Complex Linear Algebra: Time:MWF 1:00-1:50 PM; Location: 2017 ECEB ECE298-CLA Register
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 | |
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)}$} and its utility in Engineering applications Read: Class-notes | |
13 | 16 | M | 4/15 | Lecture: Multi-valued complex functions; Riemann sheets; Branch cuts Read: Class-notes |
14 | W | 4/17 | Lecture: Complex analytic mapping (Domain coloring) Visualizing complex valued functions Colorized plots of rational functions Software: Matlab: zviz.zip, python | |
15 | F | 4/19 | Lecture: Riemann’s extended plane: The Riemann sphere (1851); pdf Mobius Transformation (youtube, HiRes), pdf description Mobius composition transformations, as matrices Read: Class-notes | |
16 | 17 | M | 4/22 | Lecture: Cauchy’s Integral theorem & Formula
Read: Class-notes |
17 | W | 4/24 | Lecture: Cauchy Residue theorem; Green’s theorem in the plane
Read: Class-notes | |
18 | F | 4/26 | Lecture: Inverse Laplace transform {$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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