Prerequisites: Graduate Standing or Consent of Instructor

Fundamental theorems of Arithmetic, of Algebra
(without rigorous proofs, but with counterexamples. Notions of Ring, Field, Group. Fields F_{p}.)

Elements of Linear Algebra.
The purpose of this Section 1 is the wide generalization of Superposition Principle. To be covered:
 Linear Space over the fields of a) real numbers, b) complex numbers as an expression of unsensitivity to a phase shift, c) shortly – over other fields of numbers.
 Linear and bilinear forms. Definition of Scalar (Dot) Product in real and complex spaces; definition of Euclidean Space as a linear space with a scalar product. Only after that, definition of Symmetric and Hermitian Operators/matrices in real and complex Euclidean spaces, respectively. Definition of Orthogonal and Unitary Operators. Connection with conservation of Energy or Probability.
 Eigenvalues and eigenvectors of a linear operator. Possibility of reduction of a Symmetric / Hermitian operator to diagonal form via rotation of coordinate system. Completeness of the system of eigenvectors of a general operator: when it takes place and when it possibly does not. “Additional” Voigt’s waves.
 Matrices, powers of matrices and functions of matrices. Thesis: “Functions of matrices are not so much different from the functions of one variable, x or z = x + iy.” Characteristic polynomial and characteristic equation for eigenvalues; Cayley–Hamilton’s theorem. Solving Coupled Wave Equations via “Interpolation Lagrange Polynomial”.
 Use of software to handle matrices.

Orthogonal Expansions.
The purpose of this section is to familiarize students with particular systems of functions used for the representation of solutions of various physical problems.
 Different possibilities to define scalar product in the linear space of functions. GrammSchmidt orthogonalization of polynomials. Study of particular “orthogonal polynomials”. Useful trick: generating functions.
 Systems of eigenfunctions of Hermitian operators in various linear Euclidean spaces of functions. Orthogonality and completeness of various trigonometric series. Study of examples of expansion.
 Discussion and study: which special functions are included into various software packages.

Fourier series (in narrow sense), Fourier integrals.
The purpose of this section is to refresh the knowledge of Fourier Transformations. In particular, the following thesis will be emphasized: “Fourier lives on singularities”.
 Fourier transformations from function g(x) to h(k), from g(t) to h(?), which uses exp(ikx) or exp(i?t), with inverted dimensions: [k] = [1/x], [?] = [1/t]. Variants to distribute the (1/2p) coefficient.
 Fourier operator F acting as
h(y) = F[g](y) = (1/2p)^{1/2} ? exp(iyy’) g(y’)d y’,
with functions g(y), h(y) being the functions of the same dimensionless argument y and belonging to the same (infinitedimensional) linear space. Properties
F^{1} = F^{+}, F^{4} = 1
of that operator. Numerous eigenfunctions of that operator, including HermitGaussian modes / quantum oscillator wave functions. Hyperbolic secant as an eigenfunction of that operator.
 Integration by parts as the way to elucidate the contributions of singularities. Notions of edge waves. Apodization. Singularities in complex plane will be studied later, in Section 6.
 Notion of discrete Fourier transform the particular finitedimensional unitary transformation.
 Use of software packages to calculate Fourier Transform. Advantages and dangers of Fast Fourier Transform (FFT) implementations, including those in various software packages. Advantage: fast. Danger: remapping of the interval, easy for mathematicians, may be clumsy for consumers (for us).

Ordinary Differential Equations (ODEs).
The purpose of this section is to refresh the knowledge of the solving methods for the most frequently encountered types of ODEs.
 Linear ODEs without righthandside.
 Number of linearly independent solutions; analogs of Wronsky’s theorem. Physical analogies: conservation of brightness; preservation of phase space, LagrangeHelmholtz invariant, Second Law of Thermodynamics.
 Equations with constant coefficients; see Lagrange Interpolation Formula from Section 1.4.
 1d wave equation and WKB approach (approximation) to solving it.
 Equations describing parametric resonance and their approximate solutions. Step aside into electrodynamics: role of impedance.
 Equation
da/dt + [G(t) + i?(t)]a(t) = f(t).
and its most general solution.
 General (nonlinear) ODEs.
 The case when all boundary conditions are prescribed at one end, i.e. Cauchy problem. Use of software to solve Cauchy problem for a system of ODEs.
 Boundary conditions are distributed between different points: what to do? Separate linear and nonlinear ODEs.
 “Analytically solvable” nonlinear ODEs: DEs with separable variables.
 Use of software packages to solve Cauchy problem.
Note to the material of Section 4. Invariance of DEs with respect to some or other transformations of coordinates and functions helps to solve these DEs and often leads to meaningful conservation laws, like energy, pressure flux, etc.

Partial Differential Equations (PDEs).
The purpose of this section is to discuss the frequently encountered PDEs and their solutions. General view of “characteristics” of PDEs will be discussed as well.
 Preliminary info: integration by parts in ndimensional integrals.
 Helmholtz Equation (HE):
[ (? – ?) + k^{2}(r) ] u(r) = 0.
 Spatially homogeneous case.
 General solution with the use of Green’s function: exact and approximate Huygens’ principle.
 Parabolic / paraxial approximation; estimation of the accuracy of that approximation.
 Thermal conductivity equation
 Notion of PDE’s “characteristics” and their relationship to Fourier expansion.
 LorentzInvariance (LI) of D’Alembert equation is almost evident – hence LI of Maxwell equations.

Analytic Functions of Complex Variable.
The purpose of this section is not (repeat, not) to show the beauty of underlying mathematics. To the contrary, the purpose is to help solving particular ODEs and PDEs and to provide technical tools for asymptotically calculating certain integrals.
 Definition; independence of integral upon the contour; poles and residues.
 Stirling’s formula for n!
 Special functions (Airy, Bessel, and other) as solutions of ODEs. “Stitching” and choice out of two linearly independent solutions in different regions of argument.
 Steepest descent method of asymptotic calculation of integrals.
 Critical point is at the boundary.
 Critical point is inside the integration interval.
 Asymptotic solution of the problem of adiabatic passage (LandauZener theory).
Time permitting:
 Dispersion relations
 Tunnel effect and FrantzKeldysh effect: asymptotic theory. Keldysh theory of multiphoton / tunnel ionization.
Necessary topic (probably at the beginning of the course): Newton’s binomial formula
(1 + x)^{a} = 1 + ax + a(a – 1)x^{2}/2! + a(a – 1)(a – 2)x^{3}/3! + a(a – 1)(a – 2)(a – 3)x^{4}/4! + …
for arbitrary (positive/negative, integer/fractional, real/complex) values of parameter a. Power series for other functions (to be known by heart): exp(x), ln(1 – x), sin(x), cos(x), sinh(x), cosh(x), arctan(x), arctanh(x). Formulas for the sum of arithmetic progression, for the sum of geometric progression.