
1
The matrix Dirac operators
175
2
Factorization of the Laplace operator on Wm- valued functions
176
1
The fundamental solution of the matrix Dirac operators
177
2
Borel-Pompeiu formulas for Wm-valued functions
179
1
Monogenic Wm-valued functions
180
1
Cauchy's integral representations for monogenic Wm-valued functions
180
1
Clifford algebra with the Witt basis and differential forms
181
2
Relation between the two matrix algebras
183
6
Operators D and D
185
4
Cauchy's integral representation for left-hyperholomorphic matrix-valued differential forms
189
1
Hyperholomorphic theory and Clifford analysis
190
5
Bibliography
195
6
Index
201