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, ¿)

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

21. More About Bessel Functions

A. Generating Function and Addition Theorems

B. Integral Representations in Terms of Bessel Functions

C. The Indices n + ¿ and n ± ¿3

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

Partial Differential Equations in Physics: Lectures on Theoretical Physics, Volume VI is a series of lectures in Munich on theoretical aspects of partial differential equations in physics.
This book contains six chapters and begins with a presentation of the Fourier series and integrals based on the method of least squares. Chapter II deals with the different types of differential equations and boundary value problems, as well as the Green's theorem and Green's function. Chapter III addresses the classic problem of heat conduction and the intuitive method of reflected images for regions with plane boundaries. Chapters IV and V examine the Bessel functions, spherical harmonics, and the general method of eigenfunctions. Chapter VI highlights the problems in radio waves propagation, always considering the earth as a plane.
This book is of great benefit to mathematicians, physicists, and physics teachers and undergraduate students.