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 L W D Date Lecture and Assignment Part I: Introduction to complex 2x2 matricies (6 Lectures) 1 11/13 M 3/23 Lecture: Introduction & Overview: 1) Integers, fractionals, rationals, real vs. complex, vectors and matrices;2) Common Math Notation symbols3) Matlab tutorial: pdf4) 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, ROC7) Historical notes on complex numbers: Solution of the quadratic (Brahmagupta, 628), cubic (c1545), quartic (Tartaglia et al..., 1535), quintic cannot be solved (Abel, 1826)Read: Class-notes as discussed during lecture; Homework 1 (NS-1): pdf, Due on Lec 4; 2 W 3/25 Lecture: Complex analytic functions;Read: Class-notes 3 F 3/27 Lecture: Pyth triplets; Gaussian Elimination Read: Class-notes 4 13/14 M 3/30 Lecture: Analysis of simple LRC circuits by matrix composition: ABCD (transmisison matrix methodRead: Class-notes: $$\S$$ 3.7 (pages 110-115)NS-1 DueHomework 2 (NS-2): pdf, Due on Lec 7 5 W 4/1 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 Lec 8 and Lec 9
 L W D Date Lecture and Assignment Part II: Complex analytic analysis (9 Lectures) 6 F 4/3 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 14/15 M 4/6 Lecture: Laplace transforms and Causality; Residue expansions Read: Class-notesNS-2 DueHomework 3: AE-1: Polynomials and related problems pdf, Due on Lec 10 8 W 4/8 Lecture: The 10 system postulates of Systems (aka, Networks) pdf;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) Read: Class-notes 9 F 4/10 Lecture: The important role of the Laplace transform re impedance: $$z(t) \leftrightarrow Z(s)$$; Fundamental limits of the Fourier vs. the Laplace Transform: $$\tilde{u}(t)$$ vs. $$u(t)$$The matrix formulation of the polynomial and the companion matrixComplex-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 Read: Class-notes: Pages 71-75 10 15/16 M 4/13 Lecture: Complex analytic functions; History: Beginnings of modern mathematics: Euler and Bernoulli, The Bernoulli family; ; natural logarithms Euler's standard circular-function package (Logs, exp, sin/cos); Brune Impedance $$Z(s) = {P_m(s)}/{P_n(s)}$$ and its utility in Engineering applicationsRead: Class-notesAE-1 DueHomework 4: DE-1: Series, differentiation, CR conditions, Branch cuts: pdf, Due on Lec 13 11 W 4/15 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 4/17 Exam I: 12:00-3:00 PM; via Zoom NS1-NS2, AE1, Lec 11; 2x2 complex matrix analysis; pdf 13 16/17 M 4/20 Lecture: Multi-valued complex functions; Riemann sheets; Branch cutsRead: Class-notesHomework 5: DE-2: Integration, differentiation wrt $$s$$; Cauchy theorems; LT; Residues; power series, RoC; LT;DE2 (pdf), due on Lec 16;DE-1 due 14 W 4/22 Lecture: Complex analytic mapping (Domain coloring)Visualizing complex valued functions Colorized plots of rational functions Software: Matlab: zviz.zip, pythonRead: Class-notes 15 F 4/23 Lecture: Riemann’s extended plane: The Riemann sphere (1851); pdfMobius Transformation (youtube, HiRes), pdf description Mobius composition transformations, as matricesRead: Class-notes 16 17/18 M 4/27 Lecture: Cauchy’s Integral theorem & FormulaRead: Class-notesHomework 6: DE-3: Inverse LT; Impedance; Transmission lines; (pdf); due on Lec 20; DE2 due 17 W 4/29 Lecture: Train-mission problem (ABCD matrix method); More on the Cauchy Residue theorem; Read: Class-notes (ABCD matrix method) 18 F 5/1 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 5/4 Lecture: Inverse Laplace transform via the Residue theorem $$t > 0$$ Read: Class-notes; 20 W 5/6 Lecture: Properties of the Laplace Transform: Modulation, convolution; ReviewDE-3 Due - W 5/6 Last day of instruction. - R 5/7 Reading Day - M 5/?/2020 Time and place to be confirmed: Official: Final Exam May 13, 7-10PM 3081ECEB