J. Rauch
4834 East Hall
Email: rauch@umich.edu
Web page: http://www.math.lsa.umich.edu/~rauch/
Office Hours: Wednesdays 10-11, 2-3.
| Week | Meeting | Date | In Class | Remarks/Web Postings |
|---|---|---|---|---|
| 1 | Lecture 1 | Tues, Sept. 8 | Complex numbers, Weierstrass M test, e^z, Derivative of R^n maps, Polar decomposition of matrices, Conformal matrices. |
Course info., hw1., Read Chapter 1. |
| 1 | td>Lecture 2 | Thurs, Sept. 10 | Conformality continued, Complex derivative, Cauchy-Riemann, mapping by z^2, exp(z). Harmonic functions. | Sections 12, 13, 14, 15, 16, 18, 19!, 20, 21!, 22!, 24, 25, 26. Skip polar coords. Postpone sections 17, 27 and 28. Conformal matrices handout. |
| 2 | Lecture 3 | Tues, Sept. 15 | Harmonic conjugate. Conjugate of ln r. Inverse function theorem for analytic functions. exp(z) and ln z continued. |
Sections 29, 30, 31, 32, 103. hw1 due. hw2. |
| 2 | Lecture 4 | Th. Sept. 17 | Derivative and integral of vector valued functions. Mean value inequality. Bound on modulus of integral. Contours. |
37, 38, 39, 43 |
| 3 | Lecture 5 | Tu. Sept. 22 | Smoothness of contours. Line integrals dx, dy, ds. Domain on the left orientation. Green's Theorem. Cauchy's Theorem (pg. 151). | Sections 40, 41, 42, 46, Skip 47. hw 2 due. hw 3 posted. |
| 3 | Lecture 6 | Th. Sept. 24 | More on ortientation. Antiderivatives. Simple connectivity. | Sections 44, 45, 48 |
| 4 | Lecture 7 | Tu. Sept. 29 | Cauchy Integral Formulas for f(z). Leibniz rule for differentiating an integral depending on a parameter. CIF for f'(z). Higher derivs of analytic functions. Cauchy inequalities. Liouville Thm. |
49, 50!, 51!, 52!. 53! hw3 due. hw4. |
| 4 | Lecture 8 | Th. Oct. 1 | Fundamental Thm. of Algebra. Mean Value Theorem. Max. Modulus Thm. | 53!, 54! |
| 5 | Lecture 9 | Tu. Oct. 6 | Taylor Series. | 55, 56, 57!, 58, 59. hw4due. hw5. |
| 5 | Lecture 10 | Th. Oct. 8 | Unique Continuation. Reflection Principal, Strong Max. Modulus Thm. Morera's Theorem (pg 169, overlooked, oops!) and applications. |
27, 28, 52 (oops). |
| 6 | Lecture 11 | Tu. Oct. 13 | Laurent representation. Theory of power series. Fourier series of periodic analytic funcions (from Laurent). |
60, 61, 62, 63, 64, 65, 66, 67. hw 5 due. hw6. Laurent yields Fourier. |
| 6 | Lecture 12 | Th. Oct 15 | Isolated singularities. | 68, 72!. |
| 7 | Study Day | Tu. Oct. 20 | No Class | |
| 7 | MIDTERM Ex | Th. Oct 22 | In class. Closed book. Two sides of a 3x5 cards of notes. No electronics. | |
| 8 | Lecture 13 | Tu. Oct 27 | Residues. Residue Theorem. | 69, 70. Skip 71, 73, 75. hw6 due. |
| 8 | Lecture 14 | Th. Oct. 29 | Zeros. Return to Isolated singularities. | 75, 76, 77. |
| 9 | Lecture 15 | Tu. Nov. 3 | ||
| 9 | Lecture 16 | Th. Nov. 5 | ||
| 10 | Lecture 17 | Tu. Nov. 10 | ||
| 10 | Lecture 18 | Th. Nov. 12 | ||
| 11 | Lecture 19 | Tu. Nov. 17 | ||
| 11 | Lecture 20 | Th. Nov. 19 | ||
| 12 | Lecture 21 | Tu. Nov. 24 | ||
| 12 | Thanksgiving | Th. Nov. 26 | No Class | |
| 13 | Lecture 22 | Tu. Dec. 1 | ||
| 13 | Lecture 23 | Th. Dec. 3 | ||
| 14 | Lecture 24 | Tu. Dec. 8 | ||
| 14 | Lecture 25 | Th. Dec. 10 | Last Class | |
| 15 | Study Day | Tu. Dec. 15 | No Class | |
| Sun. Dec. 20 | 10:00-12:00, Room to be announced. | ATTN. Differs from time schedule! |