作者介紹
(德)阿諾德·索末菲|責編:陳亮
阿諾德·索末菲（Arnold Sommerfeld，1868-1951），Sommerfeld是德國偉大的理論物理學家、應用數學家、流體力學家、教育家、原子物理與量子物理的創始人之一。他對理論物理多個領域，包括力學、光學、熱力學、統計物理、原子物理、固體物理（包括金屬物理）等有重大貢獻，在偏微分方程、數學物理等應用數學領域也有重要貢獻。他引進了第二量子數（角量子數）、第四量子數（自旋量子數）和精細結構常數，等等。20世紀最偉大的物理學家之一Planck在獲得1918年度諾貝爾物理學獎的頒獎典禮的儀式上的演講中指出：「Sommerfeld…便可以得到一個重要公式，這個公式能夠解開氫與氫光譜的精細結構之謎，而且現在最精確的測量……一般地也能通過這個公式來解釋……這個成就完全可以和海王星的著名發現相媲美。早在人類看到這顆行星之前Leverrier就計算出它的存在和軌道。」
Sommerfeld思想深刻，研究成果影響深遠。例如，他去世后發展起來的數值廣義相對論和新近崛起的引力波理論研究中，還引用「Sommerfeld條件」，該條件在求解中發揮了重要作用。這再次彰顯了他的科學工作的巨大價值。
目錄
CHAPTER Ⅰ. FOURIER SERIES 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. Anhar-monic 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 NormalForm
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 HeatConduction
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. Thrce-Dimensional Case f = f(r,P,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 ±
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 Reciprocai Radii
B. The Boundary Value Problem of Potential Theory for the Sphere, ThePoisson 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 Acous-tics 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 Con-dition of Radiation
29. The Eigen Value Spectrum of Wave Mechanics. Balmer's Term
30. Green's Function for the Wave Mechanical Scattering Problem. TheRutherford Formula of Nuclear Physics
Appendix Ⅰ. Normalization of the Eigenfunctions in the Infinite Domain
Appendix Ⅱ. A New Method for the Solution of tlie Exterior BoundaryValue Problem_ of the Wave Equation Presented for the Special Case of che 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 Con-ductive 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 Horiz
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
Indcx
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