- The page size for these files was chosen so that
if you print with the page scaling option
"multiple pages per sheet" and select 2 pages per sheet,
then there is no reduction of the print size on
- If you choose "none" and "auto rotate and center",
there will be no reduction in the print size and
you will have very wide margins. You will get the
same result if you choose "Shrink to printable area".
- If you want to enlarge the print,
use the option "Fit to printable area".
Short Tutorial Notes.
For Lecture Notes Associated with Courses, click
and search for the word "Notes".
- Bonds and CDs.
Updated 3/6/2017, 3/3/2017, 2/21/2017, 2/20/2017.
A brief introduction is given to compound interest, certificates
of deposit, and bonds. The focus is on determining a fair price,
yield to maturity, accrued interest, and duration. Matlab code
is given to compute the accrued interest with
the 30/360 US method, which is used for US corporate
bonds and many US agency bonds.
certificate of deposit,
days between dates,
- Magnitude and
Phase of Complex Numbers.
Updated 7/12/2017, 7/6/2017.
Every nonzero complex number can be expressed in terms of its magnitude
and angle. This angle is sometimes called the phase or argument of the
complex number. Although formulas for the angle of a complex number
are a bit complicated, the angle has some properties that are
simple to describe. In particular, when the complex number is a function
of frequency, we derive a simple formula for the derivative of the
- Why Are Minimum-Phase Systems Called Minimum Phase?
Updated 7/12/2017, 7/9/2017.
The main reason is that such systems have minimum phase lag (phase lag
is the negative of the phase). The key to deriving this result is to
study the derivative of the phase, which is closely related to the group
delay. This further requires a deeper-than-usual understanding of the
angle or argument of a complex number. This background is included.
- Gaussian Quadrature and the Eigenvalue Problem. (Printing options do not apply.)
Introduction to Gaussian quadrature (numerical integration) and orthogonal polynomials.
Boole's rule, Newton–Cotes formulas, Simpson's rule, three-term recurrence.
- Introduction to Frequency Analysis and the DFT (Short Version).
Updated 6/1/2017, 5/25/2017.
The continuous-time Fourier transform is defined.
It is shown that windowing has the effect of blurring
the transform. Then sampling is introduced as a way to
approximate the transform integral. The resulting approximation
is seen to be periodic.
Windowing and sampling are then combined to obtain the
discrete Fourier transform (DFT). Matlab code for
plotting the fast Fourier transform (FFT) on the appropriate
frequency axis is provided.
sampling rate, sinc function, spectrum.
- Introduction to Frequency Analysis and the DFT (Long Version).
Updated 5/25/2017, 4/23/2016.
The relationships between the continuous-time Fourier transform,
the continous-time Fourier series, and the discrete Fourier
transform (DFT) are presented.
fast Fourier transform (FFT), line spectrum, Nyquist rate,
sampling rate, sinc function, spectrum.
- Derivation of the Fourier Inversion Formula,
Bochner's Theorem, and Herglotz's Theorem.
Summability kernels are introduced in order to rigorously
approximate the Dirac delta function and its sifting property.
It is then possible to establish the convolution property
of the Fouier transform and the inverse Fourier transform
formula. The inversion formula is then used to motivate
Bochner's theorem. Bochner's theorem is proved by appealing
to well-known properties of limits of sequences of
cumulative distributions in probability theory.
Herglotz's theorem is handled similarly.
- The Intermediate-Value Theorem.
Updated 2/15/2015, 11/9/2014.
A simple proof of the intermediate-value theorem is given.
As an easy corollary, we establish the existence of $n$th
roots of positive numbers.
- Basic Properties of Power Series.
Updated 11/9/2014, 10/4/2014, 10/3/2014, 10/2/2014, 9/30/2014, 9/23/2014, 9/21/2014.
These notes provide a quick introduction (with proofs)
to the basic properties of power series, including
the exponential function and the
fact that power series can be differentiated term by term.
discrete Fubini theorem,
differentiation term by term,
multiplication of power series,
radius of convergence,
uniform Cauchy criterion,
- The Fundamental Theorem of Algebra and the Minimum Modulus Principle.
Updated 11/8/2014, 10/11/2014, 10/2/2014.
A direct proof of the fundamental theorem of algebra is given.
In other words, we show that every polynomial of degree greater
than or equal to one has at least one root in the complex plane.
A slight modification of the proof yields the minimum modulus
- Permutations, the Parity Theorem,
Updated 10/15/2014, 9/14/2014, 9/12/2014, 9/11/2014, 9/2/2014.
The Parity Theorem says that
whenever an even (resp. odd) permutation
is expressed as a composition of transpositions,
the number of transpositions must be even (resp. odd).
The purpose of this article is to give a simple definition
of when a permutation is even or odd, and develop just enough
background to prove the parity theorem. Several examples
are included to illustrate the use of the notation and
concepts as they are introduced. We then define the determinant
in terms of the parity of permutations. We establish basic
properties of the determinant. In particular, we show
that $\det B A = \det B \, \det A$, and we show that
$A$ is nonsingular if and only if $\det A \ne 0$.
Block Matrix Formulas.
Updated 7/7/2015, 7/6/2015, 7/5/2015.
matrix inversion formula,
upper left block.
Updated 3/18/2017, 10/6/2016, 8/11/2015, 8/22/2013.
strongly convex function,
Updated 9/19/2014, 9/13/2014.
contraction mapping theorem,
linear differential equation,
nonlinear differential equation,
variation of parameters,
A simple lemma is proved giving sufficient conditions for an operator
to be closable.
last modified Sun 08 Apr 2018, 11:46 AM, CDT
Web Page Contact: John (dot) Gubner (at) wisc (dot) edu