Chapter I Fourier Series and Integrals 1.1 Fourier Series 1.2 Example of a discontinuous function. Gibbs' phenomenon and non-uniform convergence 1.3 On the convergence of Fourier series 1.4 Transition to Fourier Integral 1.5. Expansion in terms of spherical functions 1.6 Generalizations: Oscillating and osculating approximations. An-harmonic analysis. An example of non-final determination of coefficients 1.6.1 Oscillating and osculating Approximation 1.6.2 An-harmonic Fourier analysis 1.6.3 An example of non-final determination of coefficients Chapter II About partial differential equations 2.7 Occurrence of the simplest partial differential equations 2.8 Elliptic, hyperbolic, parabolic types. Characteristics theory 2.9 Differences between hyperbolic, elliptic and parabolic equations. The analytic character of their solutions 2.9.1 Hyperbolic Differential Equation 2.9.2 Elliptic Equation 2.9.3 Parabolic differential equation 2.10 Green's Theorem and Function for Linear, especially Elliptic Differential Equations 2.10.2 Normal Form of Green's Theorem, especially for Elliptic Equations 2.10.3 Definition of Unit Source and Principal Solution 2.10.4 The Analytic Character of the Solution of an Elliptic Differential Equation 2.10.5 Principal Solution in an Arbitrary Number of Dimensions 2.10.6 Definition of the adjoint differential expression0.6 Definition of Green's Function for Self-adjoint Equations 2.11 Riemann's Integration of the Hyperbolic differential Equation 2.12 Green's Theorem in Heat Conduction. The Principal Solution of the Heat Conduction Equation Chapter III Boundary Value Problems in Heat Conduction 3.14 The problem of Earth's temperature 3.15 The problem of the ring 3.16 The linear heat conductor with two ends 3.17 Reflection in a plane and in space 3.18 Uniqueness of the solution in the case of an arbitrarily formed heat conductor Chapter IV Cylinder and Sphere Problems 4.19 Bessel and Hankel functions 4.19.1 The Bessel function and its integral representation 4.19.2 The Hankel Functions and their Integral Representation 4.19.3 Series expansions at zero 4.19.4 Recursion Formulae 4.19.5 Aysmptotic Representation of the Hankel Functions 4.20 Heat Compensation in a Cylinder 4.20.1 One-dimensional case f = f (r) 4.20.2 Two-dimensional case f = f(r, j) 4.20.3 The Three-dimensional Case f = f(r, j, z) 4.21. More about Bessel functions 4.21.1 Generating Function and Addition Theorems 4.21.2 Integral Representations in Terms of Bessel Functions 4.21.3 Half-integer and third integer subscripts 4.21.4 Generalization of the saddle point method according to Debye 4.22 Spherical Functions and Potential Theory 4.22.1 The Generating Function 4.22.2 Differential and difference equations 4.22.3 The Associate Spherical Functions 4.22.4 About the Associate Functions with Negative superscript m 4.22.5 Surface Spherical Functions and Representation of Arbitrary Functions 4.22.6 Representation of the Spherical Functions 4.22.6 Integral Representation of the Spherical Functions 4.22.7 A Recursion Formula for the Associate Functions 4.22.8 Normalization of the Associate Functions 4.22.9 The Addition Theorem of the Spherical Functions 4.23. The Green Function of Potential Theory for the sphere. Sphere and Circle Problems for other Differential Equations 4.23.1 The Geometry of Reciprocal Radii 4.23.2 The Boundary Value Problem of Potential Theory for the Sphere, Poisson's Integral 4.23.3 General remarks regarding the transformation by reciprocal radii: 4.23.5 Failure of spherical reflection for the wave equation 4.24 More about Spherical Functions: 4.24.1 Plane Wave and Spherical Wave in Space 4.24.2 Asymptotic Matters 4.24.3 The spherical function as electrical multi-pole 4.24.4 Details of hypergeometric functions 4.24.5 Spherical functions with non-integer subscripts 4.24.6 Spherical functions of the second kind Appendix 4.1 Reflection in a circular cylindrical or spherical mirror 4A1.1 Circular Cylindrical Metal Mirror 4A1.2 The segment of a sphere as an elastic reflector Appendix 4.2 Supplement to Riemann's problem of sound waves in 2.11 Chapter V Eigen-functions and Eigen-values 5.25 Eigen-value and Eigen-functions of the oscillating membrane 5.25.1 The rectangle 0 x a, 0 y b 5.25.2 Circle, Circular Ring, Circular Sector 5.25.3 Ellipse and Elliptic-Hyperbolic Curve Quadrangle 5.26 General Remarks about the Boundary Value Problems of Acoustics and Heat Conduction 5.27 Free and Forced Vibrations. Green's Function of the Vibration Equation 5.28 Infinite Region and Continuous Spectrum of Eigen-values. Radiation-condition 5.29 The Eigen-value Spectrum of Wave Mechanics. The Balmer Term 5.30 The Green function of the wave-mechanical scattering problem. Rutherford's formula of nuclear physics Appendix 5.1 Normalization of eigen-functions in an unfinitely expanded region Appendix 5.2 A new kind of method for the solution of the external boundary value problem of teh wave equation, explained by the example of the sphere. Appendix 5.3 The wave mechanical eigen functions of the dispersion problem in polar co-ordinates Appendix 5.4 Plane and spherical wave in unlimited space of any number of dimensions 5A1 Co-ordinate System and Notation 5A.2 The Eigen-functions of the Unbounded Poly-dimensional Space 5A.3 The Spherical wave and Green's Function in the poly-dimensional space 5A.4 Transition from the spherical to the plane wave 6.31 Hertz's Dipole in a Homogeneous Medium and above a Perfectly Conducting Earth 6.31.1 Introduction of Hertz's Dipole 6.31.2 Integral Representation of Primary Excitation 6.31.3 Vertical- and Horizontal Antenna over an Infinitely Well Conducting Earth 6.32 The Vertical Antenna over an arbitrary Earth 6.33 The Horizontal Antenna over an Arbitrary Earth 6.34 Errors during taking bearings of an electrical horizontal antenna 6.35 The Magnetic or Frame Antenna 6.36 Radiation Energy and Earth's Absorption Index