Chapter I
Fourier Series and Integrals

1.1 Fourier Series 
1.2 Example of a discontinuous
function. Gibbs' phenomenon and nonuniform 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. Anharmonic analysis. An example of nonfinal determination of coefficients 
1.6.1 Oscillating and
osculating Approximation 
1.6.2 Anharmonic Fourier analysis 
1.6.3 An example of nonfinal
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 Selfadjoint 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 Onedimensional case f
= f (r) 
4.20.2 Twodimensional case f
= f(r, j) 
4.20.3 The
Threedimensional 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 Halfinteger
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 multipole 
4.24.4 Details of
hypergeometric functions 
4.24.5
Spherical functions with noninteger 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
Eigenfunctions and Eigenvalues

5.25 Eigenvalue and
Eigenfunctions 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 EllipticHyperbolic 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 Eigenvalues. Radiationcondition 
5.29
The Eigenvalue Spectrum of Wave Mechanics. The Balmer Term 
5.30
The Green function of the wavemechanical scattering problem. Rutherford's formula of nuclear physics 
Appendix 5.1 Normalization
of eigenfunctions 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 coordinates 
Appendix 5.4
Plane and spherical wave in unlimited space of any number of dimensions 
5A1 Coordinate System and Notation 
5A.2
The Eigenfunctions of the Unbounded Polydimensional Space 
5A.3
The Spherical wave and Green's Function in the polydimensional space 
5A.4 Transition
from the spherical to the plane wave 
Chapter VI
Problems of wireless
telegraphy

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 
Appendix 6
Wireless
Telegraphy on the Spherical Earth

Exercises
Hints and Answers

Index
