## Complex Variables with Applications

NOTE: The "Main Topics" entries are not exhaustive, and all entries referring to future dates are speculative.
Date Main topics Book sections Homework due (to my Kerchof mailbox by 11AM the following day)
8/22 Hello. Five great things about complex analysis. Course syllabus. Basic terminology and arithmetic of complex numbers. Conjugates. 1.1 --
8/24 Graphing with complex numbers. Modulus (as length/distance). Argument and its principal value. Classwork. 1.2-3 --
8/29 Discussion of previous classwork. Modulus and argument for products. Polar form using cosine and sine. The complex exponential ez and polar form. Classwork. 1.3-4 --
8/31 Formulas of Euler and De Moivre; using these to derive trig identities. Computing and graphing (primitive) roots of unity. Classwork. 1.4-5 1.1/6,16,20b,22,25
1.2/4(just 3-2i),7abe(derive Cartesian descriptions and graph),16
1.3/12
9/5 Roots of complex numbers. ez is not one-to-one on C. The quadratic formula, understood for C. Terminology for planar sets.

9/5: Last day to add
9/6: Last day to drop
1.5-6 --
9/7 Reviewing roots and terminology for planar sets. A version of [zero derivative implies constant] for real functions on planar sets. Visualizing complex functions and using u(x,y)+iv(x,y) form. Classwork. 1.6,2.1 1.2/8
1.3/5c,13a(give ctrex)
1.4/2,4,7,8,12b,20a,23a
1.5/4a
Give an algebraic proof of the triangle inequality for complex numbers.
9/12 Limits of sequences and functions. Continuity. Variations involving infinity. Classwork. 2.2 --
9/14 Discussion of previous classwork: where Arg is continuous, geometric meaning of certain functions. Electric fields. Complex differentiability: terminology and examples. Cauchy-Riemann equations. 2.3-4 1.4/17ad,18ad
1.5/5df,6b,7b,11,16,19
1.6/2-8,12
2.1/1ab
Use the formula for the sum of a geometric series to compute the sum of all the nth roots of unity.
9/19 Some homework solutions. Review of differentiability and the Cauchy-Riemann equations. Using C-R to determine where a function is differentiable. Classwork. 2.4 --
9/21 How to find the image of a set under a map, with examples. Reviewing the connection between C-R and differentiability, with an example. Zero derivative on a domain implies constant. Visualizing the linear approximation to a complex differentiable function. 2.4 2.1/3d,4,5,6(include a drawing for c),10,13bc
2.2/7cdef,13,25
2.3/4a,10
9/26 In-class essay: Explain the proof of Theorem 5, starting on p. 74 of the text. General review. Definition of a harmonic function of two variables. 2.5 --
9/27
7-8:30 PM
Monroe 122
Midterm 1
Thoughts on Midterm 1
Midterm 1
Solution to Midterm 1
The median was 32.5 out of 48. Grade breaks are as follows: A/B 37, B/C 27.5, C/D 18.
9/28 Review of C-R and differentiability. Analytic functions depend on z and not on real and imaginary parts "separately". Some conditions that imply an analytic function is constant, or that two analytic functions differ by a constant. A harmonic function is a steady-state temperature function, and (on a disk) the real part of an analytic function. Properties of harmonic conjugates and how to find them. Level curves of a harmonic function and its conjugate are orthogonal wherever f' is nonzero. 2.4-5 --
10/3 READING DAY
10/5 Some review. Derivation of the heat equation in 1D at least. How physical intuition for steady-state temperature functions can suggest behavior of harmonic functions. Basic terminology and manipulations for polynomials and rational functions. The Fundamental Theorem of Algebra. Classwork. 2.6,3.1 2.4/1c,2,5,11,14
2.5/1,3bc,6,8(give pf/ctrex),11
10/10 Again, how/why physical intuition for steady-state temperature functions suggests properties of harmonic functions. The Fundamental Theorem of Algebra; factoring into irreducble polynomials over R and C. Properties of the function ez. 3.2 --
10/12 Visualizing some properties of ez. Defining log z and its principal branch Log z, which is analytic on the slit plane with derivative 1/z. An example of a harmonic function on a domain with no harmonic conjugate. Branches of multivalued functions (like log z and arg z). Classwork. 3.3 This homework
10/17 Being a harmonic conjugate is not a reversible relation. The multiplicity of the zero of a sum of polynomials. Recap and development of last lecture. Complex powers. Starting classwork.

Last day to withdraw
3.5 --
10/19 Review of complex power functions. Terminology for contours. Classwork. 4.1 3.2/5abc,6,14b,15,17bc,25
3.3/1,5
--Factor z5+32 into irreducibles over C and over R (the second answer should not involve any complex numbers).
--Explaining things in your own words, give an example of a harmonic function on a domain that does not have a harmonic conjugate. (You are welcome to use the example from class.)
10/24 Complex powers live on (possibly degenerate) spirals. Review of contours. Contour integrals of complex functions, done by parameterizing and reducing to real integrals. Archimedes. ML-estimates. Independence of Path. 4.2-3 --
10/26 Donut Day. Review of contour integrals. Some integrals computed using antiderivatives, being careful to avoid branch cuts. "Theorem 7": [having an antiderivative], [integrating to zero over loops], [having path independendent integrals] are equivalent. Simply connected domains. Cauchy Integral Theorem. 4.3-4 3.5/1ad,4,5(not the book's example),8(from def of sin, not Ex. 3),19(use the quadratic formula)
4.1/1,4,10
Do #2,3 on the October 19 classwork. For #2(ii), give a short proof.
10/31 Quiz on Cauchy's Integral Theorem. Review of Theorem 7, DIT, and CIT, plus two corollaries. How these theorems appeared in Calculus III. Examples computed via one of CIT and Theorem 7 but not the other. Winding numbers. Computing integrals using winding numbers and partial fractions. 4.4 --
11/2 Trick or treat. The vector field versions of our recent theorems. Summary of winding numbers. Integrating powers of (z-z0) using winding numbers and theorems. Computing integrals when there are multiple singularities: can do them separately and sum. Bounding the modulus of a complex function on a curve: basic techniques and practice. 4.4 4.2/3,5,7,9,14
4.3/1aceg,4,5
4.4/3,9,10cde,13
11/7 Reviewing. Derivatives and integrals of branches. More practice bounding functions. The Cauchy Integral Formula. 4.5 --
11/8
7-8:30 PM
Monroe 122
Midterm 2
Thoughts on Midterm 2
Midterm 2
Solution to Midterm 2
The median was 34.5 out of 53. Grade breaks are as follows: A/B 40, B/C 28, C/D 21.
11/9 Guest lecturers with candy. Musical interlude. The Generalized Cauchy Integral Formula: proof, examples, and consequences. Analytic functions and harmonic functions automatically have derivatives of all orders. If you have an antiderivative, you have a derivative. Classwork. 4.5 --
11/14 Starting with a function on a simple loop, defining a new function off it that is automatically C: how this relates to CIF and GCIF. An example done two ways. GCIF+: the version with winding numbers. The relationship between having a derivative and having an antiderivative. Cauchy estimates. Liouville's theorem. Mean Value Property. 4.5-6 --
11/16 Review of GCIF+ and its consequences, including the MVP for harmonic functions. The Maximum Modulus Principle: two versions, three remarks. Basic concepts for sequences and series. Classwork. 4.6,5.1 --
11/21 What we did last time. Power series have a radius of convergence. The Taylor series of an analytic function equals it on the largest disk where the function is analytic. Some computations and manipulations with series. A real series might diverge because of problems in the complex plane. 5.2-3 --#3 from the Nov 9 classwork handout
--4.5/2,3,5,7,9(now optional: and give examples of functions f that make each of the inequalities an equality)
--4.6/7,8,10,14,15
11/23 THANKSGIVING HOLIDAY
11/28 Functions analytic on disks are the same thing as Taylor series; derivatives, coefficients, and integrals all determine each other. Functions analytic on annuli are the same thing as Laurent series; the coefficients are integrals. Examples of Laurent series and strategies for finding them. Residues, and a first pass at the Residue Theorem. 5.5,6.1 --
11/30 Review of Laurent series. Order of a(n isolated) zero. Three flavors of isolated singularities. Residue Theorem (p. 312 of the text). A few ways to compute residues. Examples. 5.6,6.1 5.1/1b (21 is bonus -- remit separately)
5.2/5cef,11b,13
5.3/4,5,6,7
5.5/1,6
12/5 Residue Theorem, without and with winding numbers. Residues: patterns, computation techniques, examples. Demonstration: computing a hard real integral using residues. How complex functions can be graphed using colors: Velleman, Wegert/Semmler. Movie: Moebius Transformations Revealed. 5.6/1abcdeg,2,5a(give ctrex)
6.1/1abcfg,3abce,7
(may be remitted to my box by noon Wed 12/6)
12/11 FINAL EXAM, 2-5 PM
Thoughts on the final
Office hours after Tues 12/5: Wed 3-4, Thurs 11-12:30, Fri 3-4.
Final exam scores

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