Colorado State University, Pueblo; Spring 2010
Math 425 — Complex Variables,
Homework Assignments & Course Schedule

Here is a link to the current week, below.

Here is a link back to the course syllabus/policy page.

In the following all sections and page numbers refer to the required course textbook, Complex Variables and Applications, 8^th Edition, by James Ward Brown and Ruel V. Churchill.

This schedule is subject to change, but should be accurate at any moment for at least a week into the future.

For each day, please read the section(s) named in the plan, before that day — we will have discussion in class on those sections for which you will have to have read the book.

Week of January 11:

• The plan for this week:
  • M: Mostly bureaucracy and introductions.
     Content:
    • definition of the complex numbers C
    • operations on C:
      • addition
      • multiplication
      • division
      • conjugation z
      • modulus (absolute value) |z|
      • Re z and Im z
    • geometric interpretation of complex addition and conjugation
    • caution: z<w makes no sense — inequalities only work for real numbers
  • W: Have read §§1-6
     Content:
    • note: conjugation is a ring homomorphism, i.e., z . w= z . w
    • found a condition for a complex number z actually to be real: z=z
    • found the "complex-only formulæ" Re z=(z+z)/2 and Im z=(z-z)/2i
    • While inequalities of complex numbers are meaningless, they are fine for moduli of complex numbers, where there is the usual triangle inequality: |z+w|≤|z|+|w|
    • If a is a fixed complex number and r a fixed real number, then |z-a|=r is the equation of a circle with center a and radius r (in the variable z); likewise, |z-a|<r describes the interior of that circle.
    • defined the polar or exponential form of a complex number, z=r(cosθ+i sinθ) where r=|z| and θ=arg(z) is the argument of z
    • defined the principal value Arg(z) of the argument of z: it is the one with value always in the interval (-π,π].
    • mentioned Euler's formula e^iθ=cosθ+i sinθ (with the power series for the e^x, sin x and cos x as motivation)
    • described the geometric interpretation of complex multipcation: multiply moduli and add arguments
  • F: Have read §§1-10. HW0: Send me e-mail (at jonathan.poritz@gmail.com) telling me:
    1. Your name.
    2. Your e-mail address. (Please give me one that you actually check fairly frequently, since I may use it to contact you during the term.)
    3. Your year/program/major at CSUP.
    4. What you intend to do after CSUP, in so far as you have an idea.
    5. Past math classes you've had.
    6. Other classes you're taking at the moment.
    7. The reason you are taking this course.
    8. Your favorite mathematical subject.
    9. Your favorite mathematical result/theorem/technique/example/problem.
    10. Anything else you think I should know (disabilities, employment or other things that take a lot of time, etc.).
    11. [Optional:] The best book you have read recently.
     Content:
    • powers of complex numbers: zⁿ has modulus which is |z|ⁿ and argument which is n . arg(z) — and this is in fact well-defined: it doesn't matter which choice of arg(z) is used, the complex nth power will be the same
    • complex roots: z^1/n has modulus which is |z|^1/n and argument which is arg(z)/n — but this is more ambiguous: there are n such roots
    • the principle nth root is the above root, when using the principle argument of z, i.e., it is the complex number |z|^1/ne^{i Arg(z)/n}
    • the nth roots of unity are the n nth roots of the complex number 1; they are of the form e^2πik/n where k=0,1,...,n-1; they form the vertices of a regular n-gon on the unit circle of the complex plane, one of whose vertices is the real number 1.
    • [Aside for those who know these terms: the nth roots of unity under multiplication are isomorphic to the group Z/nZ under addition.]
    • the set of nth roots of any complex number z is the set formed by picking one such root (say the principle root) and mulitplying it by all n nth roots of unity. Therefore, geometrically, it is the set of vertices of a regular n-gon on the circle in the complex plane of radius |z|^1/n centered at the origin, rotated so that one of the vertices is at angle coordinate Arg(z)/n.
• NOTE: Friday is the last day to add classes

Week of January 18:

• The plan for this week:
  • M: Have read §§11-14.  Content:
    • a whole passle of definitions from §11, along with a few examples:
      • an open set of complex numbers
      • a close set of complex numbers
      • the interior of a set of complex numbers
      • the exterior of a set of complex numbers
      • the boundary of a set of complex numbers
      • a connected set of complex numbers (really, "piecewise linear path-connected set" would be a better term; it amounts to the same thing as other definitions of the word "connected", however, in the context of open sets of complex numbers)
      • a domain — in the sense of a nice (open and connected) set of complex numbers, not the same thing (at least not automatically) as the "domain of definition" of a complex function
    • complex-valued functions of a complex variable: some basic notation and terminology (and a few examples)
      • a real graph would be hard to visualize, as it would be 4-dimensional
      • writing a complex function f(z) as real and complex parts of the output corresponding to real and complex parts of input: f(x+iy)=f(z)=u(x,y)+i v(x,y)
      • some three-dimensional graphs (well, graphs which are 2-dimensional surfaces inside 3-dimensional space) built out of a complex function f(z):
        • z=u(x,y)
        • z=v(x,y)
        • z=|f(x+i y)|
        • z=Arg(x+i y)
  • W: Keep (re)reading §§12-15. Hand in HW1:
    • p.5: 2, 11
    • p.8: 2 (use eqns (6) and (9) in the book)
    • p.12: 6
    • p.14: look at exercises 11 and 12. Now prove that the roots of real polynomials occur in conjugate pairs, i.e,. z is a root of a polynomial with real coefficients if and only if z is a root.
    • p.22: 2, 4
    • p.29: 3, 6
     Content:
    • student volunteers (if necessary, chosen by your instructor) presenting some of their solutions to HW1 problems
    • fiddling a bit with on-line graphics for complex functions, for example:
      • a graphing tool called "ComplexTool"
      • or here 's a simpler one
      • here is a whole page with graphics mixed with explanatory text
      • or just search yourself, such as looking for applet, graph, complex, and function on a search engine
    • definition of a limit of a complex-valued function
  • F: Have read §§16-19.
     Content:
    • theorem on complex limits (the "limit laws"), including limits of sums, products, compositions, etc.
    • the Riemann sphere, its identification (excluding the north pole) with the complex plane by stereographic projection
    • neighborhoods of infinity in the complex plane, and as small neighborhoods of the north pole on the Riemann sphere
    • a complex limit as z goes to infinity, or a limit as z goes to a finite value (in the complex plane) equalling infinity, or both — definition, and picture in terms of the Riemann sphere.
    • complex functions thought of in terms of their actions on the Riemann sphere: rotating, fixing one point of the sphere and pulling the rest along in one direction, etc.
    • the definition of the complex derivative

Week of January 25:

• The plan for this week:
  • M: Have read §§19 & 20
     Content:
    • more on the definition of the complex derivative
    • examples of complex-differentiable and non-differentiable functions
    • FACT: a complex-differentiable funtion defined on a domain and which takes on only real values must be constant (and sketch of proof)
    • rules for (complex) differentiation (just like for real differntiation): sum, product, quotient, chain, etc.
  • W: Have read §21. Hand HW2:
    • p.33: 1, 5
    • p.37: 3
    • p.44: 3, 7, 8
    • p.55: 1, 5, 11
     Content:
    • student volunteers (if necessary, chosen by your instructor) presenting some of their solutions to HW2 problems
    • reminder of the definition of the partial derivative u_x of a (real) function u(x,y) of two (real) variables
    • continuing with complex differentiability: connection with the Cauchy-Riemann Equations, which tell us that the real and imaginary parts, u and v of a complex differentiable function must satisfy u_x= v_y and u_y= -v_x.
    • brief foreshadowing of what is to come: the CR eqns imply that the real and imaginary parts of a complex differentiable function are actually harmonic, i.e., they satisfy the Laplace Equation u_xx+ u_yy=0 (and likewise for v).
  • F: Have read §§22-24
     Content:
    • conditions for differentiability
    • the Cauchy-Riemann equations in polar coordinates: r u_r= v_&theta and u_&theta= -r v_r.
    • analytic (sommetimes called holomorphic) functions: definitions and examples
    • definition of an entire function
• NOTE: Monday is the last day to drop classes without a grade being recorded

Week of February 1:

• The plan for this week:
  • M: Have read §§ 24-27
     Content:
    • an analytic function with vanishing derivative throughout a domain is constant there
    • if both a function and its conjugate are analytic then it (they) must be (both) constant
    • if an analytic function has constant modulus, then it must actually be constant
    • Laplace's equation, harmonic functions, harmonic conjugates, definitions, examples, and techniques
  • W: Have read §26 Hand HW3:
    • p.62: 3, 8, 9
    • p.71: 1, 4, 10
    • p.77: 1, 6
    Content:
    • harmonic conjugates: definitions, finding them, examples
  • F: Have read §§27 & 28
     Content:
    • unique analytic continuation of analytic functions
    • The Reflection Principle (start)

Week of February 8:

• The plan for this week:
  • M: Have read §§28-31
     Content:
    • The Reflection Principle (end)
    • your friend the exponential function
    • your acquaintance the logarithm
  • W: Have read §§30-33. Hand HW4:
    • p.81: 1, 2, 7, 9
    • p.87: 1, 4
    • p.92: 9
     Content:
    • more on the logarith: the principal value of log z
    • other branches of the logarithm, their branch cuts
    • algebraic identities with the logarithm: the usual ones we are used to from the real case, but sometimes off by 2πni
    • calculus properties of the logarithm: its derivative (away from the branch cut) is indeed 1/z
    • complex powers — properties, power rule for differentiation
  • F: Have read §§33, 34 & 36
     Content:
    • some discussion of issues with the most recent homework: reminder of the gradient, dot products, etc.
    • definition of complex trigonometric functions, some elementary properties, such as the Pythagorean identity sin²x+cos²x=1
    • inverse trigonometric functions, briefly

Week of February 15:

• The plan for this week:
  • M: Have read §§37-39
     Content:
    • curves in the complex plane and their tangent vectors in complex notation
    • definite integrals of complex functions of a real variable
    • complex contours
  • W: Have read §40. Hand HW5:
    • p. 97: 1, 3, 10
    • p. 100: 1, 2, 4
    • p. 104: 2, 8
     Content:
    • an arc; a simple or simple closed curve (or Jordan curve)
    • the Jordan Curve Theorem: a simple closed curve divides the plane into two distinct parts, one bounded (called the interior) and one unbounded (called the exterior).
    • a positively or counterclockwise oriented simple closed curve
    • a differentiable and smooth arc
    • a contour is a piecewise smooth arc; we often work with simple closed contours
    • contour integrals — only the definition
    • review for Midterm I; here is a review sheet
  • F: Midterm I today in class.

Week of February 22:

• The plan for this week:
  • M: Have read §§37-41
     Content:
    • examples of computing integrals of complex-valued functions of a real variable — in particular, for n and m integers, ∫₀^2πe^inθe^-imθdθ=0 unless n=m, in which case the integral has value 2π
    • examples of complex contour integrals — in particular, for n an integer and along the contour C which goes around the unit circle in the complex plane once counterclockwise, ∫_Czⁿdz=0 unless n=-1, in which case it has value 2πi.
    • consequence: contour integrals of polynomials around the unit circle are always zero.
    • another example: integrating polynomials around other circles (still get zero).
    • intuition: analytic functions should have nice power series, power series are like big polynomials, and closed curves are a lot like circles, so we hope that it will turn out that the contour integral of any analytic function around any closed contour will give zero.
    • another example: integrating the function f(z)=z around the unit square in the corner of first quadrant; yields zero.
  • W: Going over the midterm
  • F: Have read §§41&42
     Content:
    • fact: contour integrals are independent of (orientation-preserving) reparametrizations of the contour; examples
    • examples of integrals along contours which are not closed, some which depend upon the particular contour, some which depend only upon its endpoints

Week of March 1:

• The plan for this week:
  • M: Have read §§43&44. Hand in one or two well-written midterm solutions per student for inclusion in a full solution set.
     Content:
    • bounds on contour integrals: a theorem and examples
    • antiderivatives and contour integrals
  • W: Have read §§45&46
     Content:
    • proof of the theorem on the use of antidervatives in contour integrals
    • The Cauchy-Goursat Theorem
  • F: Have read §§47—49. Hand HW6:
    • p. 135: 1, 2, 5, 6
    • p. 149: 3
     Content:
    • a proof of a special case of Cauchy-Goursat

Week of March 8:

• The plan for this week:
  • M: Have read §§50&51
     Content:
    • Cauchy-Goursat in simply and multiply connected domains
    • the Cauchy Integral Formula
    • an extension of the Formula
  • W: Have read §§52&53
     Content:
    • applications of the Formula
    • Liouville's Theorem and the Fundamental Theorem of Algebra
  • F: Have read §50-52. Hand HW7:
    • p. 160: 1, 4, 7
    • p. 170: 1, 3, 5, 6
     Content:
    • the intuition behind, and applications and examples of, the (extended) Cauchy Integral Formula
• NOTE: Friday is the last day to withdraw (with a W) from classes

Week of March 15:

• The plan for this week:
  • M: Have read §54
     Content:
    • close reading of the book's proof of the Maximum Modulus Principle
  • W:
     Content:
    • working through applications of the xCIF (the Extended Cauchy Integral Formula) to doing various kinds of complex integrals around closed contours
  • F:
     Content:
    • working out some problems, such as, for example, these

Week of March 22:

• Spring Break! No classes, of course. Please be working on the write-ups of the five extended problems which were assigned last Wednesday. Don't hesitate to contact me during the break (e-mail is best) for clarification of the problem, help on its solution, or advice on exposition.

Week of March 29:

• The plan for this week:
  • M: Hand in as early in the day as you can (definitely at least 1/2 hour before class — since they will be copied and handed out to your classmates as Midterm II review materials!)
     Content:
    • review for Midterm II; here is a review sheet; here is one of the proofs (or, actually, main ideas of a proof) with which you should be comfortable
  • W: Midterm II today in class.
  • F:
     Content:
    • (quick) post-Midterm discussion
    • definition of convergence of a sequence of complex numbers, hence also the words convergent and divergent
    • equivalence of the convergence of a sequence of complex numbers with the convergence of the sequences of real and imaginary parts
    • definition of convergence of a series of complex numbers, convergent and divergent again
    • equivalence of the convergence of a series of complex numbers with the convergence of the series of real and imaginary parts
    • proposition: if a series converges then the sequence of individual terms must converge to zero, for complex numbers just as the same was true for real numbers
    • definition of absolute convergence of a series of complex numbers
    • proposition: absolutely convergent sequences are (plain-old) convergent

Week of April 5:

• The plan for this week:
  • M: Have read §§55-59. Hand HW8:
    • p. 188: 6-8
      You may hand in re-worked problems from the last midterm, if you so choose, for extra credit; I recommend you do so if your score was less than 100.
     Content:
    • the statement of Taylor's Theorem
    • the big idea of the proof of Taylor's Theorem
    • examples of Taylor series
    • the statement of Laurent's Theorem
  • W: Have read §§60-65.
     Content:
    • working with Laurent series, including examples, finding them, etc.
    • continuity, differentiation and integration of power series
  • F: Have read §§66, 67
     Content:
    • examples of computing more series, usually with some algebra and a few series we already know, like the geometric series, cos z, sin z, and e^z
    • uniqueness of series representations
    • (starting) multiplication and division of power series

Week of April 12:

• The plan for this week:
  • M:Read §§67—69 Hand HW9:
    • p. 195: 3, 5, 7
    • p. 205: 1, 3, 4, 6
     Content:
    • (more) multiplication and division of power series
    • definition of isolated singular point, residue
  • W:Have read §68—70
     Content:
    • isolaed singular point
    • principal part
    • removable singularity
    • pole of order m (also simple pole)
    • essential singularity
    • residue
    • Cauchy's Residue Theorem
  • F: CLASS CANCELED Please use this extra time:
    • to read carefully the sections of chapter 6 we are covering; and
    • to start the last homework set, due the last day of class — it is a large set, and I will be unable to give you feedback on it unless you get it to me on time

Week of April 19:

• The plan for this week:
  • M:Have read all of Chapter 6.
     Content:
    • general form of an analytic function with a zero or pole of finite order at some z₀ in terms of a function which is analytic and non-zero at z₀ and a power of (z-z₀).
  • W: Read §78, 79
     Content:
    • behavior near a pole: the limit of the function is ∞
    • behavior near essential singularities: the Casorati-Weierstrass Theorem
  • F: Hand HW10:
    • p. 219: 1, 3
    • p. 225: 1, 5
    • p. 239: 1, 2, 5
    • p. 243: 1, 4
    • p. 248: 1, 3, 5
    • p. 267: 3, 4, 8
     Content:
    • using the Residue Theorem
    • a method to compute improper integrals: idea and examples
    • final discussion and review

• Exam week, no classes.
• Hand in any outstanding homework assignments, by Monday at the latest.
• here is a review sheet for the final
• Thursday, April 29th: from 8:00-10:20 we will have a comprehensive in-class final, in our usual classroom