Final exam coverage

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