Complex variables I (G63.2450/V63.0393)
Fall 2006
Professor Stephen Childress
(childress@cims.nyu.edu,
X3135,
717 WWH)
Tuesday, 5:10-7:00 pm in WWH 1302.
Office Hours: Tuesdays 4-5pm, Thursdays 10-11am,
or
by appointment.
This course will be taught as a first course
in
Complex variables using a well-established textbook.
Other useful texts are listed below.
The basic theory will be treated and there will be
discussion
of applications.
Homework will be regularly assigned, graded, and returned with answers.
The grade in the course will be based upon homework and a final exam.
Text: Complex
Variables
and Applications, by Brown and Churchill (7th additiion)--Available
at
the NYU bookstore and on reserve in the Courant Library.
Other reference texts: Complex Analysis by Ahlfors. A more
absatract
take on the subject, the text used in the one-semester graduate course.
Complex
Variables by Ablowitz and Fokas. A newer text, accessible and
complete.
Introduction
to Complex Analysis by Nehari. A good basic text.
Conformal
Mapping by Nehari. An advanced treatment, primarily for the second
semester.
All of the above available on reserve in the
Courant
Library.
Tentative Syllabus
- Lecture 1 (Sept. 5): Basic properties of complex
numbers. Ch. 1 of text, pp. 1-29. Download
Problems not to be handed in.
- Lecture 2: (Sept. 12): Functions of a complex
variable. Limits and continuity, derivatives. pp. 29-59 of text.
- Lecture 3: (Sept. 19):
Derivatives continued, Cauchey-Riemann equations, analyticity, pp.
60-78of text. (Note: We will delay discussion of
sections 26 and 27 of Chapter 2 until later.)
- Lecture 4: (Sept. 26):
Properties of some elementary functions. Chapter 3 of text.
- Lecture 5: (Oct. 3): Integrals
I: Preparation for the Cauchy-Goursat theorem. pp. 111-141 of
text.
- Lecture 6: (Oct. 10): Integrals
continued. Cauchy-Goursat theorem with applications.
142-156. Download example of how to break up an arbitrary closed
contour into simple closed contours. Download
example of non-smooth contour.
Download an integral considered in class but not
finished. Download Homework 3 answers.
- Lecture 7: (Oct. 17): Cauchy
integral formula and applications. pp. 156-173. Download hint for problem 6, set 6.
- Lecture 8: (Oct. 24): Sequences,
Taylor series, Laurent series. pp. 175-200 of text. Download using Laurent series to find contour integrals.
- Lecture 9: (Oct. 31): Properties
of power series, pp. 200-220 of text.
- Lecture 10: (Nov. 7): Residue
theory,Chap. 6 of text, skip sec. 70.
- Lecture 11: (Nov.14):
Applications of residue theory
- Lecture 12: (Nov. 21):
Applications of residue theory continues. Worked out examples of
integration on a branch cut.Download pdf file.
Note corrections to this pdf file: In figure A, interchange 1and 3 and
also 2 and 4. In the limits epsilon goes to zero, not infinity.
Also remove the minus sign from the equation fourth row from the
bottom.
- Lecture 13: (Nov. 28): Finish
applications of residue theory. Skip sections 81 and 82. Begin
mapping of functions.
- Lecture 14: (Dec. 5):
Mapping of functions continued.
- FINAL EXAM: It will be on
Tuesday, December 19, 1302 WWH, 5:10-7pm. The exam is closed
book, but you bring one 8.5x11 inch paper with notes on both
sides. The exam will cover all
material through lecture 13. This amounts to chapters 1-7 except
for sections 26,27,70,81,82. OFFICE HOURS THIS WEEK: THURSDAY 14
10-12AM, FRIDAY DEC. 15 10-12AM.