Exam 1 topics

Exam 1 covers the same content as Homework Sets 1–5. Sections from the text, and the topics included, are as follows:

  • Complex numbers (Chapter 1)
    • How cubic equations lead to complex numbers (but will not be on exam!)
    • Definition of complex numbers as ordered pairs of reals and of their addition and multiplication operations
    • Geometric representation of complex numbers, and of their addition and multiplication
    • Algebra of complex numbers
    • Real part, imaginary part, conjugate; reciprocals and quotients
    • Modulus, polar representation, ez and exponential form, Euler’s formula, Arg and arg
    • DeMoivre’s formula, nth roots
    • Curves and their parametrization; closed and simple closed curves; Jordan curve theorem
    • Open, closed, and punctured disks
    • Interior and boundary points; interior, boundary, and closure of a set; bounded and unbounded sets; connected sets
  • Complex functions (Chapter 2 except section 2.4.1;
    Section 10.2 except Theorem 10.3 and except subsection 10.2.1

    • Complex functions in cartesian and polar forms
    • Idea of a complex function as a mapping of the plane; image of sets; one-to-one and onto functions
    • Affine linear functions and their mapping properties; finding images of sets under such functions
    • The nth power function and the principal nth root function; other branches of square-root
    • The reciprocal function (1/z); the extended complex plane and the Riemann sphere
  • Limits and continuity of complex functions (Sections 3.1–3.2)