Complex Analysis 2 - Spring 2004
COURSE: MATH 5520
Call # 14015
TIME AND PLACE: 10:25-11:45 MW in Memorial Center - East Side, Room 134.
INSTRUCTOR: Dr. Robert Gardner
OFFICE: Room 308G of Gilbreath Hall
OFFICE HOURS: 1:00--2:00 MW
PHONE: 439-6977 (308G Gilbreath), Math Department Office 439-4349
E-MAIL: gardnerr@etsu.edu
WEBPAGE:
www.etsu.edu/math/gardner/gardner.htm (see my webpage for
a copy of this course syllabus and updates for the course). For information about Complex Analysis 1 (MATH 5510), see the online syllabus for that course.
TEXT: Complex Analysis, 2nd Edition, by John Conway.
PREREQUISITE: Complex Analysis 1 (MATH 5510), or permission of instructor.
GRADING: Homework will be assigned and collected regularly.
Later in the term, we wll attempt to solve some new resrach problems, and part of your grade may be based on "class participation" in these projects. Grades will be assigned based on a 10 point scale with "plus" and "minus" grades being assigned as appropriate (based on grade points assigned by the university, on a plus/minus 3 point system).
SCHEDULE: Our tentative outline is:
Chapter 4. Complex Integration: Riemann-Stieltjes integrals, power series, zeros of analytic functions,
Fundamental Theorem of Algebra, Maximum Modulus Theorem, winding number, Cauchy's Integral Formula, properties of path integrals, Open Mapping Theorem.
Chapter 5. Singularities: classification of singularities, Laurent series, residues, integrals, meromorphic functions, argument principle, Rouche's Theorem.
Chapter 6. Maximum Modulus Theorem versions of Max Mod Theorem, Schwarz's Lemma,
Hadamard's Three Circles Theorem (maybe), Pragmen-Lindelof Theorem (maybe).
Research Topics in Polynomials: location of zeroes in terms of coefficients, rate of growth results, Bernstein type inequalities.
IMPORTANT DATES:
Monday, January 19 = Martin Luther King, Jr. Day, no class.
Monday, March 1 to Friday March 5 = Spring Break, no class.
Friday, April 9 = Good Friday, no class.
Friday, April 23 = Last day of class.
HOMEWORK.The following homework is assigned:
| Assignment | Problems | Due Date | Credit | Cumulative Credit |
| HW1 | 4.1.6, 4.1.8, 4.1.11 | Friday January 23 | 3+3+3=9 | 9 |
| HW2 | 4.1.21, 4.1.22 | Friday January 30 | 3+3=6 | 15 |
| HW3 | 4.2.7a, 4.2.9a, 4.2.9c | Friday February 6 | 3+3+3=9 | 24 |
| HW4 | 4.2.5, 4.3.1, 4.3.3, 4.3.5, BONUS: 4.3.9, Bonus 1 | Monday February 16 | 3+3+3+3+(3)+(3)=12 | 36+(6) |
| - | Bonus 2 | Monday February 23 | (1) | 36+(7) |
| - | Bonus 3 | Friday February 27 | (2) | 36+(9) |
| HW4' | Problems 1, 2 | Wednesday March 10 | 3+2=5 | 41+(9) |
| HW5 | 4.4.2, 4.4.3 | Friday March 19 | 3+3=6 | 47+(9) |
| HW6 | 4.5.3, 4.5.6, 4.5.8 | Friday March 26 | 3+3+3=9 | 56+(9) |
| - | Bonus 4 | Friday April 2 | (3) | 56+(12) |
| HW7 | 4.6.1, 4.6.2, 4.6.3, 4.6.7, 4.6.8a, 4.6.10, 4.6.11 (do 4) | Wednesday April 7 | 3+3+3+3=12 | 68+(12) |
| HW8 | 4.7.1, 4.7.3, 4.7.4 (do 2), BONUS: 4.7.7 | Wednesday April 14 | 3+3+(3) | 71+(15) |
| HW9 | 5.1.1 b, c, h; 5.1.4a, 5.1.6, 5.1.11, BONUS: 5.1.4b, 5.1.4c | Wednesday April 21 | - | - |
| HW10 | 5.2.1a, 5.2.2d, 5.2.3, BONUS: 5.2.4 | Wednesday April 28 | - | - |
Bonus points are in parentheses.
Problems
Problem 1. Give a direct proof of Cao and Gardner's Corollary 2.1 by going though the proof of Theorem 2.1 with the appropriate choice of the betas, t, k, and l.
Problem 2. State a corollary of Cao and Gardner's Theorem 2.1 with the hypotheses of monotone INCREASING for the coefficients (similar to Corollary 2.1).
Bonus Problems
Bonus 1. Show that a real polynomial can be factored into a product of linear terms and irreducible quadratic terms.
Bonus 2. In the Joyal, Labelle, Rahman paper, show that (an-a0+|a0|)/|an| is greater than or equal to 1.
Bonus 3. Show that if |z| is less than or equal to 1, then |z+K+1| is less than or equal to (K an-a0+|a0|)/|an|.
Bonus 4. Show that the limit as p approaches 0 of ||P||p is as claimed.
Research Problems
Research Problem 1. Take the Aziz and Zargar's Theorem 3 and drop the hypothesis that the coefficients are positive and find a (slight) generalization.
Research Problem 2. Take the hypotheses of Cao and Gardner's Corollary 2.1 and modify so that coefficients are monotone INCREASING and add the hypothesis that... either (1) an-1 is less than or equal to Ka n, or (2) an-2 is less than or equal to Ka n and
an-3 is less than or equal to La n-1. Get a result.
Research Problem 2'. Take the hypotheses of "Mokak2" and find a disc NOT centered at 0 that contains all the zeroes of P (HINT: Mimic AZ2).
Research Problem 3. Add the "t condition" to the above problems and get a further generalization.
Research Problem 4. Modify the hypotheses of Research Problem 2' to include a reversal of the inequalities (as done in Theorem 4 of Aziz and Zargar).
Research Problem 5. Combine the hypotheses of Gardner and Govil with those of Gardner and Weems to get a single unifying result.
Other Important Dates
Monday March 8: NO CLASS THIS DAY. Instead, we will meet on Friday March 12 at the usual time.
My Publications Accessible through the Sherrod Library (online)
PDF files of several of my complex analysis publications can be accessed through the Sherrod Library reserve area. Go to the
reserve listing for me and this will lead you to all of my reserve material, including my complex papers. On reserve are:
- "Inequalities Concerning the Lp Norm of a
Polynomial and its Derivative," with N. K. Govil, Journal of
Mathematical Analysis and Applications, 179(1) (1993) 208-213
(MR# 94h:41025).
- "On the Location of the Zeros of a Polynomial," with N. K. Govil,
Journal of Approximation Theory, 78 (1994)
286-292 (MR# 95f:30006).
- "An Lp Inequality for a Polynomial and its
Derivative," with N. K. Govil, Journal of Mathematical Analysis and
Applications, 193 (1995) 490-496 (MR# 96e:26017), and
194(3) (1995) 720-726 (MR# 96m:26018).
- "Some Generalizations of the Enestrom-Kakeya
Theorem," with N. K. Govil, Acta Mathematica Hungarica,
74(1-2) (1997) 125-134 (MR# 97k:30009).
- "Functions of Exponential Type Not Vanishing in
a Half-Plane," with N. K. Govil, Analysis, 17 (1997) 395-402 (MR# 98m:30044).
- "A Bernstein Type Lp
Inequality for a Certain Class of Polynomials," with A. Weems,
Journal of Mathematical Analysis and its Applications, 219
(1998) 472-478 (MR# 99g:41012).
- "Restrictions on the Zeros of a Polynomial as a Consequence of Conditions on the Coefficients of Even and Odd Powers of the Variable," with Jiansheng Cao, Journal of Computational and Applied Mathematics, 155(1) (2003) 153-162.
Other Information
Here is some additional information concerning the course:
- The reference for the Joyal Labelle Rahman result is: On the Location of Zeros of Polynomials, A. Joyal, G. Labelle, and Q.I. Rahman, Canadian Mathematics Bulletin, 10(1) (1967), 53-63.
- First paper handed out is: Some Extensions of the Enestrom-Kakeya Theorm, A. Aziz and B. A. Zargar, Glasnik Matematicki, 31(51) (1996), 239-244.
- Second paper handed out: Restrictions on the Zeros of a Polynomial as a Consequence of Conditions on the Coefficients of Even Powers and Odd powers of the Variable, J. Cao and R. Gardner, Journal of Computational and Applied Mathematics, 155 (2003), 153-162.
- The third and fourth papers handed out deal with Bernstein-type inequalities and were:
Inequalities Concerning the Lp Norm of a Polynomial and Its Derivative, R. Gardner and N. Govil, Journal of Mathematical Analysis and Applications, 179(1) (1993), 208-213; and
A Bernstein Type Lp Inequality for a Certain Class of Polynomials, R. Gardner and A. Weems, Journal of Mathematical Analysis and Applications, 219, (1998), 472-478.
- A webpage of John Conway (a LaTeX file):
Corrections for my book "Functions of One Complex Variable." (Second edition, fourth printing). The seventh printing, which exists, incorporates these corrections. (He has not compiled a set of corrections for later printings.)
- A webpage of John Conway (a LaTeX file):
Additions and Changes for "Functions of One Complex Variable."
Some of these are comments on the exercises and some are references to the literature.
- A webpage of John Conway (a LaTeX file):
Additions and Changes for "Functions of One Complex Variable,II."
- A nice reference on our research area: "Topics in Polynomials : Extremal Problems, Inequalities, Zeros" by G.V.
Milovanovi´c, D.S. Mitrinovi´c, and Th.M. Rassias. River Edge, NJ: World Scientific, 1994. The Sherrod Library has a copy of this book (call number: QA241.M535 1994).
- A nice reference on our research area: "Analytic Theory of Polynomials" by Q.I. Rahman and G. Schmeisser,
London Mathematical Society Monographs, New Series, Number 26, New York: Oxford University Press, 2002.
The Sherrod Library has a copy of this book (call number: QA161.P59 R34 2002), though I probably have it checked out.
- Check out the webpage for the SIAM meeting at ETSU on April 2-3, 2004.
Return to
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