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