Developing Timetable for MATH 3340, Fall 2017 - Sherman
Complex Variables with ApplicationsNOTE: 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
|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
|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
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
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
|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||--|
Thoughts on 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||--|
|10/5||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
|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
|10/19||Review of complex power functions. Terminology for contours. Classwork.||4.1||3.2/5abc,6,14b,15,17bc,25
--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)
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
|11/7||Reviewing. Derivatives and integrals of branches. More practice bounding functions. The Cauchy Integral Formula.||4.5||--|
Thoughts on 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)
|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)
|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)
(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