Syllabus

HW 1 (due Mon 1/23): 1.17*,2.8*, 2.17,2.18*,2.20* (solutions, courtesy of Ricardo Reyes-Grimaldo)

HW 2 (due Mon 2/6): 3.8*,3.10, 3.13,3.14*,3.18*,3.19,3.44(e)*,3.45,3.46,3.53*(solutions)

HW 3 (due Mon 2/20): 4.3, 4.8(f), 4.27*,4.30*,5.3,5.4*,5.12,5.14* Also: required reading: sections 7.2 and 7.3 (review of sequences and series)(solutions, courtesy of Ricardo Reyes-Grimaldo)

HW 4 (due Mon 3/6):7.30, 7.31*,7.34,7.37*, 8.6, 8.8,8.10, 8.18,8.23*,8.26* (solutions)

HW5 (due 3/22, day of final exam): 8.17*, 8.18, 8.30, 8.31,8.35*, 9.2(c)* and 9.2(d)*, 9.6,9.7,9.8(c)* and 9.8(e)*,9.15* (hint: use 9.14), 9.16, 9.17 (hint: use 9.16)

**Announcements:**

- The midterm on Fr 2/10 covers the first 4 chapters of the text. Specifically: Chap 2: Continuous, differentiable and holomorphic complex functions and Cauchy-Riemann equations. Ch 3: Mobius transfmormations as conformal mappings, the Riemann sphere and cross ratio (but not the stereographic projection). Examples of complex functions: exponential, trig and hyperbolic trig functions, the multivalued logarithm and the principal logarithm; the multi-valued and principal power functions. Chap 4: Definition and properties of path integrals; antiderivatives: definition, properties, existence and construction. Cauchy's Theorem and consequences, Cauchy's integral formula (integration along circle and along simple, closed contractible paths). You will not be asked to prove any of the results proved in class, but you need to be able to apply these results to problems which will be similar to those assigned as homework. Best preparation: solve the problems from the text in chap1-4. Note: the midterm is closed book, and no formula sheets, calculators, electronic devices etc are allowed. Bring only white paper and pen/pencil.
- The comprehensive final exam will be in our usual classroom (BEXL 207) on Wed March 22nd, 9:30-11:20am. You are allowed to bring a pre-approved 1 sided page containing not more than 15 formulas/theorem statements etc (but not worked-out solutions to problems). You should send this page to me via email by Sun March 19th. I will approve your list or send it back to you with edits. I will then print the approved sheet and bring it with me to the final exam, so you don't need to bring it yourself.
- The final is comprehensive and covers everything mentioned above for the midterm, as well as the following topics: Chap 5: Analogues of Cauchy's Integral Formula, for higher order derivatives. Antiderivatives and Morera's Theorem. The Fundamental Theorem of Algebra, and Liouville's Theorem. Closed path integrals and their relevance to compute real improper integrals. We skipped Chap 6, and as mentioned in class Chap 7 is important only for subsequent chapters. It will not be tested for specifically. However, you should understand the notions of pointwise, absolute and uniform convergence, especially for power series (Thm 7.31 is crucial), as well as how these notions impact term-by-term integration etc. Chap 8: The important relationship between holomorphic functions and analytic functions (i.e. functions that are representable as power series). The classification of zeros of holomorphic functions, and, as a consequence, the identity principle. The maximum and minimum modulus theorem (but skip Cor 8.20). Laurent series, and annular regions of convergence. You should be able to find the Laurent series of given functions, and understand their relevance to path integration of those functions. Chap 9: Classifying isolated singularities (removable,pole, essential), and various criteria to identify them, e.g. directly using the function, or using Laurent series (Prop 9.5 and 9.8). Definition of the order of a pole. You may skip results about essential singularities like the Casorati-Weierstrass theorem. The notion of residues and knowing ways to calculate them (see Prop 9.11), and their use in integration, i.e. the Residue Theorem (Thm 9.10). We did NOT cover the argument principle, nor Rouche's Theorem, so that section can be skipped.