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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