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.

*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

- definition of the complex 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*(with the power series for the^{iθ}=cosθ+i sinθ*e*and^{x}, sin x*cos x*as motivation) - described the geometric interpretation of complex multipcation: multiply moduli and add arguments

- note: conjugation is a
*F:*Have read §§1-10.**HW0:**(at jonathan.poritz@gmail.com) telling me:*Send me e-mail*- Your name.
- 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.)
- Your year/program/major at CSUP.
- What you intend to do after CSUP, in so far as you have an idea.
- Past math classes you've had.
- Other classes you're taking at the moment.
- The reason you are taking this course.
- Your favorite mathematical subject.
- Your favorite mathematical result/theorem/technique/example/problem.
- Anything else you think I should know (disabilities, employment
or other things that take a lot of time,
*etc.*). - [Optional:] The best book you have read recently.

*Content:*- powers of complex numbers:
*z*has modulus which is^{n}*|z|*and argument which is^{n}*n . arg(z)*— and this is in fact**well-defined**: it doesn't matter which choice of*arg(z)*is used, the complex*n*th power will be the same - complex roots:
*z*has modulus which is^{1/n}*|z|*and argument which is^{1/n}*arg(z)/n*— but this is more ambiguous: there are*n*such roots - the
*principle n*th root is the above root, when using the principle argument of*z*,*i.e.,*it is the complex number*|z|*^{1/n}e^{i Arg(z)/n} - the
are the*n*th roots of unity*n n*th roots of the complex number 1; they are of the form*e*where^{2πik/n}*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
*n*th roots of unity under multiplication are isomorphic to the group**Z**/*n***Z**under addition.] - the set of
*n*th 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 n*th 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|*centered at the origin, rotated so that one of the vertices is at angle coordinate^{1/n}*Arg(z)/n*.

*NOTE:*Friday is the last day to add classes

*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

- an
- 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)*

- a real graph would be

- a whole passle of definitions from §11, along with a few
examples:
*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

- theorem on complex limits (the "limit laws"), including limits
of sums, products, compositions,

*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*of a (real) function_{x}*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*and_{x}= v_{y}*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*(and likewise for_{xx}+ u_{yy}=0*v*).

*F:*Have read §§22-24

*Content:*- conditions for differentiability
- the Cauchy-Riemann equations in polar coordinates:
*r u*and_{r}= v_{&theta}*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

*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)

*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
*sin*^{2}x+cos^{2}x=1 *inverse trigonometric functions*, briefly

- some discussion of issues with the most recent homework: reminder
of the gradient, dot products,

*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.

*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,*∫*unless_{0}^{2π}e^{inθ}e^{-imθ}dθ=0*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,*∫*unless_{C}z^{n}dz=0*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.

- examples of computing integrals of complex-valued functions of a real
variable — in particular, for
*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

*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*

*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*

- applications of the
*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

*The plan for this week:**M:*Have read §54

*Content:*- close reading of the book's proof of the
*Maximum Modulus Principle*

- close reading of the book's proof of the
*W:*

*Content:*- working through applications of the
**xCIF**(the*Extended Cauchy Integral Formula*) to doing various kinds of complex integrals around closed contours

- working through applications of the
*F:*

*Content:*- working out some problems, such as, for example, these

**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.

*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:**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

*The plan for this week:**M:*Have read §§55-59. Hand**HW8**:*p. 188*: 6-8

*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

- working with Laurent series, including examples, finding them,
*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

- examples of computing more series, usually with some algebra and
a few series we already know, like the geometric series,

*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*(also**m***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

*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
*z*in terms of a function which is analytic and non-zero at_{0}*z*and a power of_{0}*(z-z*._{0})

- general form of an analytic function with a zero or pole of finite
order at some
*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