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
|