In this chapter, we explored complex functions. We began with the complex exponential, which takes a complex number #z=a+b\cdot\ii# and maps it to #\e^{z}=\e^{a}\cdot\left(\cos\left(b\right)+\sin\left(b\right)\cdot \ii\right)#. This allowed us to learn Euler's formula, #\e^{\theta\cdot \ii}=\cos\left(\theta\right)+\sin\left(\theta\right)\cdot \ii#, which we then used to introduce the polar-exponential formula of a complex number
\[\begin{array} {rcl}
z&=& r\cdot \e^{\theta\cdot \ii} \\ a&=& r\cdot \cos\left(\theta\right) \\ b&=& r\cdot\sin\left(\theta\right)\end{array}\]

This representation simplifies many complex calculations and offers clear advantages in understanding rotations and transformations within the complex plane.

We also studied the complex logarithm function. This function is only defined for the principal value #\varphi# of a complex number, where #-\pi<\varphi\leq\pi#. Then, for #z=r\cdot \e^{\varphi\cdot\ii}#, we have #\log\left(z\right)=\log\left(r\right)+\varphi \cdot \ii#. Following the logarithm, we studied De Moivre's theorem, which allowed us to simplify powers of complex numbers to #z^n=r^n\cdot\e^{n\cdot \theta\cdot \ii}#. We also used the theorem to discuss roots of unity and roots of complex numbers. We thus learned that an equation of the form #z^n=w# always has #n# solutions.

Finally, we studied complex polynomials. This led us to the fundamental theorem of algebra, which showed us that a complex polynomial of degree #n# always has #n# roots (not necessarily distinct). Moreover, we learned that if the polynomial has real coefficients, the complex roots will always come in complex conjugate pairs, following which we concluded that odd degree polynomials always have at least one real root. These insights enriched our comprehension of how complex numbers unify various elements in algebra, offering a cohesive framework for tackling polynomial equations and understanding their roots, particularly the role of conjugate pairs.