Need help Clarifying the meaning to something in Complex Analysis

Question

My professor has lately been writing $\text{Re }z$ and $\text{Im }z$ for my complex analysis class and I'm confused to what this means. Could someone explain? Does it mean the real parts, so just $x$ and the imaginary parts, so just $yi$?

Answer 1

A complex number is expressed by two real numbers $x$ and $y$ such that:

$$z=x+iy \in \mathbb{C}$$

and by definition:

• $\text{Re}(z)=x\in \mathbb{R}$
• $\text{Im}(z)=y\in \mathbb{R}$

Note also that by the definition of complex conjugate $\bar z= x-iy$, we have that:

• $\text{Re}(z)=\frac{z+\bar z}{2}=x\in \mathbb{R}$
• $\text{Im}(z)=\frac{z-\bar z}{2i}=y\in \mathbb{R}$

Answer 2

Given a complex number $z=x+iy$, $$ \text{Re}(z)=x,\quad \text{Im}(z)=y $$

Answer 3

Let $z = x + i\cdot y$ be a complex number.

Then $\text{Re}\ z = x$, $\text{Im}\ z = y$.