Professor of Mathematics
Apr 19 2015:
This week: steady plane-parallel 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 | Cauchy-Riemann 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 5.3-5.4). 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 term-by-term 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 Cauchy-Hadamard 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 multi-valued functions. Chapter 13. Boundary-value 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 half-plane 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 Schwarz-Christoffel mapping formula. | ||
| Week 15 | Lecture 24 | Monday, April 13 | Chapter 15. Elements of fluid dynamics. Steady plane-parallel 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 Kutta-Joukowski 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 |