Complex Linear Algebra: Time:MWF 1:00-1:50 PM; Location: 3081 ECEB ECE298-CLA; (Register here)
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 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 8 |
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); 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 (DE1): Series, differentiation, CR conditions, Branch cuts: pdf, Due on Lec 13 |
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 DE-2: Integration, differentiation wrt \(s\); Cauchy theorems; LT; Residues; power series, RoC; LT; (pdf), due on Lec 16; DE-1 due |
14 | W | 4/17 | Lecture: Complex analytic mapping (Domain coloring) Visualizing complex valued functions Colorized plots of rational functions | |
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 DE-3: Inverse LT; Impedance; Transmission lines; (pdf; due on Lec 20; DE-2 due |
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; Review DE-3 Due | |
- | R | 5/2 | Reading Day | |
- | M | 5/6/2019 | Final Exam tentative: 7-10PM ECE 3081 |
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