Building upon the operations and representations of complex numbers, this chapter delves deeper into the interactions and applications of these numbers within more specific mathematical contexts. Students will extend their skills to work with complex-valued functions like exponentials, logarithms, and polynomials.

This chapter covers more advanced topics than the ones presented in "Operations with complex numbers," beginning with the complex exponential function. This leads to Euler's formula, enabling complex numbers to be expressed in polar-exponential form as #z=r\cdot \e^{\theta\cdot \ii}#. This section is particularly important for analyzing waveforms and oscillations. We then explore the complex logarithm, De Moivre's theorem, and the roots of unity.

The latter part of this chapter addresses complex polynomials, introducing the fundamental theorem of algebra and the notion of conjugate roots for complex polynomials with real coefficients. Understanding these concepts is key for those working with systems and applications involving stability and control in engineering.

The key learning objectives for this chapter are listed below. They outline the essential skills and knowledge to be developed and provide a reference point for what is expected by the end of the chapter.

Learning objectives

The student...

1. works with complex-valued functions, such as exponentials and logarithms, and reproduces and applies Euler’s representation of complex numbers and De Moivre’s theorem;
2. understands complex polynomial functions, how to factorize them and obtain their zeros, as well as reproduces and applies the fundamental theorem of algebra.