The topics listed below are covered by Problem Sets 1–3 and 3 bis

• How cubic equations lead to complex numbers
• 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, e^z 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 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 of complex functions; continuity of complex functions