Introduction to Complex Analysis

complex numbers. The set of complex numbers ${\Complex}$ is the set

Given a complex number ${c = (a,b)}$ we say that ${a}$ is the real component of ${c,}$ denoted ${\text{Re}(c)= a,}$ and that ${b}$ is the complex component of ${c,}$ denoted ${\text{Im}(c) = b.}$ We may express a complex number ${(a,b) \in \Complex,}$ with the notation ${a + bi \in \Complex.}$

imaginary unit. The tuple ${i = (0,1)}$ is called the imaginary unit.

definition. Given ${(a,b), (c,d) \in \Complex,}$ the following operations are are defined:

where ${a,b,c,d \in \reals.}$

properties of complex arithmetic. Let ${\alpha, \beta \in \Complex.}$ Then the following properties hold:

1. ${\alpha + \beta = \beta + \alpha}$ for all ${\alpha, \beta \in \Complex.}$
2. ${\alpha\beta = \beta\alpha}$ for all ${\alpha, \beta \in \Complex.}$
3. ${(\alpha + \beta) + \lambda = \alpha + (\beta + \lambda)}$ for all ${\alpha, \beta, \lambda \in \Complex.}$
4. ${(\alpha\beta)\lambda = \alpha(\beta\lambda)}$ for all ${\alpha, \beta, \lambda \in \Complex.}$
5. ${\lambda + 0 = \lambda}$ for all ${\lambda \in \Complex.}$
6. ${\lambda \by 1 = \lambda}$ for all ${\lambda \in \Complex.}$
7. For all ${\alpha \in \Complex,}$ there exists a unique ${\beta \in \Complex}$ such that ${\alpha\beta = 1.}$
8. ${\lambda (\alpha + \beta)= \lambda\alpha + \lambda\beta}$ for all ${\alpha, \beta, \lambda \in \Complex.}$