Tuesday-Thursday 1:00-2:30. 1068 East Hall
J. Rauch
4834 East Hall
Email: rauch@umich.edu
Web page: http://www.math.lsa.umich.edu/~rauch/
Office Hours: Wednesdays 10-12, Thursday 2:40-3:40.
Email office hours at all hours.
| Week | Meeting | Date | In Class | Remarks/Web Postings |
|---|---|---|---|---|
| 1 | Lecture 1 | Tues, Sept. 6 | Complex numbers. Complex plane. Complex conjugate. Modulus. Argument. Argument in product. Inverses. Solutions of quadratic equations. Roots. Image of a large circle by a polynomial. Limits and Cauchy criterion. |
Course info. Chapter 1, Chapter 2, Chapter 6. |
| 1 | td>Lecture 2 | Thurs, Sept. 8 | Weierstrass M test. Exponential function. Second derivation of de Moivre. Differentiation of functions from R^2 to itself. Imgae of circles by linear transformations of the plane. |
Chapter 1,2,6. Image of Circles by 2x2 Matrices handout. HW 1. |
| 2 | Lecture 3 | Tu, Sept. 13 | Linear transformations of the plane and definition of complex derivative. Cauchy-Riemann. Sums products, quotients, composition. Examples. f'=0. Local expansion and contraction by analytic functions. |
Chapter 4. Image of Circles by 2x2 matrices handout section 2. |
| 2 | Lecture 4 | Th., Sept. 15 | Mappings by exponential and trigonometric functions. Conformality. Inversion. Liouville theorem in R^3. Bad behavior of complex exponents. Definitions of line integrals dx, dy, ds, and dz. | Section 5.1, 5.2. Inverse function theorem neither in text nor handout. HW 2. |
| 3 | Lecture 5 | Tu, Sept. 20 | Independence of parameterizaton. Fundamental theorem of calculus, real and complex version. Existence of a primitive and independence of path. | Section 5.5. |
| 3 | Lecture 6 | Th., Sept. 22 | Computing primitive. Green's Theorem. Cauchy integral formula. | Green's theorem. Cauchy integral formula. Section 5.6 is an alternate proof. The alternate works without assuming continuity of derivatives. Uses 5.3. HW 3. |
| 4 | Lecture 7 | Tu., Sept. 27 | Higher differentiability. Cauchy integral formula for derivatives. Leibniz' rule. Antiderivatives of analytic functions. Morera Theorem. Harmonic functions. | Section 5.5, 5.6, 5.7, 5.8. |
| 4 | Lecture 8 | Th., Sept. 29 | Harmonic conjugate. Mean Value property Maximuml modulus principal. Maximum and minimum principals for harmonic functions. Cauchy inequalities. Liouville thm. Fundamental Theorem of Algebra. | Sections 5.8, 10.3, 7.1. HW 4. |
| 5 | Lecture 9 | Tu., Oct. 4 | Intro to Taylor expansion. Radius of convergence. Weierstrass M test. Taylor expansion of analytic functions. Weierstrass' Theorem. |
Ch 6. Section 6.39. |
| 5 | Lecture 10 | Th., Oct. 6 | Isolated zeroes and unique continuation principal. Behavior where f"=0. Open mapping theorem. | Section 10.2. Open mapping handout posted Oct 12. |
| 6 | Lecture 11 | Tu. Oct. 11 | Laurent expansion. Classification of isolated singularities. | 11.1, 11.2 |
| 6 | Lecture 12 | Th. Oct 13 | End of classification. Easy computation of Laurent coefficients. Partial fractions. Fourier series of analytic functions. | 11.2. Partial fractions handout. Laurent yields Fourier handout. Open mapping handout posted. |
| 7 | Study Day | Tu., Oct. 18 | No Class. Study day. | |
| 7 | MIDTERM Ex | Th. Oct 20 | In class. One 3"x5" card (two sides) of notes from home. No electronics. | No Fourier series. Sections in book not covered: 1.4, 5.3, 5.4, 8.2, 9.3, 11.3. Otherwise we have coverd the topics of Chapters 1 through Chapter 11 plus handouts. Note that our treatment of many topics, e.g. Cauchy's integral theorem is quite different. So we did not do Sections 5.3, 5.4, but did prove Cauchy's Theorem. It used Green's Theorem, that is not in the text. Some topics are neither in handouts nor the text, for example the Inverse Function Theorem. In my opinion, the texts "Comments" are very well done. |
| 8 | Lecture 13 | Tu. Oct 25 | End of Fourier series. | Laurent yields Fourier handout. Midterm solutions posted. |
| 8 | Lecture 14 | Th. Oct. 27 | Residue theorem. Counting zeroes and poles. Argument Principal. Rouche Theorem. Fundamental Thm. of Algebra. | 11.3, 12.1, 12.2. Man and Dog handout. 12.24 omitted. |
| 9 | Lecture 15 | Tu, Nov. 1 | Rouche example 12.22. Improper integrals, absolute and conditional convergence. Evaluation of integrals by method of residues. Ex 12.31. |
12.22, 12.31, 12.32. |
| 9 | Lecture 16 | Th, Nov. 3 | Evaluation of integrals involving either principal value integrals and/or multiple valued functions. The thermometer contour. | 12.33, 12.41, 12.42. |
| 10 | Lecture 17 | Tu, Nov. 8 | Laplace transform is analytic in s. Fourier transform pairs. 1/z preserves lines and circles. Fractional linear transformations. | 8.1, 8.2. (8.26 not covered). |
| 10 | Lecture 18 | Th. Nov. 10 | Examples of mappings. Derivation of heat equation. Examples of solutions of Dirichlet problems. Mean value property and maximum/minimum prinicipals for harmonic functions. Reflection principal. Uniqueness for Dirichlet problem in a slab. |
8.2, 13.1, 13.2, 13.5. The reflection principal in class is a special case of 13.51-13.52. The statement of 13.52 is long to digest. |
| 11 | Lecture 19 | Tu. Nov. 15 | Conformal maps of disk to itself. Schwarz lemma. Dirichlet problems with step function data and the arg function. Poisson kernel. | 153/prob.28,29. 13.1. Dirichlet problem handout. |
| 11 | Lecture 20 | Th. Nov. 17 | Analytic Continuation. Riemann surfaces. | Section 13.4. Also 9.2. Boundary value problems with Neumann conditions. |
| 12 | Lecture 21 | Tu. Nov. 22 | Riemann suface of (z^2-1)^{1/2. Flows in half spaces and corners. | 15.1, 15.2. Fluid flow handout. |
| 12 | Thanksgiving | Th. Nov. 24 | No Class | |
| 13 | Lecture 22 | Tu. Nov. 29 | Flow over a half disk. Lift and drag using Bernoulli. D'Alembert Theorem or Paradox. | 15.1, 15.2. The flow over an airfoil presented in 15.2 is nice and NOT covered in class because we did not discuss the Riemann mapping theorem. We do a set of complementary problems. For Bernoulli I recommend the Feynman Lectures on Physics, Vol. II, sections 40.2-40.3. |
| 13 | Lecture 23 | Th. Dec. 1 | Fourier series, Fourier integrals and complex analysis | handout. |
| 14 | Lecture 24 | Tu. Dec. 6 | Fourier continued. Sampling Theorem. | handout continued. |
| 14 | Lecture 25 | Th. Dec. 8 | Dirichlet problem in a disk. | HW 12 due. 13.1, 13.2. Dirichlet problem in the disk handout. |
| 15 | Lecture 26 | Tu. Dec. 13 | Electrostatics. Electrostatic screening. | Maxwell's Treatise on Electricity and Magnetism, Vol I, Article 203-205 and Figure XIII. Feynmann Lectures on Physics,Vol. II, Section VII-5. Screening not on final. |
| Mon. Dec. 19 | 4:00PM-6:00PM. In our classroom. Two 3"x5" cards (four sides) of notes from home. No electronics. 30% premidterm, 70% postmidterm. |