MTH 335 (sec 201) Syllabus Spring 2004
MTH 461 (sec 201) Syllabus Spring 2009
CRN 3390
Prerequisites: MTH 229, 230 , 231 and 460 with a grade of C or higher in each, fairly recently.
Learning Objectives: To learn the theory of complex variables and its application to computations.
Meeting time: M W 5:00 - 6:15 pm , Room 518 Smith Hall
Instructor: Dr. Alan Horwitz Office: Room 741 Smith Hall Phone: (304)696-3046 Email: horwitz@marshall.edu
Text: Complex Variables and Applications, Brown and Churchill , 8th edition, Brooks/Cole
Recommended: Lab Manual for running Mathematica 3.0 on Windows at Marshall University, Rubin
Recommended Calculator: TI-83 series or HP 48G , HP 48G+ , or HP 48GX or anything you want.
We won’t be using it much.
Grading : homework (and possibly labs) 22.72% (125points)
3 major exams (each worth 18%) 54.54% (300 points)
final( comprehensive ) exam 22.72% (125 points)
total : 550 points
Final exam date: Wednesday May 6, 2009 5 - 7 pm
Course Policies:
Attendance is required . You are responsible for reading the text, working the exercises, coming to office hours for help when you’re stuck, and being aware of the dates for homework assignments and major exams as they are announced. Late homework will be penalized. Graduate students will have slightly more challenging homework questions.
Exam dates will be announced at least one week in advance. Makeup exams will be given only if you have an acceptable written excuse with evidence and/or you have obtained my prior permission. On exam days, we may have a shortened lecture period followed by the exam. Graduate students will have slightly more challenging exam questions.
Makeups are likely to be more difficult than the original exam and must be taken within one calendar week of the original exam date. You can’t make up a makeup exam: if you miss your appointment for a makeup exam, then you’ll get a score of 0 on the exam.
If you anticipate being absent from an exam due to a prior commitment, let me know in advance so we can schedule a makeup. If you cannot take an exam due to sudden circumstances, you must call my office and leave a message or email me on or before the day of the exam! Makeups may be more difficult than the original exam and must be taken within one calendar week of the original exam date. You can't make up a makeup exam: if you miss your appointment for the makeup exam, then you
get a score of 0 on the exam.
In borderline cases, your final grade can be influenced by factors such as your record of attendance, whether or not
your exam scores have been improving during the semester, and your class participation. For example, if your course point total is at the very top of the C range or at the bottom of the B range , then a strong performance on the final exam can result in getting a course grade of B, while a weak performance can result in a grade of C.
Attendance Policy: This is NOT a distance learning course !
Regular attendance is expected ! Attendance will be checked daily with a sign-in sheet. If your grade is borderline, then good attendance can result in attaining a higher grade. Likewise, poor attendance can result in a lower grade.
Having more than 3 weeks worth of unexcused absences (i.e., 6 of 29 lectures ) will automatically result in a course grade of F! Being habitually late to class will count as an unexcused absence for each occurrence. Carrying on conversations with your neighbor could be counted as being absent. Walking out in the middle of lecture is rude and a distraction to the class ; each occurrence will count as an unexcused absence. If you must leave class early for a doctor’s appointment , etc., let me know at the beginning and I’ll usually be happy to give permission. Absences which can be excused include illness, emergencies, or official participation in another university activity.
Documentation from an outside source ( eg. coach, doctor, court clerk…) must be provided to ME directly. If you lack documentation, then I can choose whether or not to excuse your absence.
HEED THIS WARNING:
Previously excused absences without documentation can always, later, instantly change into the
unexcused type if you accumulate an excessive number ( eg. more than 2 weeks worth ) of absences of any kind ,
both documented and undocumented .
You are responsible for keeping track of the number of times you've been absent and whether or not the absence was excused. I tally this information up at the end of the semester, so don't count on me to give you a warning when you've reached the threshold of failing from being excessively absent.
MTH 461 (sec 201) Syllabus Spring 2009
CRN 3390
(continued)
Cell-Phone and Pager Policy: Shut off those damned things !
Unless you are a secret service agent, fireman, or paramedic on call, all electronic communication devices such as pagers and cell phones should be shut off during class. Violation of this policy can result in confiscation of your device and forced participation in a study of the deleterious health consequences of frequent cell phone use.
Sleeping in Class:
Habitual sleeping during lectures will be considered as an unexcused absence for each occurrence. If you are that tired,
go home and take a real nap! You might want to change your sleeping schedule, so that you can be awake for class.
The major exams will be roughly on the 4th, 8 th , and 12th weeks, plus or minus one week.
Their precise dates will be announced at least one week in advance and the topics will be specified.
We may not have the time to cover all the topics listed on the Topics sheet, and we won(t necessarily
cover the sections in order. In some chapters, we will focus on specific topics, rather than covering everything.
Some topics will be covered with material outside of the text. Come regularly and you(ll know where we are.
|Week |Date |Sections Covered |
|1 |8/25 - 8/29 | |
|2 |9/2 - 9/5 | |
| |( Labor day on 9/1 ) | |
|3 |9/8 - 9/12 | |
|4 |9/15 - 9/19 |Exam 1 |
|5 |9/22 - 9/26 | |
|6 |9/29 - 10/3 | |
|7 |10/6 - 10/10 | |
|8 |10/13 - 10/17 |Exam 2 |
|9 |10/20 - 10/24 | |
|10 |10/27 - 10/31 | |
| |(last day to drop) | |
|11 |11/3 - 11/7 | |
|12 |11/10 - 11/14 |Exam 3 |
|13 |11/17 - 11/21 | |
| |(Thanksgiving Break next week ) | |
|14 |12/1 - 12/5 | |
| |(WEEK OF THE DEAD begins on 12/3) | |
|15 |12/8 - 12/9 | |
| |(final exam on 12/10) | |
MTH 461 (sec 201) Syllabus Spring 2009
CRN 3390
(continued)
Topics
1. complex number as ordered pair, defining i , real and imaginar parts
sums and products of complex numbers
algebraic properties: complex numbers are a field
differences and quotients
complex numbers viewed as position vectors, modulus and Triangle Inequality
applications of complex conjugates
polar form of a complex number, principal value of argument
using Euler’s Formula to write a complex number in exponential form
products , quotients and powers of complex numbers in exponential form
finding roots of complex numbers, plotting them on a circle
topological concepts in the complex plane
2. complex valued functions
polynomial and rational functions, roots as multiple valued functions
domain and range
translation, rotation and reflection mappings
examples of images of curves and regions under polynomial and exponential mappings
finding limits of complex valued functions
how a complex valued function can fail to have a limit
theorems on finding infinite limits and limits at infinity
properties of continuous complex valued function
definition of derivative of complex valued function
how a complex valued function can fail to be differentiable
rules for differentiation
deriving the Cauchy Riemann equations and using them to determine whether or not a
complex valued function is differentiable at a given point
restating the Cauchy Riemann equations using polar coordinates
definition of analytic functions, entire functions, singular points
using Cauchy Riemann equations to determine if a function is analytic at a point or entire
a derivative of zero implies being constant
harmonic functions satisfy Laplace’s equation
component functions of an analytic function are harmonic
uniqueness property of analytic functions in a connected, open domain
the Reflection Principle Theorem
3. properties of exponential functions of a complex variable, including periodicity
solving an exponential equation for a complex valued unknown
multiple values of a logarithm function of a complex variable
finding the principle value of a logarithm
principle branch, branch cut and branch points for multiple-valued logarithmic functions
algebraic properties of logarithms
derivatives, principal value and principal branch of a complex exponent
raised to a complex power
derivatives of exponential functions with a complex number base
using Euler's Formula to define sine and cosine of a complex variable in terms of
exponential functions
identities for trigonometric functions of a complex variable
derivatives of hyperbolic functions of a complex variable
MTH 461 (sec 201) Syllabus Spring 2009
CRN 3390
(continued) Topics
4. familiar rules for derivatives and integrals of a complex valued function of a real variable
examples of simple arcs, simple closed curves, positive orientation
examples of contour integrals of complex functions of a complex variable
contour integral when limit of integration lies on branch cut by using improper integration
estimating an upper bound for a contour integral of a bounded complex valued function contour integrals of continuous functions with antiderivatives are path independent
how to compute a contour integral when a function has branch cuts along the path
positively oriented closed paths are counterclockwise
Cauchy-Goursat Theorem: closed-path contour integral of an analytic function is zero
closed-path contour integral of analytic function on simply connected domain is zero
how to compute contour integrals of analytic functions on multiply connected domains
Cauchy Integral Formula , applications to finding derivatives and other consequences
using Liouville’s Theorem for entire bounded functions to
prove Fundamental Theorem of Algebra
using Gauss’s Mean Value Theorem to help prove the Maximum Modulus Principle
5. definition of a convergent sequence of complex numbers
definition of convergence for an infinite series of complex numbers
Taylor’s Theorem and Laurent’s Theorem
circle of convergence and absolute convergence and uniform convergence of power series
integrating and differentiating power series
uniqueness of Taylor series and Laurent series representations
Cauchy Product of two power series
long division of power series
6. isolated singular point at a finite point and at infinity
finding the residue of a function at an isolated singular point , using it to evaluate integrals
using Cauchy’s Residue Theorem to evaluate a contour integral when there are finitely
many isolated singular points within the closed curve
residues at infinity
order of a pole, simple pole, removable singular point, essential singular point
calculating residues at poles of order m
representing a function with a zero of order m
rational functions with poles of order m
calculating residue of a rational function with a simple pole
limit of a function at a pole is infinite
Riemann’s Theorem on removable singularities
Casaroti-Weierstrass Theorem on values of a function in
a neighborhood of an essential singularity
7. definitions of different types of improper integrals, including Cauchy Principal Value
example of evaluating an improper integral of an even function by using residues
using residues to evaluate improper integrals associated with Fourier coefficients
using Jordan’s Lemma to find integrals associated with Fourier coefficients
using contours with indented paths to evaluate improper integrals when
the function has a singularity or branch point on the real axis
using Cauchy Residue Theorem when path of integration lies along a branch cut
meromorphic functions and winding numbers
the Argument Principle
using Rouche’s Theorem to estimate the number of zeros of a function within a region
8.
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