Math 421, Autumn, 2022, University Park

Complex numbers

\[\begin{gather*} 1 = e^{2 \pi i}, \quad m \ddot{z} = -z | z | - i, \quad i \frac{\partial \psi}{\partial t} = k \frac{\partial^2 \psi}{\partial x^2} + V(x) \psi \\ \mathscr{F}[f](s) = \int_{-\infty}^{\infty} f(x) e^{-2 \pi i s x} dx, \quad f'(w) = \frac{1}{2 \pi i} \oint_{\partial \Omega : \Omega \ni w} \frac{f(z)}{(z-w)^2} dz \\ \ddot{z} = \frac{-z}{|z|^3} \end{gather*}\]

• Taught by Tim Reluga, 424 McAllister Hall
• Textbook: Introduction to Complex Analysis by H. A. Priestley, 2003 Warning: see Typos section below.
• Syllabus (see for details missing above!)
• Typo account: $ 5.50

Final exam Monday from 6:50 to 8:40 pm in Thomas 119.

Project information

Topics due October 31st. See details here

Presentation video uploads ...

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Homeworks

• Homework 1, due Wednesday, August 31, answers
• Homework 2 part a and part b, due Wednesday, September 14, answers (updated 4 pm, Tuesday, 13 Sept.)
• Homework 3, Chapter 5 of Priestley, problems 1-10. Due Wednesday, September 21
• Homework 4, Chapter 6 of Priestley problems 1-10, due Wednesday, September 28
• Homework 5, Chapter 7 of Priestley, 1-5,9,10,14,15,16, due Wednesday, October 5.
• Homework 6, here, due Wednesday, October 12
• No homework due Wednesday, October 19
• Homework 7, Priestly, Chapter 4, #1 & 2, Chapter 10, #1-2. due Wednesday, October 26
  • Update: don't do 10.3. We haven't covered quite enough yet!
• Homework 8, here, due Wednesday, November 2
• Homework 9, here due Wednesday, November 9
• Homework 10, Ch 17, {1,2,5i-iii,8a(i-iv)8bi,9i-iii,11,18}, due Wednesday, November 16 partial solutions
• Homework 11, here (Posted Tuesday, Nov. 29, updated Monday, not collected)

Quiz solutions

• Quiz 0, pretest of background knowledge.
• Quiz 1, Wednesday, August 31 Answers
• Quiz 2, Wednesday, September 14 Answers
• Quiz 3, Wednesday, September 21 Answers
• Quiz 4, Wednesday, October 5 Answers
• Quiz 5, Monday, October 10 Answers
• Quiz 6, Wednesday, October 28,Answers
• Quiz 7, Wednesday, November 13,Answers
• Quiz 8, Wednesday, November 16,Answers
• Quiz 9, Wednesday, December 7,Answers

Resources

• References on asymptotic analysis
• Week 3 lecture supplement
• Javascript practice with complex arithmetic (in beta)
• Javascript Practice with complex polar arithmetic (in beta, feedback encouraged)
• My complex variables quick reference sheet
• MIT version of a related course

Computing and Visualizations

• Hutchinson-Wright example of solving a complex transcendental equation.
• Cosine map \(\cos(z)\) illustrating how cosine transforms the complex plane.
• Exponential function \(\exp(z)\) argument plot with code.

• Visual demonstration of the properties of \(\exp(-1/z^{2})\) for real and complex inputs. For complex inputs, this function exhibits whats called an essential singularity at \(z=0\).

• Argument plot for a random 5-root polynomial. Roots are where the color map is singular. Sample python code to draw this plot is here.

• An online interpretter for the python programming language that can be used as a basic complex-number calculator.

• You may also be interested in installing the excellent, free, and widely used program Spyder, which gives a nicer environment for working with math in python.

• Sympy is a free interactive symbolic algebra programming package that can can be used to study many of our equations. You can experiment with it using Sympy live before installing and using your own version in Spyder.

• Virtual Math Museum visualizations of complex plane transformations

• A winding number visualization, for when we get to winding numbers...

Books Online

• Complex analysis.com is a traditional textbook rewritten with interactive plots and diagrams placed online. For example, they have some nice interactive plots for \(f(z)=1/z\).

• Complex analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka

• Complex Analysis Project

• Complex analysis by George Cain, 2001

• Free classic textbook The Theory Of Functions Of A Complex Variable by A. G. Sveshnikov, A. N. Tikhonov

Linkstream of math news and thoughts for the semester

• Joukowski Airfoil at Bell labs 2022-11-20
• Are Real Numbers Real? - PLATO AND THE NERD 2022-10-24
• AI Language Models Are Struggling to "Get" Math - IEEE Spectrum 2022-10-12
• Seven Sins of Numerical Linear Algebra - Nick Higham 2022-10-12
• New integer types I'd like to see 2022-09-30
• Oh, we got a good old fashioned fight between Gary Marcus and Yann Lecun because of a recent ZDNET article. 2022-09-28
• Posits, a New Kind of Number, Improves the Math of AI - IEEE Spectrum 2022-09-28
• Turns are better than radians or degrees 2022-09-26
• Philosophy of mathematics - a reading list - Logic Matters 2022-09-20
• Kripke obituary 2022-09-20

Textbook Typos

We've encountered a few typographical issues with our textbook. This is the 2003 2nd edition of "Introduction to Complex Analysis" by H. A. Priestley, ISBN 0198525621.

• If you are using an electronic version of the textbook, be aware that some versions lose some of the overlines indicating conjugation. This particularly effects Exercises 5.11(i) and 6.6(iii)
• Page 76, equation line 3 is missing a summation for n from 1 to \(\infty\).
• Page 124, in the first line in the main equation of the fundamental theorem of calculus proof, we should be integrating \(F'(z)\), not \(F(z)\).
• Page 263, Figure 21.3 caption: should be contour for \(\mathcal{L}^{-1}(1/\sqrt{p})\)