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.)
|10/26||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 by first writing z in polar form with an appropriate angle.
|12/11||FINAL EXAM, 2-5 PM|