MATHEMATICS 603: Mathematical Principles in Science III
SP 2012, 10:30 a.m. MWF,
Rodica D. Costin, office MW 436
Ulrich Gerlach, office
Rodica D. Costin: W, F 11:30-12:20 a.m. or by appointment
policy and syllabus
Gerlach's online book
Further reading: you
may find useful to consult also
Fourier Series and Boundary Value Problems, by
Brown and Churchill
analysis: an introduction, by Stein and Shakarchi
March 26 Lecture 1 notes
March 30 pages 1 to 5b from the notes here.
The HW assignment
handed in in class
today is due next Friday April 6, at the beginning of class.
April 2 ...continuation
April 4 The Fourier
NOTE: the HW is due Monday April 9 (since
we are behind with the material taught in class).
the proof of the Fourier Theorem, pages C5-C8 in the notes above.
Today we do:
The Fourier Integral.
Dirac's delta function and other
Please read these notes
(including the material not covered in class) and solve the
Friday, April 6 (i)
Fourier Integral Theorem and its proof reviewed
(ii) Fourier transform as an isometric transformation
Apr. 9 (i) Fourier transform of a periodic function
(ii) Construction of
an orthonormal wave packet basis for L^2
(iii) Handout on
four applications of O.N. wavepackets.
April 11 Lecture topics were:
- Fourier Series Theorem + Fourier Integral Theorem ==> The
set of wave packets form a complete (and O.N.) basis for the
space of square integrable (= finite energy) functions on the
- The phase space representation of such functions.
- Parceval's relation as a sum of wave packet energies occupying
their respective phase space cells.
- A musical score's (say, Mozart's or Beethoven's sonatas; see
aggregate of notes on a 2-D sheet of music represented as wave
packet energies occupying their respective 2-D phase space
Homework 2 is due this Friday,
The purpose of the lecture is to set the context for Monday, when we
will decompose L^2 into the hierarchy of multi resolution subspace,
hoping to open the curiosity about multi resolution analysis.
Today's material is not yet in the typeset notes, but see the
handwritten notes: a
quick study to wavelets, notes
Multi Scale Analysis:
a) Translated sinc functions: its application to the
Whittaker-Shannon sampling theorem.
b) Translated and dilated sinc functions as a multi resolution basis
for L^2 .
c) The "father" wavelet as the fountainhead for these basis
Here is a handout for HW3.
April 16 Ideas developed today:
"Mother" wavelet was only mentioned, with no illustrative example.
Same thing for the pyramid algorithm.
- Spaces of wavelets generated by translation and dilation of
the father wavelet.
- Dilation hierarchy of father generated wavelets.
- Stepwise Multi Resolution Analysis into bulk ("father
") and detail ("mother") wavelets generated subspaces of L^2;
MRA <--> father wavelet.
- Projection of an L^2 function into these subspaces.
Today's development consisted of
April 21 ...continuation
- Input-output relation as the generalization based on an
observed causal source-response relation for a linear system.
- Linear algebra review of the existence and uniqueness of
a linear input-output relation expressed equivalently in terms
(i) the existence of right and left inverses as prototypes for
the unit impulse response (a.k.a. the Green's function) for the
system, (ii) onto and one-to-one mapping properties, (iii)
spanning and linear independence properties, and hence the
solution to the inhomogeneous linear system.
- Definition of the concept of a given operator and its adjoint
to include the given and the (in general different) adjoint
boundary conditions characterizing their domains.
- (See 4.1.1. in the text and an algebra review.)
- The reciprocity theorem: proof and application
(end of Section 4.2)
- Poisson's equation for the simple string
- Continuity and jump conditions for a string imbedded in an
- The Fundamental Theorem of Green's function Theory: discussion
of its featues and its proof;
- Uniqueness of the Green's function, if it exists; a preview of
the importance of uniqueness.
- Construction of the Green's function for separated boundary
condition stated as a theorem; geometrical and analytical reason
for when the construction is successful, and when it isn't.
- Worked example: the amplitude profile of a static simple
- Solution to the totally inhomogeneous boundary value problem
in terms of the system's Green's function.
- Construction of the Green's function for arbitrary homogeneous
- Solution to the inhomogeneous boundary value problem in terms
of solutions to the Sturm-Liouville eigenvalue problem.
Solutions to a non-selfadjoint inhomogeneous boundary
value problem in terms of the eigenfunctions of the associated
given S-L problem and its adjoint, whose eigenfunction are
"bi-orthonormal" to those of the given one.
- Integrate the Green's function in the complex lambda plane to
obtain the orthonormalized eigenfunction of a self-adjoint
boundary value problem
- Worked example of a simple string with free boundary
conditions at both ends.
- Comments about the infinite string boundary value problem as
the limit of a finite string boundary value problem.
- Review: branches, branch cuts, and Riemann sheets for the
square root function
You and Your Research
(by Richard Hamming), a guide for doing great work in
your chosen field"; Integral
equations via Green's functions"
- Harmonically driven semi-infinite string: (i) Its amplitude
profile, (ii) outgoing and incoming boundary conditions, (iii)
Green's function for the amplitude profile, (iv) square
integrability and non-integrability on the two Riemann sheets of
the square root function.
- Comment about section 4.10.4: The infinite string as the limit
of one which is finite: the mixed Dirichelet-Neumann conditions
become the outgoing or incoming boundary conditions as the
string length becomes arbitrarily large.
May 9 Helmholtz's equation as a highly degenerate
- Outgoing and square integrability boundary conditions of the
Green's function for a simple string infinite in both
- Additional comment about section 4.10.4: The infinite string
as the limit of one which is finite: the aggregate of simple
poles of the coalesce into a branch cut.
- Fourier Sine Theorem via complex integration of the Green's
function for a semi-infinite string.
- Students should start reading Chapter 5
- Helmholtz's equation: its plane wave solutions relative to
- Helmholtz's equation as a highly degenerate eigenvalue problem
- Eigenvalues of a complete set of commuting operators as an
Translation and rotation symmetries of the Euclidean plane
Their representations in the space of solutions of the Helmholtz
equation: translation vs. rotation eigenfunctions
Goal: for the Helmholtz equation construct a rotation
eigensolution which is a linear combination of translation
- Rotation eigenfunction (a.k.a. cylinder harmonic) as a
superposition of plane wave solutions to the the Helmholtz
- The defining and convergence properties of the complex contour
integral representations of the two Hankel functions
- Selections from the set of 23 properties of Hankel and Bessel
- Solution to the wave equation on a cylindrically symmetric
background in terms of cylinder normal modes (Property 6)
- The asymptotic behaviour and its physical meaning for the
Hankel, Bessel, and Neuman functions (Properties 8-10)
- The Bessel function defined in compliance with the reflection
principle of complex variables theory (Property 11)
- Integral representation of Bessl functions of integral order
- "Property 14": Complex contour integral proof of a Hankel
function as a linear combination of Bessel functions
- "Property 15-16": Complex conjugates of Hankel and Neumann
- The Contiguity Relations (Property 16)
- Plane wave as a superposition of cylinder waves (Property 18)
- The Cylindrical Addition Theorem (Property 19)
May 25 Methods for finding the asymptotic behavior of
integrals with a large parameter:
- The exterior boundary value ("scattering") problem
- The interior boundary and initial value ("normal modes in a
Stationary phase method.
Steepest descent method.
operators (p.9 revised)
Green's Function; uniqueness
(Note: "fundamental solution" F is for linear operators with
constant coefficients, and solves P(d/dx)F=delta.)
Green's function of the adjoint
problem (older- revised p.3, 5B,5C,6), and newer (revised). Supplement.
Simpler forms for second
order linear differential operators
Review of the resolvent and elements of
Non-self-adjoint boundary value
problems (p.2B, 7B added)
resources: recall the dual space (p.4), the adjoint (p.15),
for the adjoint boundary conditions (p.20), the Green's function for
non-selfadjoint problems (p. 21)
Sampling Theorem - the case of a compact interval.