The Final Exam covers the entire semester’s work. Below are the *additional *topics (since Exam 2) for the Final Exam.

- Evaluating contour integrals (secs. 6.5 and 7.5):
- Cauchy’s Integral Formula and Cauchy’s Integral Formula for Derivatives
- The Residue Theorem

- Series expansions of functions (secs. 7.2–7.3):
- Uniqueness of power series expansion in a disk
- Taylor series of a function that’s holomorphic in a disk
- Laurent series expansion on an annulus of a function holomorphic on that annulus
- Manipulation of geometric series to find Laurent series expansion
- Sum of a Laurent series (of of a Taylor series) may be differentiated term-by-term—and hence integrated term-by-term—in the annulus where it is defined

- Zeros and singularities (sec. 7.4):
- Order of a zero of a function
- Classification of singularities: isolated vs. non-isolated; for isolated singularities—removable, pole, and essential
- Order of a pole: definition; finding from Laurent series; finding the order of a pole z0 when the function has form g(z)/h(z) and you know the order of the zero z0 of g(z) and h(z)

- Residues (sec. 8.1):
- Definition of residue in terms of Laurent series
- Formula for finding residue at a pole of order k; special case of the formula when the pole is simple (i.e., of order 1)
- The Residue Theorem