|
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 |
|
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
|