I Cauchy-Type Integrals in the Theory of a Plane Geopotential Field.- 1 Cauchy-Type Integral.- 1.1 Definition.- 1.1.1 Cauchy Integral Formula.- 1.

1.2 Concept of the Cauchy-Type Integral.- 1.1.3 Piecewise Analytical Functions.- 1.2 Main Properties.- 1.

2.1 Hölder Condition.- 1.2.2 Calculation of Singular Integrals in Terms of the Cauchy Principal Value.- 1.2.3 Sokhotsky-Plemelj Formulas.

- 1.2.4 Generalization of the Sokhotsky-Plemelj Formulas for Piecewise Smooth Curves.- 1.2.5 Cauchy-Tpye Integrals Along the Real Axis.- 1.3 Cauchy and Hubert Integral Transforms.

- 1.3.1 Integral Boundary Conditions for Analytical Functions.- 1.3.2 Determination of a Piecewise Analytical Function from a Specified Discontinuity.- 1.3.

3 Inversion Formulas for the Cauchy-Type Integral (Cauchy Integral Transforms).- 1.3.4 Hilbert Transforms.- 2 Representation of Plane Geopotential Fields in the Form of the Cauchy-Type Integral.- 2.1 Plane Potential Fields and Their Governing Equations.- 2.

1.1 Vector Field Equations.- 2.1.2 Concept of a Plane Field.- 2.1.3 Plane Field Equations.

- 2.1.4 Plane Field Flow Function.- 2.2 Logarithmic Potentials and the Cauchy-Type Integral.- 2.2.1 Logarithmic Potentials.

- 2.2.2 Logarithmic Potentials in Complex Coordinates.- 2.2.3 Cauchy-Type Integral as a Sum of the Logarithmic Potentials of Simple and Double Layers.- 2.3 Complex Intensity and Potential of a Plane Field.

- 2.3.1 Concept of Complex Intensity of a Plane Field.- 2.3.2 Complex Intensity Equations.- 2.3.

3 Representation of Complex Intensity in Terms of Field Source Density.- 2.3.4 Complex Potential.- 2.4 Direct Solution of the Equation for Complex Field Intensity.- 2.4.

1 Two-Dimensional Ostrogradsky-Gauss Formula in Complex Notation.- 2.4.2 Pompei Formulas.- 2.4.3 Solution to the Equation for Complex Intensity.- 2.

5 Representation of the Gravitational Field in Terms of the Cauchy-Type Integral.- 2.5.1 Complex Intensity of the Gravitational Field.- 2.5.2 Representation of the Gravitational Field of a Homogeneous Domain in Terms of the Cauchy-Type Integral.- 2.

5.3 Representation of the Gravitational Field of a Domain with an Analytical Mass Distribution in Terms of the Cauchy-Type Integral.- 2.5.4 Case of Vertical or Horizontal Variations in the Density.- 2.5.5 Case of Linear Density Variations Along the Coordinate Axis.

- 2.5.6 General Case of Continuous Density Distribution.- 2.5.7 Calculation of the Gravitational Field of an Infinitely Extended Domain.- 2.6 Representation of a Fixed Magnetic Field in Terms of the Cauchy-Type Integral.

- 2.6.1 Complex Potential of a Polarized Source.- 2.6.2 Complex Intensities and Potential of a Magnetic Field.- 2.6.

3 Representation of the Magnetic Potential of a Homogeneous Domain in Terms of the Cauchy-Type Integrals.- 2.6.4 General Case of Magnetization Distribution.- 2.6.5 Analytical Distribution of Magnetization.- 3 Techniques for Separation of Plane Fields.

- 3.1 Separation of Geopotential Fields into External and Internal Parts Using Spectral Decomposition.- 3.1.1 Statement of the Problem of Plane Field Separation.- 3.1.2 Spectral Representations of Plane Fields.

- 3.1.3 Determination of the External and Internal Parts of the Scalar Potential and Field (Gauss-Schmiedt Formulas).- 3.2 Kertz-Siebert Technique.- 3.2.1 Problem of Separation of Field Complex Intensity.

- 3.2.2 Field Separation at Ordinary Points of the Line L.- 3.2.3 Field Separation at Corners of the Line L.- 3.2.

4 Kertz-Siebert Formulas.- 3.2.5 Equivalence Between the Kertz-Siebert and the Gauss-Schmiedt Formulas.- 4 Analytical Continuation of a Plane Field.- 4.1 Fundamentals of Analytical Continuation.- 4.

1.1 Taylor Theorem.- 4.1.2 Uniqueness of an Analytical Function.- 4.1.3 Concept of Analytical Continuation.

- 4.1.4 Concept of the Riemann Surface.- 4.1.5 Weierstrass Continuation of an Analytical Function.- 4.1.

6 Singular Points of an Analytical Function.- 4.1.7 Penleve Continuation of an Analytical Function (Principle of Continuity).- 4.1.8 Conformai Mapping.- 4.

2 Analytical Continuation of the Cauchy-Type Integral Through a Path of Integration.- 4.2.1 Analytical Continuation of a Real Analytical Function of a Real Variable.- 4.2.2 Concept of an Analytical Arc; the Herglotz-Tsirulsky Equation for the Arc.- 4.

2.3 Analytical Continuation of a Function Specified Along an Analytical Curve.- 4.2.4 Continuation of the Cauchy-Type Integral Through a Path of Integration; Singular Points of the Continued Field.- 4.3 Analytical Continuation of a Plane Magnetic Field into a Domain Occupied by Magnetized Masses.- 4.

3.1 Analytical Continuation of a Magnetic Potential into a Domain of Analytically Distributed Magnetization.- 4.3.2 Continuation Through a One-Side Herglotz-Tsirulsky Analytical Arc.- 4.3.3 Analyticity Condition for the Boundary of a Domain Occupied by Magnetized Masses.

- 4.3.4 Singular Points of Analytical Continuation of the Magnetic Potential.- 4.3.5 Determination of Complex Magnetization of a Body from its Magnetic Potential.- 4.4 Analytical Continuation of a Plane Gravitational Field into a Domain Occupied by Attracting Masses.

- 4.4.1 Characteristics of the Gravitational Field of a Homogeneous Domain Bounded by an Analytical Curve.- 4.4.2 Continuation of the Gravitational Field into a Domain with an Analytical Density Distribution.- 4.4.

3 Case of a Homogeneous Domain Bounded by a Piecewise Analytical Curve.- 4.4.4 Singular Points of the Continued Field, Lying on the Boundary of a Material Body.- 4.5 Integral Techniques for Analytical Continuation of Plane Fields.- 4.5.

1 Forms of Analytical Continuation of Plane Fields in Geophysics.- 4.5.2 Reconstruction of a Function Analytical in the Upper Half-Plane from Its Real or Imaginary Part.- 4.5.3 Analytical Continuation of Plane Fields into a Horizontal Layer Using Spectral Decomposition of the Cauchy Kernel.- 4.

5.4 Case of Field Specification on the Real Axis. The Zamorev Formulas.- 4.5.5 Downward Analytical Continuation of Functions Having Singularities Both in the Lower and in the Upper Half-Planes.- 4.5.

6 Analytical Continuation into Domains with Curvilinear Boundaries.- 4.5.7 Bateman Formula; Continuation of Complex Intensity of the Field into the Lower Half-Plane Using Its Real Part.- II Cauchy-Type Integral Analogs in the Theory of a Three-Dimensional Geopotential Field.- 5 Three-Dimensional Cauchy-Type Integral Analogs.- 5.1 Three-Dimensional Analog of the Cauchy Integral Formula.

- 5.1.1 Vector Statements of the Ostrogradsky-Gauss Theorem.- 5.1.2 Vector Statements of the Stokes Theorem.- 5.1.

3 Analog of the Cauchy-Type Integral.- 5.1.4 Relationship Between the Three-Dimensional Analog and the Classical Cauchy Integral Formula.- 5.1.5 Gauss Harmonic Function Theorem.- 5.

1.6 Cauchy Formula Analog for an Infinite Domain.- 5.1.7 Three-Dimensional Analog of the Pompei Formulas.- 5.2 Definition and Properties of the Three-Dimensional Cauchy Integral Analog.- 5.

2.1 Concept of a Three-Dimensional Cauchy Integral Analog.- 5.2.2 Evaluation of Singular Integrals in Terms of the Cauchy Principal Value.- 5.2.3 Three-Dimensional Analogs of the Sokhotsky-Plemelj Formulas.

- 5.3 Integral Transforms of the Laplace Vector Fields.- 5.3.1 Integral Boundary Conditions for the Laplace Field.- 5.3.2 Piecewise Laplace Vector Fields.

Determination of a Piecewise Laplace Field from a Specified Discontinuity.- 5.3.3 Inversion Formulas for the Three-Dimensional Cauchy Integral Analog.- 5.3.4 Three-Dimensional Hilbert Transforms.- 5.

4 Cauchy Integral Analogs in Matrix Notation.- 5.4.1 Matrix Representation of the Differentiation Operators of Scalar and Vector Fields.- 5.4.2 Matrix Representations of Three-Dimensional Cauchy Integral Analogs.- 6 Application of Cauchy Integral Analogs to the Theory of a Three-Dimensional Geopotential Field.

- 6.1 Newton Potential and the Three-Dimensional Cauchy Integral Analog.- 6.1.1 Newton Potential.- 6.1.2 Newton Potential of Simple Field Sources.

- 6.1.3 Newton Potential of Polarized Field Sources.- 6.1.4 Three-Dimensional Cauchy-Type Integral as a Sum of a Simple and a Double Layer Field.- 6.2 Representation of the Gravitational Field in Terms of the Cauchy Integral Analog.

- 6.2.1 Gravitational Field Equations.- 6.2.2 Representation of the Gravitational Field of a Three-Dimensional Homogeneous Body in Terms of the Cauchy-Type Integral.- 6.2.

3 Gravitational Field of a Body with an Arbitrary Density Distribution.- 6.2.4 Case of Vertical or One-Dimensional Horizontal Variations in the Density.- 6.2.5 Some Special Cases of Density Distribution.- 6.

2.6 Calculation of the Gravitational Field of a Three-Dimensional Infinitely Extended Homogeneous Domain.- 6.2.7 Field of an Infinitely Extended Domain Filled with Masses of a Z-Variable Density.- 6.3 Representation of a Fixed Magnetic Field in Terms of the Cauchy Integral Analog.- 6.

3.1 Intensity and Potential of a Fixed Magnetic Field.- 6.3.2 Representation of a Magnetic Field with an Arbitrary Distribution of Magnetized Masses.- 6.3.3 Potential Distribution of Magnetization.

- 6.3.4 Laplace Distribution of Magnetization.- 6.3.5 Magnetic Field of a Uniformly Magnetized Body.- 6.4 Generalized Kertz-Siebert Technique for Separation of Three-Dimensional Geopotential Fields.

- 6.4.1 Statement of the Problem of Separation of a Three-Dimensional Field.- 6.4.2 Separation of Fields at Ordinary Points of the Surface.- 6.4.

3 Separation of Fields at Singular Points of the Surface.- 6.4.4 Generalized Kertz-Siebert Formulas.- 7 Analytical Continuation of a Three-Dimensional Geopotential Field.- 7.1 Fundamentals of Analytical Continuation of the Laplace Field.- 7.

1.1 Analytical Nature of Laplace Vector Fields.- 7.1.2 Uniqueness of Laplace Vector Fields and Harmonic Functions.- 7.1.3 Concept of Analytical Continuation of a Vector Field and Its Riemann Space.

- 7.1.4 Continuation of the Laplace Field Using the Taylor Series.- 7.1.5 Stal Theorem (Principle of Continuity for the Laplace Field).- 7.2 Analytical Continuation of the Three-Dimensional Cauchy In.