This course begin in the Brown and Churchill
text where CV I left off.

Other useful texts are listed below. The format of the course will be
as in CV I, but other texts, all on reserve in the CIMS Library, will be

used for sources of material. The book by Ablowitz and Fokas is
one you might want to purchase as a secondary text. It has not been
ordered for the bookstore, but you can obtain it from online sources.

Text: *Complex
Variables
and Applications*, by Brown and Churchill (7th addition)--Available
at
the NYU bookstore and on reserve in the Courant Library.

Other reference texts: *Complex Analysis* by
Ahlfors. A more
abstract
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 this
semester.

All of the above available on reserve in the
Courant
Library.

**Lecture 1 : Review of linear fractional transformations and examples of mappings of a half-plane. The map w=sin z. Powers and frational powers of z. pp. 307-328 of text. Riemann surfaces. (Skip sections 91and 93 for now.)**

- Lecture 2: Conformal mappings.
Harmonic functions. Applications from heat conduction. (pp.
343-365)

- Lecture 3: Applications to heat
condition. Electrostatics. Begin applications to fluid flow. (pp.
365-385)

- Lecture 4: Applications of
conformal mapping to fluid flow. Bernoulli and Blasius theorems.

- Lecture 5: The lifting flat plate. Airfoil theory using conformal maps. (Note: Anyone interested in an elementary book on fluid dynamics should look at
- Elementary Fluid Dynamics, D.J. Acheson, Oxford University Press, available in paperback.) Begin Schwarz-Christoffel transformation pp. 391-396 of text. Download pdf file of some examples of flow past Joukowski airfoils.
- Lecture 6: The Schwarz-Christoffel transformations: triangles, rectangles, degenerate polygons, pp. 387-406.
- Lecture 7: Applications of Schwarz-Christoffel transformation, pp.
406-415.

- Lecture 8: Poisson-type integral formulas.
- NOTES FOR LECTURES 9-14: DOWNLOAD
(updated) PDF FILE

- Lecture 9: The Schwarz lemma.
The Riemann mapping theorem.

- Lecture 10: Asymptotic
expansions and the evaluation of integrals. The method of stationary phase.

- Lecture 11: The method of
steepest descent.

- Lecture 12: Symmetry,
continuation, and the gamma function.

- Lecture 13: The
Riemann zeta function.

- Lecture 14: Analysis and
pplication of transform methods.

- THE FINAL EXAM WILL TAKE PLACE
MAY 8, 5-7pm PLEASE NOTE THAT THE TIME OF
THE FINAL EXAM TIME STAYS AS I ANNOUNCED, AT 5-7 PM. THERE HAS BEEN
CONFUSION ABOUT THIS AS THE MATH WEB PAGE LISTS THE EXAM AT 6-7:50,
WHICH COINCIDES WITH UNDERGRADUATE SCHEDULES.

- Note: The graders for the
class are Stanley
Zhang (Rm 503 WWH, stanley@cims.nyu.edu, odd-numbered sets) and
Qi Zhang (912 WWH, zhangqi@cims.nyu.edu, even-numbered sets). They will
be available in their respective offices for 4-5pm Tuesdays for
questions regarding problems.

- Homework 1. Download pdf file.
- Homework 2. Download pdf file.
- Homework 3. Download pdf file.
- Misprints
in problem 5: "log(b)" should be "m log(b)" and "log(z-b')" should be
"log(b'-z)".

- Homework 4. Download pdf file.
- Typo in problem 5. Should be "y=0, |x|<2a". Also problem 2 was already in set 3, so disregard. You have 4 problems this week
- Homework 5. Download pdf file.
- Homework 6. Download pdf file.
- Homework 7 (Due in THREE weeks, March 27, incorrectly stated as March 20 on problem sheet). In problem 7 "suitable" should be read as "bounded on the real line".Download pdf file.
- Homework 8. Typo in problem 4: g(x) should be (1-e^(-x))(1+x)^(-1). Download pdf file.
- Homework 9. Typos: The exponent in prob. 2 should be a-b-1, not a-b+1. In prob. 1 , line above equation, "Re(z) < 0". In pro. 5, in the sum, "(-i)^n" replaces "(-1)^n". Download pdf file.
- Homework
10. (This is the last homework assignment). Typos: In 3 "|u|=R". In
4(b). On the RHS of
the equatiion is should be 2^{-2z} not 2^{-2z-1}. In 6 "equation (135)"
not"(130)". Download
pdf file.