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Home Page for Math 555

Textbook:

Complex Analysis with Applications, by R. A. Silverman, Dover Publications, Mineola, New York, 1984.

Another useful book:

Applied Complex Variables, by John W. Dettman, Dover Publications, Mineola, New York.

Class Meetings:

Tuesdays and Thursdays 1:10 PM - 2:30 PM in 1068 East Hall.

Office hours:

Tuesdays and Wednesdays 2:30 PM - 4:00 PM in 3086 East Hall.

Grading and Course Policies:

Students will be evaluated on the basis of
Tentative Schedule
Week Meeting Date In Class Homework
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.
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.
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.		  
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.
Chapter 12. Applications of the Residue Theorem. Root finding: the argument principle, Rouché's Theorem, and the Fundamental Theorem of Algebra.

 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