- Lecture 1, sections 1.1-1.5Lecture 1, sections 1.1-1.5: Numbers, arrays, matrices, vectors, linear operators, also on YouTube.
- Lecture 2, part 1 and part 2, sections 1.6-1.9Inner products, the Cauchy-Schwarz inequality, linear independence, completeness, the dimension of a vector space, also on YouTube part1 and part 2.
- Lecture 3,Sections 1.10-1.13: orthonormal vectors, outer products, Dirac notation, the adjoint of a linear operator, also on YouTube.
- Lecture 4Sections 1.14-1.20, Self-adjoint or hermitian operators; real, symmetric linear operators; unitary operators; Hilbert spaces; antilinear, antiunitary operators; symmetry in quantum mechanics; determinants; and the inverse of a matrix. also on YouTube.
- Lecture 5,Sections 1.20-1.25: determinants, inverse of a matrix, linear equations, linear least squares, Lagrange multipliers, eigenvectors, eigenvectors of a square matrix, also on YouTube.
- Lecture 6,Sections 1.25-1.28: Eigenvectors of a square matrix, a matrix obeys its characteristic equation, functions of matrices, hermitian matrices. Also on YouTube.
- Lecture 7, Sections 1.28-1.31: Hermitian matrices, normal matrices, compatible normal matrices, the singular-value decomposition. Also on YouTube.
- Lecture 8, Sections 1.30-1.34: Compatible normal matrices, the singular-value decomposition, the Moore-Penrose pseudoinverse, the rank of a matrix, software, also on YouTube.
- Lecture 9,Sections 2.1-2.7: Complex Fourier series, the interval, where to put the 2pi's, real Fourier series for real functions, stretched intervals, Fourier series in several variables, how Fourier series converge. Also on YouTube.
- Lecture 10,sections 2.7-2.8: How Fourier series converge, quantum-mechanical examples. also on YouTube.
- Lecture 11,Sections 2.8-2.10 and 3.1: Quantum-mechanical examples, Dirac notation, Dirac's delta function, and the Fourier transform, also on YouTube.
- Lecture 12, sections 3.1-3.3: Fourier transforms, the Fourier transform of a real function, some results due to Dirac, Parseval, and Poisson, also on YouTube.
- Lecture 13,sections 3.3-3.6: Some results due to Dirac, Parseval, and Poisson, derivatives and integrals of Fourier transforms, Fourier transforms of functions of several variables, convolutions. Also on YouTube.
- Lecture 14, sections 3.6-3.13 Convolutions, the Fourier transform of a convolution, Fourier transforms and Green's functions, Laplace transforms, derivatives and integrals of Laplace transforms, Laplace transforms and differential equations, inversion of Laplace transforms, applications to differential equations. Also on YouTube.
- Lecture 15, sections 4.1-4.9, Convergence, tests of convergence, convergent series of functions, power series, factorials and the gamma function, Taylor series, Fourier series as power series, the binomial series and theorem, logarithmic series. Also on YouTube.
- Lecture 16,sections 4.10-4.12 and 5.1-5.3: Dirichlet series and the zeta function, Bernoulli numbers and polynomials, asymptotic series, analytic functions, Cauchy's integral theorem, Cauchy's integral formula Also on YouTube.
- Lecture 17,sections 5.3-5.10: Cauchy's integral formula, the Cauchy-Riemann conditions, harmonic functions, Taylor series for analytic functions, Cauchy's inequality, Liouville's theorem, the fundamental theorem of algebra, Laurent series Also on YouTube.
- Lecture 18, sections 5.10-5.13: Laurent series, singularities, analytic continuation, the calculus of residues. Also on YouTube.
- Lecture 19, sections 5.13-5.14: The calculus of residues and ghost contours. Also on YouTube.
- Lecture 20, sections 5.14-5.16: Ghost contours, logarithms and cuts, powers and roots. Also on YouTube.
- Lecture 21, sections 5.17-5.19: Conformal mapping, Cauchy's principal value, dispersion relations. Also on YouTube.
- Lecture 22, sections 5.19-5.22 and 6.1: Kramers-Kronig relations, phase and group velocities, the method of steepest descent, and ordinary linear differential equations. Also on YouTube.
- Lecture 23, sections 6.1-6.5: Ordinary linear differential equations; linear partial differential equations; gradient, divergence, and curl; separable partial differential equations. Also on YouTube.
- Lecture 24, sections 6.5--6.8: Separable partial differential equations, wave equations, first-order differential equations, separable first-order differential equations. Also on YouTube.
- Lecture 25, sections 6.8--6.16: Separable first-order differential equations, hidden separability, exact first-order differential equations, the meaning of exactness, integrating factors, homogeneous functions, the virial theorem, homogeneous first-order ordinary differential equations, linear first-order ordinary differential equations. Also on YouTube.
- Lecture 26, sections 6.16--6.19: Linear first-order ordinary differential equations, systems of differential equations, singular points of second-order ordinary differential equations, Frobenius's series solutions. Also on YouTube.
- Lecture 27, sections 6.20--6.27: Fuch's theorem, even and odd differential operators, why not three solutions? boundary conditions, a variational problem, self-adjoint differential operators. Also on YouTube.
- Lecture 28, sections 6.27-6.33: Self-adjoint differential operators, self-adjoint differential systems, making operators formally self adjoint, wronskians of self-adjoint operators, first-order self-adjoint differential operators, a constrained variational problem, eigenfunctions and eigenvalues of self-adjoint systems. Also on YouTube.
- Lecture 29, sections 6.33-6.38, 6.40, and 8.1-8.9: Eigenfunctions and eigenvalues of self-adjoint systems, unboundedness of eigenvalues, completeness of eigenfunctions, the inequalities of Bessel and Schwarz, Green's functions, eigenfunctions and Green's functions, nonlinear differential equations, the Legendre polynomials, the Rodrigues formula, the generating function, Legendre's differential equation, recurrence relations, special values of Legendre's polynomials, Schlaefli's integral, orthogonal polynomials, the azimuthally symmetric laplacian. Also on YouTube.
- Lecture 30, sections 8.9-8.13 and 9.1: The azimuthally symmetric laplacian, Laplace's equation in two dimensions, Helmholtz's equation in spherical coordinates, the associated Legendre functions/polynomials, spherical harmonics, Bessel functions of the first kind. Also on YouTube.
- Lecture 31, sections 9.1-9.4: Bessel functions of the first kind, spherical Bessel functions of the first kind, Bessel functions of the second kind, spherical Bessel functions of the second kind. Also on YouTube.

- Lecture 1, sections 11.1-11.9
- Lecture 2, sections 11.9-11.13
- Lecture 3,
sections 11.13-18

- Lecture 4,
sections 11.18-11.26

- Lecture 5, sections 11.27-11.35
- Lecture 6, sections 11.35-11.38
- Lecture 7, sections 11.38-11.44
- Lecture 8, sections 11.44-11.48
- Lecture 9, sections 11.48-11.49
- Lecture 10,
sections 11.49-11.50 and 13.1

- Lecture 11,
sections 13.1-13.5

- Lecture 12,
sections 13.5-13.10

- Lecture 13,
sections 13.10-13.14

- Lecture 14,
sections 13.13-13.17

- Lecture 15,
sections 13.17-13.18

- Lecture 16,
sections 13.18-13.21

- Lecture 17,
sections 14.1-14.4

- Lecture 18,
sections 14.4-14.6

- Lecture 19,
sections 16.1-16.5

- Lecture 20,
sections 16.5-16.9

- Lecture 21,
sections 16.9-16.11

- Lecture 22,
sections 16.11-16.12

- Lecture 23,
sections 16.12 and 15.1-2

- Lecture 24,
sections 15.1-5

- Lecture 25,
sections 16.12-14

- Lecture 26,
sections 18.1-18.3

- Lecture 27,
sections 19.1-19.7

- Lecture 28,
sections 10.1-10.15

- Lecture 29,
sections 10.15-10.19

- Lecture 30,
sections 10.19-10.33

- Lecture 1, sections 1.1-1.5
- Lecture 2 Part 1, Part 2, sections 1.5-1.12
- Lecture 3, sections 1.12-1.17
- Lecture 4, sections 1.17-1.23
- Lecture 5, sections 1.23-1.27
- Lecture 6, sections 1.28-1.31
- Lecture 7, sections 1.31-2.1
- Lecture 8, sections 2.1-2.7
- Lecture 9, sections 2.7-2.10
- Lecture 10, sections 2.10-3.1
- Lecture 11, sections 3.1-3.4
- Lecture 12, sections 3.4-3.12
- Lecture 13, sections 4.1-4.8
- Lecture 14, sections 4.8-5.1
- Lecture 15, sections 5.1-5.3
- Lecture 16, sections 5.3-5.10
- Lecture 17, sections 5.10-5.14
- Lecture 18, sections 5.14-5.16
- Lecture 19, sections 5.16-5.19
- Lecture 20, sections 5.19-6.3
- Lecture 21, sections 6.3-6.6
- Lecture 22, sections 6.6-6.14
- Lecture 23,sections 6.14-6.21
- Lecture 24,sections 6.21-6.28
- Lecture 25,sections 6.28-6.32
- Lecture 26,sections 6.32-6.36
- Lecture 27,sections 6.36-7.3 & 7.5
- Lecture 28,sections 8.1-8.9
- Lecture 29, sections 8.9-9.1
- Lecture 30, section 9.1
- Lecture 31, sections 9.1-9.2

- Lecture 1: linear algebra up to linear independence.
- Lecture 2: dimension of a vector space, inner and outer products, and Dirac notation.
- Lecture 3: laser demo on polarization vectors, identity operators, vectors and their components, linear operators and their matrix elements, determinants.
- Lecture 4: determinants, systems of linear equations, eigenvectors and eigenvalues, the adjoint of a linear operator, hermitian operators, unitary operators, anti-unitary and anti-linear operators, Wigner's theorem on symmetry in quantum mechanics, eigenvalues of a square matrix.
- Lecture 5: eigenvalues of a square matrix, functions of matrices, hermitian matrices.
- Lecture 6: hermitian matrices, normal matrices, determinant of a normal matrix, tricks with Dirac notation, compatible normal matrices.
- Lecture 7: compatible normal matrices, a matrix satisfies its characteristic equation, the singular-value decomposition.
- Lecture 8: Part 1 Part 2: applications of the singular-value decomposition, LAPACK, the rank of a matrix, the tensor or direct product, density operators.
- Lecture 9: on tensor products, density operators, correlation functions, groups, and complex Fourier series.
- Lecture 10: on the Fourier series for \( \exp(-m |x|) \), real Fourier series, the Fourier series for x, complex and real Fourier series for an interval of length L.
- Lecture 11: on Fourier series in several variables, the convergence of Fourier series, and quantum-mechanical examples.
- Lecture 12: on quantum-mechanical examples, the harmonic oscillator, non-relativistic strings, periodic boundary conditions, and the transition to the Fourier transform.
- Lecture 13: on the transition to the Fourier transform, the Fourier transform of a gaussian and of a real function, a representation of the delta-function, Parseval's identity, derivatives of a Fourier transform, and momentum and momentum space.
- Lecture 14: on the uncertainty principle, Fourier transforms in several variables, application to differential equations, Fick's law, and diffusion.
- Lecture 15: on the wave equation, diffusion, convolutions, Gauss's law, and the magnetic vector potential.
- Lecture 16: on the Fourier transform of a convolution, finding Green's functions, Laplace transforms, examples of Laplace transforms, how to measure the lifetime of a fluorophore, differentiation and integration of Laplace transforms, inverting Laplace transforms, convergence of infinite series, tests of convergence, series of functions, uniform convergence, the Riemann zeta function, and power series.
- Lecture 17: on power series, the geometric series, the exponential series, factorials and the Gamma function, Taylor series, Fourier series, the binomial theorem, the binomial coefficient, double factorials, the logarithmic series, Bernoulli numbers and polynomials, the Lerch transcendent, asymptotic series, and the exponential integrals.
- Lecture 18: on dielectrics, analytic functions, Cauchy's integral theorem, Cauchy's integral formula, the Cauchy-Riemann conditions, and harmonic functions.
- Lecture 19: on the Cauchy-Riemann conditions and the contour integral of a general function, harmonics functions, applications to two-dimensional electrostatics, Earnshaw's theorem, Taylor series, and Cauchy's inequality.
- Lecture 20: on Cauchy's inequality, Liouville's theorem, the fundamental theorem of algebra, Laurent series, poles, essential singularities, and the calculus of residues.
- Lecture 21 Part 1 Part 2: on the calculus of residues, ghost contours, third-harmonic microscopy, and several examples of the use of ghost contours.
- Lecture 22: on logarithms, cuts, roots, contour integrals around cuts, Cauchy's principal value, i-epsilon rules, and application to Feynman's propagator.