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Complex Linear Algebra: Time:MWF 1:00-1:50 PM; Location: 2017 ECEB ECE298-CLA Register

ECE 298 ComplexLinearAlg-S19 Schedule (Spring 2019)

L W D Date Lecture and Assignment

Part I: Introduction to complex 2x2 matricies (6 Lectures)
1 11 M 3/11 Lecture: Introduction & Overview:
1) Integers, fractionals, rationals, real vs. complex, vectors and matrices;
2) Common Math Notation symbols
3) Matlab tutorial: pdf
4) Polynomials and Newton's complex root finding method; Polynomial root classification by convolution;
5) Fundamental Thm of Algebra (pdf); 6) Series representations of analytic functions, ROC
7) Historical notes on complex numbers: Solution of the quadratic (Brahmagupta, 628), cubic (c1545), quartic (Tartaglia et al..., 1535),
quintic cannot be solved (Abel, 1826) and much much more
Read: Class-notes
Homework 1 (NS-1): Basic Matlab commands: pdf, Due on Lec 3; help.m
2 W 3/13 Lecture: Complex analytic functions, geometry; vector scalar (i.e., dot) products
NS-1 Due
Read: Class-notes
Homework 2 (NS-2):

pdf Due on Lec 6

3 F 3/15 Lecture: Inverse matrix via Gaussian Elimination

Read: Class-notes

12 Spring Break
4 13 M 3/25 Lecture: Analysis of simple LRC circuits by matrix composition: ABCD (transmisison matrix method, composition of polynomials)
Read: Class-notes
5 W 3/27 Lecture: Pell's equation: {$M^2-Nn^2=1 (m,n,N\in{\mathbb N})$} & Fibonacci Series {$f_{n+1} = f_n + f_{n-1}, (n,f_n \in{\mathbb N})$}
Companion matrix and eigen-analysis (eigenvalues, eigenvectors)Read: Class-notes
Homework 3 (NS-3): Pell's equation, Fibonacci sequence; pdf, Due on Lec 9
L W D Date Lecture and Assignment
Part II: Complex analytic analysis (9 Lectures)
6 F 3/29 Lecture: Fourier transforms for signals vs. Laplace transforms for systems Fourier Transform (wikipedia);
Notes on the Fourier series and transform from ECE 310 pdf including tables of transforms and derivations of transform properties;
Classes of Fourier transforms pdf due to various scalar products.

Read: Class-notes
NS-2 Due

7 14 M 4/1 Lecture: Laplace transforms and Causality; Residue expansions

Read: Class-notes

8 W 4/3 Lecture: The 10 system postulates of Systems (aka, Networks) pdf;
The important role of the Laplace transform re impedance: {$z(t) \leftrightarrow Z(s)$};
A.E. Kennelly introduces complex impedance, 1893 pdf; Fundamental limits of the Fourier vs. the Laplace Transform: {$\tilde{u}(t)$} vs. {$u(t)$}
Read: Class-notes
NS-3 Due

9 F 4/5 Exam I (In Class): 2x2 matrix analysis
10 15 M 4/8 Lecture: Integration in the complex plane: FTC vs. FTCC; Analytic vs complex analytic functions and Taylor formula
Calculus of the complex {$s=\sigma+j\omega$} plane: {$dF(s)/ds$}, {$\int F(s) ds$} (Boas, see page 8)
The convergent analytic power series: Region of convergence (ROC)
Complex-analytic series representations: (1 vs. 2 sided); ROC of {$1/(1-s), 1/(1-x^2), -\ln(1-s)$}
1) Series; 2) Residues; 3) pole-zeros; 4) Continued fractions; 5) Analytic properties
History: The amazing Bernoulli family; Fluid mechanics; airplane wings; natural logarithms
Beginnings of modern mathematics: Euler and Bernoulli, Euler's standard circular-function package (Logs, exp, sin/cos);
D'Angelo {$e^z$} & {$\log(z)$} Math 446 lecture
Inversion of analytic functions: Example: {$\tan^{-1}(z) = \frac{1}{2i}\ln \frac{i-z}{i+z}$}, the inverse of Euler's formula (1728)Read: Class-notes
Homework 4 (DE-1): Series, differentiation, CR conditions, Bi-Harmonic functions: pdf, Due Oct 30
11 W 4/10 Lecture: Differentiation in the complex plane: Complex Taylor series;
Cauchy-Riemann (CR) conditions and differentiation wrt {$s$}: {$Z^\prime(s) \equiv \frac{dZ(s)}{ds} = \frac{dZ(s)}{d\sigma} = \frac{dZ(s)}{dj\omega}$}
Differentiation independent of direction in {$s$} plane: {$Z(s)$} results in CR conditions:
{$\frac{\partial R(\sigma,\omega)}{\partial\sigma} = \frac{\partial X(\sigma,\omega)}{\partial\omega}$} and {$\frac{\partial R(\sigma,\omega)}{\partial\omega} = -\frac{\partial X(\sigma,\omega)}{\partial\sigma}$}
Cauchy-Riemann conditions require that Real and Imag parts of {$Z(s) = R(\sigma,\omega) + j X(\sigma,\omega)$} obey Laplace's Equation:
{$\nabla^2 R=0$}, namely: {$\frac{\partial^2R(\sigma,\omega)}{\partial \sigma^2} + \frac{\partial^2 R(\sigma,\omega)}{\partial \omega^2} =0 $} and {$\nabla^2 X=0$}, namely: {$\frac{\partial^2 X(\sigma,\omega)}{\partial \sigma^2} + \frac{\partial^2 X(\sigma,\omega)}{\partial \omega^2} =0$},
Biharmonic grid (zviz.m)
Discussion: Laplace's equation means conservative vector fields: (1, 2)
Read: Class-notes & Boas pages 13-26; Derivatives; Convergence and Power series
12 F 4/12 Lecture: Complex analytic functions; Brune Impedance {$Z(s) = {P_m(s)}/{P_n(s)}$} and its utility in Engineering applications
Read: Class-notes
13 16 M 4/15 Lecture: Multi-valued complex functions; Riemann sheets; Branch cuts
Read: Class-notes
14 W 4/17 Lecture: Complex analytic mapping (Domain coloring)
Visualizing complex valued functions Colorized plots of rational functions

Software: Matlab: zviz.zip, python
AE-3: Complex algebra; visualizing complex functions; Mobius transformations; (pdf due 1 week)
Read: Class-notes

15 F 4/19 Lecture: Riemann’s extended plane: The Riemann sphere (1851); pdf
Mobius Transformation (youtube, HiRes), pdf description
Mobius composition transformations, as matrices
Read: Class-notes
16 17 M 4/22 Lecture: Cauchy’s Integral theorem & Formula

Read: Class-notes

17 W 4/24 Lecture: Cauchy Residue theorem; Green’s theorem in the plane

Read: Class-notes

18 F 4/26 Lecture: Inverse Laplace transform {$t \le 0$}; Case for causality Laplace Transform,
Cauchy Riemann role in the acceptance of complex functions:
Convolution of the step function: {$u(t) \leftrightarrow 1/s$} vs. {$2\tilde{u}(t) \equiv 1+ \mbox{sgn}(t) \leftrightarrow 2\pi \delta(\omega) + 2/j\omega$}
Read: Class-notes;
19 18 M 4/29 Lecture: Inverse Laplace transform via the Residue theorem {$t > 0$}

Read: Class-notes;

20 W 5/1 Lecture: Properties of the Laplace Transform: Modulation, convolution, etc.

AE-3 Due

- R 5/2 Reading Day

- M 5/? Final Exam TBD

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