# Physics 501: Mathematical Physics I

*Fall 2018*

## Lecture Schedule

Lecture Topics
1 Course overview; introduction to PDEs; characteristic equation; types of PDE
2 Method of characteristics; boundary conditions; separation of variables in
Cartesian coordinates; HW1 out
3 Examples: normal modes, Laplace's equation; separation in cylindrical polar
coordinates; Bessel's equation; examples
4 Separation in spherical polar coordinates; Legendre's equation; examples
Associated Legendre equation; spherical harmonics; spherical Bessel functions
HW1 due; HW2 out
5 2nd order linear ODEs; singular points and series solutions; Fuchs's theorem
6 Second solutions; Wronskian development; self-adjoint operators, Sturm-Liouville theory
HW1 returned; HW2 due
7 Eigenfunctions and eigenvalues; orthogonality of eigenfunctions; eigenfunction
expansion; Euler-Fourier formula; HW3 out
8 Applications of Bessel series; mean square convergence; applications of Fourier
series; spectral methods; Hw2 returned
9 Bessel function properties: generating function; recurrence relations; normalization;
asymptotic behavior; Hankel functions; cylindrical waves; applications; HW3 due
10 Legendre polynomials; Rodrigues formula; generating function and interpretation;
Legendre series; multipole expansion; recurrence relations
11 Associated Legendre functions; spherical harmonics; the addition theorem;
multipole expansion; HW3 returned
12 MID-TERM (Thursday, November 1; coverage: HW1-HW3); HW4 out
13 Applications of spherical harmonics; Fourier series --> Fourier transforms
14 Parseval's theorem; examples of Fourier transforms; examples; Fourier transforms of
derivatives; applications to PDEs and ODEs
15 Complex analysis basics; complex functions; Caucy Riemann conditions; Cauchy's theorem;
Cauchy integral formula; Taylor and Laurent series; HW4 due; HW5 out
16 Zeros and singularities; poles; the residue theorem; examples
17 More applications of the residue theorem; convolution theorem; HW6 out
18 Greens functions; properties and examples; fundamental solutions;
HW5 due
19 Applications of Greens functions; electrodynamics
20 Catch-up, review, or preview of numerical methods -- PHYS 502
HW6 due