Complex Linear Algebra: Time:MWF 1:001:50 PM; Location: 3081 ECEB ECE298CLA; (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: Classnotes Homework 1 (NS1): 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: Classnotes Homework 2 (NS2): pdf, Due on Lec 6  
3  F  3/15  Lecture: Pyth triplets (Lec 7); Inverse matrix via Gaussian Elimination (Lec 15)
Read: Classnotes  
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: Classnotes: Lec 16 page 120125 '''Homework 3 (NS3):Pell's equation, Fibonacci sequence; pdf, Due on Lec 8 
5  W  3/27  Lecture: Pell's equation: \(M^2Nn^2=1 (m,n,N\in{\mathbb N})\) & Fibonacci Series \(f_{n+1} = f_n + f_{n1}, (n,f_n \in{\mathbb N})\) Companion matrix and eigenanalysis (eigenvalues, eigenvectors) Read: Classnotes Lec 8 and Lec 9 
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: Classnotes Homework 1 (NS1): 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: Classnotes Homework 2 (NS2): pdf, Due on Lec 6  
3  F  3/15  Lecture: Pyth triplets (Lec 7); Inverse matrix via Gaussian Elimination (Lec 15)
Read: Classnotes  
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: Classnotes: Lec 16 page 120125 '''Homework 3 (NS3):Pell's equation, Fibonacci sequence; pdf, Due on Lec 8 
5  W  3/27  Lecture: Pell's equation: \(M^2Nn^2=1 (m,n,N\in{\mathbb N})\) & Fibonacci Series \(f_{n+1} = f_n + f_{n1}, (n,f_n \in{\mathbb N})\) Companion matrix and eigenanalysis (eigenvalues, eigenvectors) Read: Classnotes Lec 8 and 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: Classnotes: Lec 19  
7  14  M  4/1  Lecture: Laplace transforms and Causality; Residue expansions
Read: Classnotes 
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: Classnotes NS3 Due  
9  F  4/5  Exam I: 7:009:30 PM; 3081 ECEB room confirmed; NS1NS3, Lec 11; 2x2 complex matrix analysis; Analysis of Exam 1 pdf Ver 1.7, Apr 18, 2019  
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) Complexanalytic series representations: (1 vs. 2 sided); ROC of \(1/(1s), 1/(1x^2), \ln(1s)\) 1) Series; 2) Residues; 3) polezeros; 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 circularfunction package (Logs, exp, sin/cos); Inversion of analytic functions: Example: \(\tan^{1}(z) = \frac{1}{2i}\ln \frac{iz}{i+z}\), the inverse of Euler's formula (1728) Read: Classnotes 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; CauchyRiemann (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 \(\frac{\partial R(\sigma,\omega)}{\partial\omega} = \frac{\partial X(\sigma,\omega)}{\partial\sigma}\) CauchyRiemann 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: Classnotes & Boas pages 1326; 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: Classnotes  
13  16  M  4/15  Lecture: Multivalued complex functions; Riemann sheets; Branch cuts Read: Classnotes DE2: Integration, differentiation wrt \(s\); Cauchy theorems; LT; Residues; power series, RoC; LT; (pdf), due on Lec 16; DE1 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: Classnotes  
16  17  M  4/22  Lecture: Cauchy’s Integral theorem & Formula Read: Classnotes DE3: Inverse LT; Impedance; Transmission lines; (pdf); due on Lec 20; DE2 due 
17  W  4/24  Lecture: Trainmission problem (ABCD matrix method); More on the Cauchy Residue theorem;
Read: Classnotes (ABCD matrix method)  
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: Classnotes;  
19  18  M  4/29  Lecture: Inverse Laplace transform via the Residue theorem \(t > 0\)
Read: Classnotes; 
20  W  5/1  Lecture: Properties of the Laplace Transform: Modulation, convolution; Review DE3 Due  
  R  5/2  Reading Day  
  M  5/6/2019  Time and place confirmed: Official: Final Exam 710PM ECE 3081 
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