Complex variables I (G63.2450/V63.0393)
Professor Stephen Childress
Tuesday, 5:10-7:00 pm in WWH 1302.
Office Hours: Tuesdays 4-5pm, Thursdays 10-11am,
This course will be taught as a first course
Complex variables using a well-established textbook.
Other useful texts are listed below.
The basic theory will be treated and there will be
Homework will be regularly assigned, graded, and returned with answers.
The grade in the course will be based upon homework and a final exam.
and Applications, by Brown and Churchill (7th additiion)--Available
the NYU bookstore and on reserve in the Courant Library.
Other reference texts: Complex Analysis by Ahlfors. A more
take on the subject, the text used in the one-semester graduate course.
Variables by Ablowitz and Fokas. A newer text, accessible and
to Complex Analysis by Nehari. A good basic text.
Mapping by Nehari. An advanced treatment, primarily for the second
All of the above available on reserve in the
- 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
- 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
- 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.