We perform arithmetic operations on complex numbers.

We learn a geometric interpretation of complex numbers.

We learn properties of the complex conjugate.

We learn to use a complex variable.

We visualize operations on complex numbers in the complex plane.

We learn to recognize and sketch special sets in the complex plane.

We learn the basic properties of the hyperbolic functions.

We introduce functions of a complex variable.

We define the multivalued complex logarithm and discuss its branches and properties. We also define complex exponential functions.

We define and discuss the complex trigonometric functions.

We determine the image of special sets.

We find limits of complex functions.

We find derivatives of complex functions

We derive the Cauchy-Riemann equations.

We determine if a function is analytic using the Cauchy-Riemann equations.

We determine and create harmonic functions.

We find limits of complex sequences.

We determine convergence of complex series.

We determine radius of convergence of complex power series.

We determine the Taylor series for a given analytic function.

We determine the Laurent series for a given function.

We compute integrals of complex functions along contours.

We compute integrals of complex functions around closed curves.

We use the Residue Theorem to compute integrals of complex functions around closed contours.