Developing Timetable for MATH 3340, Fall 2017  ShermanComplex 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.  1.23    
8/29  Discussion of previous classwork. Modulus and argument for products. Polar form using cosine and sine. The complex exponential e^{z} and polar form. Classwork.  1.34    
8/31  Formulas of Euler and De Moivre; using these to derive trig identities. Computing and graphing (primitive) roots of unity. Classwork.  1.45  1.1/6,16,20b,22,25 1.2/4(just 32i),7abe(derive Cartesian descriptions and graph),16 1.3/12 

9/5  Roots of complex numbers. e^{z} is not onetoone on C. The quadratic formula, understood for C. Terminology for planar sets. 9/5: Last day to add 9/6: Last day to drop 
1.56    
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. CauchyRiemann equations.  2.34  1.4/17ad,18ad 1.5/5df,6b,7b,11,16,19 1.6/28,12 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 CauchyRiemann equations. Using CR 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 CR 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  Inclass 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 78:30 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 CR 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 steadystate 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.  2.45    
10/3  READING DAY  
10/5  Some review. Derivation of the heat equation in 1D at least. How physical intuition for steadystate 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 steadystate temperature functions suggests properties of harmonic functions. The Fundamental Theorem of Algebra; factoring into irreducble polynomials over R and C. Properties of the function e^{z}.  3.2    
10/12  Visualizing some properties of e^{z}. 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 z^{5}+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    
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) 4.1/1,4,10 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. 

10/31  
11/2  
11/7  
11/8 78:30 PM Monroe 122 
Midterm 2.  
11/9  
11/14  
11/16  
11/21  
11/23  THANKSGIVING HOLIDAY  
11/28  
11/30  
12/5  
12/11  FINAL EXAM, 25 PM 