Professor of Mathematics
August 25, 2016:
Webpage is under construction.
Week | Meeting | Date | In Class | Homework | |
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Week 1 | Lecture 1 | Tuesday, September 6 | Chapter 1. Arithmetic and algebra of complex numbers. See the short article of Trefethen on the role of complex analysis in modern applied mathematics. |
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Lecture 2 | Thursday, September 8 | Chapter 2. Topology of complex numbers. | HW 1 Assigned. | ||
Week 2 | Lecture 3 | Tuesday, September 13 | Chapter 3. Complex functions of complex arguments. | ||
Lecture 4 | Thursday, September 15 | Chapter 4. Differentiation of complex functions. Analogies and key differences with real differentiation. | HW 1 Due. HW 2 Assigned. | ||
Week 3 | Lecture 5 | Tuesday, September 20 | Cauchy-Riemann equations. Dbar formulation. Principles of conformal mapping. | ||
Lecture 6 | Thursday, September 22 | 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 4 | Lecture 7 | Tuesday, September 23 | Indefinite contour integrals and antiderivatives. Cauchy's Integral Formula and the infinite differentiability of analytic functions (!). | ||
Lecture 8 | Thursday, September 27 | (Real) harmonic functions and their properties. 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 5 | Lecture 9 | Tuesday, October 4 | Uniform convergence of function series and continuity/analyticity of sums. Applicability of term-by-term calculus. | ||
Lecture 10 | Thursday, October 6 | Chapter 7. Power series as a (very) special case of complex function series. Radius of convergence. | HW 4 Due. HW 5 Assigned. | ||
Week 6 | Lecture 11 | Tuesday, October 11 | 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 | Thursday, October 13 | Periodicity, zeros, and mapping properties of exponentials, trigonometric functions, and hyperbolic functions. Review for midterm. | HW 5 Due. HW 6 Assigned. |
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Week 7 | Tuesday, October 18 | FALL STUDY BREAK | |||
Lecture 13 | Thursday, October 20 | MIDTERM EXAM | |||
Week 8 | Lecture 14 | Tuesday, October 25 | Chapter 8. (continued) Fractional linear transformations. | ||
Lecture 15 | Thursday, October 27 | Chapter 9. Analytic "functions" with multiple values. Branch points and Riemann surfaces. | HW 6 Due. HW 7 Assigned. | ||
Week 9 | Lecture 16 | Tuesday, November 1 | More on branch points. Chapter 10. Taylor series of analytic functions. Liouville's Theorem. |
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Lecture 17 | Thursday, November 3 | Maximum modulus principle and related topics. | HW 7 Due. HW 8 Assigned. | ||
Week 10 | Lecture 18 | Tuesday, November 8 | Chapter 11. Laurent series. Isolated singular points: removable ones, poles, and essential singularities. | ||
Lecture 19 | Thursday, November 10 |
The residue of a function at an isolated singular point. The Residue Theorem. |
HW 8 Due. HW 9 Assigned. | ||
Week 11 | Lecture 20 | Tuesday, November 15 | Evaluation of definite improper integrals by means of the Residue Theorem. | ||
Lecture 21 | Thursday, November 17 | Evaluation of integrals with multi-valued functions. Exponential integrands and Fourier/Laplace transforms. | HW 9 Due. HW 10 Assigned. | ||
Week 12 | Lecture 22 | Tuesday, November 22 | (Statement of) Riemann's Mapping Theorem. Implications thereof. Fractional linear maps (only) preserve the Riemann sphere. Analytic continuation via power series and symmetry principles. | ||
Thursday, November 24 | THANKSGIVING BREAK | ||||
Week 13 | Lecture 23 | Tuesday, November 29 | Riemann surface revisited. Branch cuts and improper integrals. | HW 10 Due. HW 11 Assigned. | |
Lecture 24 | Thursday, December 1 | Chapter 13. Boundary-value problems for harmonic functions. The Dirichlet problem in a disk. Poisson kernel. Generalization to the half-plane via conformal mapping | |||
Week 14 | Lecture 25 | Tuesday, December 6 | (Statement of) Riemann's Mapping Theorem. Implications thereof. Fractional linear maps (only) preserve the Riemann sphere. | . | |
Lecture 26 | Thursday, December 8 | Analytic continuation via power series and symmetry principles. | HW 11 Due. Optional homework assigned. . | ||
Week 15 | Lecture 27 | Tuesday, December 13 | Review; Special topic | ||
Lecture 27a | Sunday, December 18 | Review Session (3088 East Hall, To be scheduled in evening) . | |||
Week 16 | Monday, December 19 | FINAL EXAM 4:00-6:00pm |