Partial Differential Equations in Physics

Chapter I. Fourier Seribs and Integrals


1. Fourier Series


2. Example of a Discontinuous Function. Gibbs' Phenomenon and Non-Uniform Convergence


3. On the Convergence of Fourier Series


4. Passage to the Fourier Integral


5. Development by Spherical Harmonics


6. Generalizations: Oscillating and Osculating Approximations. Anharmonic Fourier Analysis. An Example of Non-Final Determination of Coefficients

A. Oscillating and Osculating Approximation

B. Anharmonic Fourier Analysis

C. An Example of a Non-Final Determination of Coefficients

Chapter II. Introduction to Partial Differential Equations


7. How the Simplest Partial Differential Equations Arise


8. Elliptic, Hyperbolic and Parabolic Type. Theory of Characteristics


9. Differences Among Hyperbolic, Elliptic, and Parabolic Differential Equations. The Analytic Character of Their Solutions

A. Hyperbolic Differential Equations

B. Elliptic Differential Equations

C. Parabolic Differential Equations


10. Green's Theorem and Green's Function for Linear, and, in Particular, for Elliptic Differential Equations

A. Definition of the Adjoint Differential Expression

B. Green's Theorem for an Elliptic Differential Equation in its Normal Form

C. Definition of a Unit Source and of the Principal Solution

D. The Analytic Character of the Solution of an Elliptic Differential Equation

E. The Principal Solution for an Arbitrary Number of Dimensions

F. Definition of Green's Function for Self-Adjoint Differential Equations


11. Riemann's Integration of the Hyperbolic Differential Equation


12. Green's Theorem in Heat Conduction. The Principal Solution of Heat Conduction

Chapter III. Boundary Value Problems in Heat Conduction


13. Heat Conductors Bounded on One Side


14. The Problem of the Earth's Temperature


15. The Problem of a Ring-Shaped Heat Conductor


16. Linear Heat Conductors Bounded on Both Ends


17. Reflection in the Plane and in Space


18. Uniqueness of Solution for Arbitrarily Shaped Heat Conductors

Chapter IV. Cylinder and Sphere Problems


19. Bessel and Hankel Functions

A. The Bessel Function and its Integral Representation

B. The Hankel Function and its Integral Representation

C. Series Expansion at the Origin

D. Recursion Formulas

E. Asymptotic Representation of the Hankel Functions


20. Heat Equalization in a Cylinder

A. One-Dimensional Case f = f(r)

B. Two-Dimensional Case f = f(r, ϕ)

C. Three-Dimensional Case f = f(r, ϕ,z)


21. More About Bessel Functions

A. Generating Function and Addition Theorems

B. Integral Representations in Terms of Bessel Functions

C. The Indices n + ½ and n ± ⅟3

D. Generalization of the Saddle-Point Method According to Debye


22. Spherical Harmonics and Potential Theory

A. The Generating Function

B. Differential and Difference Equation

C. Associated Spherical Harmonics

D. On Associated Harmonics with Negative Index m

E. Surface Spherical Harmonics and the Representation of Arbitrary Functions

F. Integral Representation of Spherical Harmonics

G. A Recursion Formula for the Associated Harmonics

H. On the Normalization of Associated Harmonics

J. The Addition Theorem of Spherical Harmonics


23. Green's Function of Potential Theory for the Sphere. Sphere and Circle Problems for Other Differential Equations

A. Geometry of Reciprocal Radii

B. The Boundary Value Problem of Potential Theory for the Sphere, The Poisson Integral

C. General Remarks about Transformations by Reciprocal Radii

D. Spherical Inversion in Potential Theory

E. The Breakdown of Spherical Inversion for the Wave Equation


24. More About Spherical Harmonics

A. The Plane Wave and the Spherical Wave in Space

B. Asymptotic Behavior

C. The Spherical Harmonic as an Electric Multipole

D. Some Remarks about the Hypergeometric Function

E. Spherical Harmonics of Non-Integral Index

F. Spherical Harmonics of the Second Kind

Appendix I. Reflection on a Circular-Cylindrical or Spherical Mirror

Appendix II. Additions to the Riemann Problem of Sound Waves in Section

Chapter V. Eigenfunctions and Eigen Values


25. Eigen Values and Eigenfunctions of the Vibrating Membrane


26. General Remarks Concerning the Boundary Value Problems of Acoustics and of Heat Conduction


27. Free and Forced Oscillations. Green's Function for the Wave Equation


28. Infinite Domains and Continuous Spectra of Eigen Values. The Condition of Radiation


29. The Eigen Value Spectrum of Wave Mechanics. Balmer's Term


30. Green's Function for the Wave Mechanical Scattering Problem. The Rutherford Formula of Nuclear Physics

Appendix I. Normalization of the Eigenfunctions in the Infinite Domain

Appendix II. A New Method for the Solution of the Exterior Boundary Value Problem of the Wave Equation Presented for the Special Case of the Sphere

Appendix III. The Wave Mechanical Eigenfunctions of the Scattering Problem in Parabolic Coordinates

Appendix IV. Plane and Spherical Waves in Unlimited Space of an Arbitrary Number of Dimensions

A. Coordinate System and Notations

B. The Eigenfunctions of Unlimited Many-Dimensional Space

C. Spherical Waves and Green's Function in Many-Dimensional Space

D. Passage from the Spherical Wave to the Plane Wave

Chapter VI. Problems of Radio


31. The Hertz Dipole in a Homogeneous Medium Over a Completely Conductive Earth

A. Introduction of the Hertz Dipole

B. Integral Representation of the Primary Stimulation

C. Vertical and Horizontal Antenna for Infinitely Conductive Earth

D. Symmetry Character of the Fields of Electric and Magnetic Antennas


32. The Vertical Antenna Over an Arbitrary Earth


33. The Horizontal Antenna Over an Arbitrary Earth


34. Errors in Range Finding for an Electric Horizontal Antenna


35. The Magnetic or Frame Antenna


36. Radiation Energy and Earth Absorption

Appendix. Radio Waves on the Spherical Earth

Exercises for Chapter I

Exercises for Chapter II

Exercises for Chapter III

Exercises for Chapter IV

Exercises for Chapter V

Exercises for Chapter VI

Hints for Solving the Exercises

Index