Complex variables II(G63.2460/V63.0394)

Spring 2007

Professor Stephen Childress

Tuesday, 5:10-7:00 pm in WWH 1013.

 Office Hours: Tuesdays 4-5pm, Thursdays  10-11am, or by appointment.

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.
  

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• 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.

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  NOTES REGARDING THE FINAL EXAM: AS FOR LAST SEMESTER , THE EXAM WILL BE CLOSED BOOK BUT ONE PAGE OF NOTES IS ALLOWED. A LIST OF REVIEW TOPICS WILL BE AVAILABLE HERE FOLLOWING LAST CLASS. FOR YOUR GRADE THE TOP 8 PROBLEM SETS WILL COUNT 60%.  GRADUATING SENIORS: I HAVE VERIFIED WITH THE ADMINISTRATION THAT YOU MAY TAKE THE EXAM MAY 8 AS PLANNED. Download set of topics for review here.
  

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