J. Rauch
4834 East Hall
Email: rauch@umich.edu
Web page: http://www.math.lsa.umich.edu/~rauch/
Office Hours: Wednesdays 2-4 PM.
Email office hours at all hours.
| Week | Meeting | Date | In Class | Remarks/Web Postings |
|---|---|---|---|---|
| 1 | Lecture 1 | Tues, Sept. 3 | 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. Weierstrass M test. |
Course info. Chapter 1, Chapter 2, Chapter 6. |
| 1 | td>Lecture 2 | Thurs, Sept. 5 | Exponential function. Second derivation of de Moivre. Differentiation of functions from R^2 to itself. Imgae of circles by linear transformations of the plane. Linear transformations of the plane and definition of complex derivative. Cauchy-Riemann. Sums products, quotients, composition. Examples. f'=0. |
Chapter 1,2,6. Image of Circles by 2x2 Matrices handout. HW 1. |
| 2 | Lecture 3 | Tu, Sept. 10 | Roots as analytic functions. Inverse function theorem, real and complex. Conformality. Local expansion and contraction and rotation by analytic functions. Mappings by c, z+c, cz, z^n. Image of circles and lines by 1/z. |
Section 1.4. Section 8.23. Chapter 4. |
| 2 | Lecture 4 | Th., Sept. 12 | Mapping by exp(z). Liouville theorem in R^3. Bad behavior of complex exponents. Definitions of line integrals dx, dy, ds, and dz. | HW1 due. HW2 posted. 5.1, 5.2. |
| 3 | Lecture 5 | Tu, Sept. 17 | Examples of line integrals. Independence of parameterization. Integral of f '(z) dz. Orientation of the plane. Orientation of the boundary of a nice domain. Green's theorem. Proved for a right triangle. CAUCHY's THEOREM. | 5.1, 5.2, 5.4. |
| 3 | Lecture 6 | Th., Sept. 19 | Antiderivatives and independence of path. Simple connectivity. | HW2 due. Section 5.5. |
| 4 | Lecture 7 | Tu., Sept. 24 | Cauchy's integral formula. Leibniz' rule for differentiating an integral depending on a parameter. Infinite differentiability of analytic functions. Morera's Theorem. Cauchy inequalities. Liouville Theorem. Fundamental Theorem of Algebra. | 5.6, 5.7, 10.12, 10.17 (different proof.), 12.23 (different proof). Comment 12.2. 188 Problem 8. |
| 4 | Lecture 8 | Th., Sept. 26 | Harmonic functions. Harmonic conjugate. Derivation of the heat equation. Mean Value property of analytic functions. Mean value property of harmonic functions. Maximal Modulus Theorem for analytic functions, begin. | HW3 due. 5.8, Page 13-14 of Parital Differential Equations, J. Rauch (you can see these pages with the google online access through the UM library. Search for heat equation then click on the page 13-14 part). 10.3. Mean Value Prop. for analytic functions is formula 26 on page 149. The mean value prop of harmonic functions seems not to be in text. |
| 5 | Lecture 9 | Tu., Oct. 1 | Maximal Modulus Theorem for analytic functions, end. Maximum/minimum principal for harmonic functions. Power series. Radius of convergence. Convergence of Taylor series of an analytic function (page 139). Singular point on boundary of disk of convergence (10.15). Example 10.16. | 10.3. Chapter 7. 10.1. |
| 5 | Lecture 10 | Th., Oct. 3 | Differentiating Taylor series. Differentiating uniformly convergent series of analytics. Unique Continuation Principal. Order of zero. Isolated zeroes. Reflection Principal. Intro to Laurent Expansions. Physical interpretation of max principal for equilibrium heat flows. Local saddle structure of harmonic functions. | HW4 due. 10.2. Reflection principal is Comment 13.5. Ch 11 up to formula (6). |
| 6 | Lecture 11 | Tu. Oct. 8 | Laurent expansions on annuli. Examples. Isolated singularities. Examples. Classification in terms of Laurent expansions. Example of the method of undetermined coefficients. Definition of Residue. | Chap 11 up to and including 11.23. |
| 6 | Lecture 12 | Th. Oct 11 | Applications of Laurent series. Partial Fractions. Fourier series of analytic functions. Inverse Laplace transforms. | HW5 due. Partial fractions handout. Laurent yields Fourier handout. Partial fractions and inverse Laplace handout. |
| 7 | Study Day | Tu., Oct. 14 | No Class. Study day. | |
| 7 | MIDTERM Ex | Th. Oct 17 | In class. One 3"x5" card (two sides) of notes from home. No electronics. | Not on exam. 2.4, 8.24-8.29. 9.3, 11.24-11.26, 11.3. The exam will cover the remaining parts of Chapters 1-11 plus handouts, plus material not in text. |
| 8 | Lecture 13 | Tu. Oct 22 | Removeable equals bounded, pole equals goes to infinity, essential equals dense values. Relation of zeroes and poles. Residue Theorem. | HW6 due. 11.2 end. 11.3. |
| 8 | Lecture 14 | Th. Oct. 24 | Evaluation of integrals by the method of residues. | 11.3 |
| 9 | Lecture 15 | Tu, Oct 29. | Evaluation of integrals by the method of residues. Indented domain, principal values. |
11.3, 11.4. HW7 Due. |
| 9 | Lecture 16 | Th, Oct. 31 | Evaluation of integrals by the method of residues. With branch cuts. | 11.4 |
| 10 | Lecture 17 | Tu, Nov. 5 | Argument Principal. Rouche's Theorem. Second pf of funcamental theorem of algebra. Example 12.22. Local mapping when df(z)/dz=0. Open mapping theorem. Approximate roots of f(z)=w for z near a zero of df/dz. | HW8 Due. 12.1, 12.2. For local mapping see section 3.3 of Ahlfors. |
| 10 | Lecture 18 | Th. Nov. 7 | Nyquist method to test asymptotic stability of ODE using argument principal. Conformal mappings. Simple examples. Fractional linear transformations. Schwarz lemma. Self maps of the disk and half plane. | 8.1 (8.15 not in the syllabus). 8.2. A weaker Schwarz lemma is problem 153/28. |
| 11 | Lecture 19 | Tu. Nov. 12 | Schwartz reflection. Application to uniqueness of the half space Dirichlet Problem. Conformal mapping and the Dirichlet Problem, examples. | HW9 due. 13.5 does a more general symmetry. The proof of Theorem 13.51 in the text is incorrect. 13.1, 13.2 gives a discussion of the Dirichlet Problem complementary to that in class. |
| 11 | Lecture 20 | Th. Nov. 14 | Conformal mapping and the Dirichlet problem, further examples. Electrostatics. | 15.3. |
| 12 | Lecture 21 | Tu. Nov. 19 | The half space and disk Dirichlet problems with step data. The Neumann a.k.a. insulator boundary condition for heat conduction. Examples. | HW10 due. Dirichlet problem handout. |
| 12 | Lecture 22 | Th. Nov. 21 | Neumann boundary condition continued. Analytic continuation. | 15.1. 15.2. 13.4. |
| 13 | Lecture 23 | Tu. Nov. 26 | Analytic continuation. Correct proof of 13.51. | HW11 due. 13.4. 13.5. |
| 13 | Thanksgiving | Th. Nov. 28 | No Class | |
| 14 | Lecture 24 | Tu. Dec. 3 | Incompressible, irrotational, planar flows. | HW12 due. 15.1, 15.2, Fluid flow handout. |
| 14 | Lecture 25 | Th. Dec. 5 | Incompressible, irrotational, planar flows. | HW 12 accepted without penalty. 15.1, 15.2, Fluid flow handout. |
| 15 | Lecture 26 | Tu. Dec. 10 | Incompressible flow. Review. | HW13 due. |
| Office hour | Dec. 17, 2:00-4:00 | |||
| Dec. 19, 1:30-3:30 | Two 3"x5" cards (four sides) of notes from home. No electronics. |