The text has three parts: Part I establishes solving the homogenous Laplace and Helmholtz equations in the three main coordinate systems, rectilinear, cylindrical, and spherical and develops the solution space for series solutions to the Sturm-Liouville equation, indicial relations, and the expansion of orthogonal functions including spherical harmonics and Fourier series, Bessel, and Spherical Bessel functions. Many examples with figures are provided including electrostatics, wave guides and resonant cavities, vibrations of membranes, heat flow, potential flow in fluids, and plane and spherical waves.
In Part II the inhomogeneous equations are addressed where source terms are included for Poisson's equation, the wave equation, and the diffusion equation. Coverage includes many examples from averaging approaches for electrostatics and magnetostatics, from Green function solutions for time independent and time dependent problems, and from integral equation methods. In Part III complex variable techniques are presented for solving integral equations involving Cauchy Residue theory, contour methods, analytic continuation, and transforming the contour; for addressing dispersion relations; for revisiting special functions in the complex plane; and for transforms in the complex plane including Green’s functions and Laplace transforms.
· Mathematical Methods for Physics creates a strong, solid anchor of learning and is useful for reference.
· Lecture note style suitable for advanced undergraduate and graduate students to learn many techniques for solving partial differential equations with boundary conditions
· Many examples across various subjects of physics in classical mechanics, classical electrodynamics, and quantum mechanics
· Updated typesetting and layout for improved clarity
This book, in lecture note style with updated layout and typesetting, is suitable for advanced undergraduate, graduate students, and as a reference for researchers. It has been edited and carefully updated by Gary Powell.
Chapter 1. The Partial Differential Equations of Mathematical Physics
Chapter 2. Separation of Variables and Ordinary Differential Equations
Chapter 3. Spherical Harmonics and Applications
Chapter 4. Bessel Functions and Applications
Chapter 5. Normal Mode Eigenvalue Problems
Chapter 6. Spherical Bessel Functions and Applications
Section II - Inhomogeneous Problems, Green’s Functions, Integral Equations
Chapter 7. Dielectric and Magnetic Media
Chapter 8. Green’s Functions: Part One
Chapter 9. Green’s Functions: Part Two
Chapter 10. Integral Equations
Section III - Complex Variable Techniques
Chapter 11. Complex Variables; Basic Theory
Chapter 12. Evaluation of Integrals
Chapter 13. Dispersion Relations
Chapter 14. Special Functions
Chapter 15. Integral Transforms in the Complex Plane
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