# Physics 501: Mathematical Physics I

## Fall 2018

[Figures following each major topic indicate the approximate number of lecture hours devoted to it.]

1. Introduction and course overview [0.5]

2. Partial differential equations [5.5]   (Riley & Hobson Chapters 10, 11)
• some familiar PDEs   (10.1, 10.4, 10.5)
• classification of PDEs   (10.3.3)
• characteristics and boundary conditions   (10.6)
• method of characteristics
• the Helmholtz equation   (11.3.2)
• solution by separation of variables   (11.1, 11.2, 11.3)
• cartesian coordinates
• cylindrical polar coordinates: Bessel's equation
• spherical polar coordinates: Legendre equation
• normal modes and general solutions
• Bessel functions; spherical Bessel functions   (9.5, 9.6)
• spherical harmonics   (9.1 - 9.3)

3. Ordinary differential equations [4.5]   (Riley & Hobson, Chapters 4, 7, 8)

• second-order systems: series solutions; the indicial equation   (7.1 - 7.4)
• second solutions; Wronskian development   (7.5)
• Fuch's theorem and properties of first and second solutions
• self-adjoint differential operators; Sturm-Liouville theory   (8.1, 8.2)
• properties of eigenfunctions of Hermitian operators   (8.3)
• orthogonality of eigenfunctions; eigenfunction expansion
• the Euler-Fourier formula   (4.1, 4.2)
• completeness and mean-square convergence of Fourier series
• uniform, pointwise, and mean-square convergence
• Bessel's inequality and Parseval's identity   (4.8)

4. Fourier series [1.5]   (Riley & Hobson, Chapter 4)

• Euler-Fourier formulae   (4.1, 4.2, 4.3)
• the Gibbs phenomenon   (4.4, 4.5)
• application to PDEs: spectral methods

5. Bessel Functions [1.5]   (Riley & Hobson Sections 9.5, 9.6)
• ODE and series solution
• orthogonality and Bessel series; Fourier-Bessel series
• generating function
• recurrence relations
• normalization
• asymptotic behavior; Hankel functions
• spherical Bessel functions
• modified Bessel functions

^^^^^^^^^^   (Mid-term coverage)   ^^^^^^^^^^

6. Legendre Functions [3]   (Riley & Hobson Sections 9.1 - 9.3)
• ODE and series solution
• orthogonality and Legendre/Laplace series
• generating function
• recurrence relations
• spherical harmonics
• multipole expansion