Professor of Mathematics
Apr 19 2015:
This week: steady planeparallel fluid flow around an obstacle (airfoil). Generation of lift. A Mathematica notebook illustrating flow about circular cylinders in various situations can be downloaded below.
Homework set 12 is available for download in the class schedule below. Due Monday, April 20.
Selected solutions to problems from recent homework sets are posted on my office door. Please feel free to look at them or copy for your own use, but please leave them there for other students.
Week  Meeting  Date  In Class  Homework 

Week 1  Lecture 1  Wedensday, January 7  Chapter 1. Arithmetic and algebra of complex numbers. The short articles of Trefethen and Rossi mentioned in class on the role of complex analysis in modern applied mathematics. Article of Deconinck et al. referred to in Rossi's article. 

Week 2  Lecture 2  Monday, January 12  Chapter 2. Topology of complex numbers.  HW 1 Assigned. 
Lecture 3  Wednesday, January 14  Chapter 3. Complex functions of complex arguments.  
Week 3  Monday, January 19  Martin Luther King, Jr., Day No Class 

Lecture 4  Wednesday, January 21  Chapter 4. Differentiation of complex functions. Analogies and key differences with real differentiation.  HW 1 Due. HW 2 Assigned.  
Week 4  Lecture 5  Monday, January 26  CauchyRiemann equations. Geometrical interpretation of analyticity. Principles of conformal mapping.  
Lecture 6  Wednesday, January 28  Chapter 5. Contour integration. Green's Theorem implies a weaker version of Cauchy's Integral Theorem (alternate approach to sections 5.35.4). Proper statement of Cauchy's Integral Theorem.  HW 2 Due. HW 3 Assigned.  
Week 5  Lecture 7  Monday, February 2  Indefinite contour integrals and antiderivatives; Morera's Theorem. Cauchy's Integral Formula and the infinite differentiability of analytic functions. (Real) harmonic functions and their properties. [Class cancelled due to weather. Here are the lecture notes for this lecture.]  
Lecture 8  Wednesday, February 4  Chapter 6. Infinite series of complex numbers and functions. Operations with series and the role of absolute versus conditional convergence.  HW 3 Due. HW 4 Assigned.  
Week 6  Lecture 9  Monday, February 9  Uniform convergence of function series and continuity/analyticity of sums. Applicability of termbyterm calculus.  
Lecture 10  Wednesday, February 11  Chapter 7. Power series as a special case of complex function series. Radius of convergence.  HW 4 Due. HW 5 Assigned.  
Week 7  Lecture 11  Monday, February 16  The CauchyHadamard formula for the radius of convergence. Analyticity of power series. Chapter 8. Special analytic functions and their elementary properties. Exponentials, trigonometric functions, and hyperbolic functions.  
Lecture 12  Wednesday, February 18  Periodicity, zeros, and mapping properties of exponentials, trigonometric functions, and hyperbolic functions. Mapping properties of fractional linear mappings.  HW 5 Due. HW 6 Assigned.  
Week 8  Lecture 13  Monday, February 23  Fractional linear mappings, continued. Chapter 9. Analytic "functions" with multiple values. Branch points and Riemann surfaces.  
Wednesday, February 25  MIDTERM EXAM  
Week 9  Monday, March 2  WINTER BREAK  
Wednesday, March 4  
Week 10  Lecture 14  Monday, March 9  Chapter 10. Taylor expansions of analytic functions. Liouville's Theorem.  HW 6 Due. HW 7 Assigned. 
Lecture 15  Wednesday, March 11  Maximum modulus principle and related topics. Chapter 11. Laurent expansions.  
Week 11  Lecture 16  Monday, March 16  Isolated singular points: removable ones, poles, and essential singularities. The residue of a function at an isolated singular point. The Residue Theorem.  HW 7 Due. HW 8 Assigned. 
Lecture 17  Wednesday, March 18  Chapter 12. Applications of the Residue Theorem. Root finding: the argument principle, Rouché's Theorem, and the Fundamental Theorem of Algebra.  
Week 12  Lecture 18  Monday, March 23  Evaluation of definite improper integrals by means of the Residue Theorem.  HW 8 Due. HW 9 Assigned. 
Lecture 19  Wednesday, March 25  Exponential integrands and Fourier transforms.  
Week 13  Lecture 20  Monday, March 30  Evaluation of integrals with multivalued functions. Chapter 13. Boundaryvalue problems for harmonic functions. The Poisson integral formula and the Poisson and Schwarz kernels.  HW 9 Due. HW 10 Assigned. 
Lecture 21  Wednesday, April 1  The Dirichlet problem in a disk. Generalization to the halfplane via conformal mapping.  
Week 14  Lecture 22  Monday, April 6  Statement of Riemann's Mapping Theorem. Implications thereof. Fractional linear maps (only) preserve the Riemann sphere. Analytic continuation via power series and symmetry principles.  HW 10 Due. HW 11 Assigned. 
Lecture 23  Wednesday, April 8  Chapter 14. Conformal mappings of polygons. The SchwarzChristoffel mapping formula.  
Week 15  Lecture 24  Monday, April 13  Chapter 15. Elements of fluid dynamics. Steady planeparallel flows. Harmonic/complex interpretation of solenoidal and irrotational flows.  HW 11 Due. HW 12 Assigned. 
Lecture 25  Wednesday, April 15  Elementary examples: flow in a corner, point sources/sinks and point vortices. The theory of flow around cylinders.  
Week 16  Lecture 26  Monday, April 20  The force exerted by a flow on an airfoil and the KuttaJoukowski Theorem. The special case of a circular cylinder.  HW 12 Due. (Due date postponed until Tuesday, April 21 at 4:30 PM in my office.) A Mathematica notebook illustrating flow around circular cylinders. 
Week 17  Thursday, April 30  FINAL EXAM 