Complex-Valued Function

Complex-Valued Functions

Definition: Complex-Valued Function

A complex-valued function on a set $X$ is a function $f:X\to \mathbb{C}$ from $X$ to the set of the complex numbers.

Definition: Real Part

The real part of $f$ is the real-valued function $\mathrm{Re}f:X\to \mathbb{R}$ which, for each $x\in X$, returns the real part of $f(x)$.

$$\mathrm{Re}f(x)\stackrel{\text{def}}{=}\mathrm{Re}(f(x))\forall x\in X$$

Definition: Imaginary Part

The imaginary part of $f$ is the real-valued function $\mathrm{Im}f:X\to \mathbb{R}$ which, for each $x\in X$, returns the imaginary part of $f(x)$.

$$\mathrm{Im}f(x)\stackrel{\text{def}}{=}\mathrm{Im}(f(x))\forall x\in X$$

Operations with Complex-Valued Functions

Definition: Function Operations

Let $f:{\mathcal{D}}_f\to \mathbb{C}$ and $g:{\mathcal{D}}_g\to \mathbb{C}$ be complex-valued functions.

• The sum of $f$ and $g$ is the function $f+g:{\mathcal{D}}_f\cap {\mathcal{D}}_g\to \mathbb{C}$ defined as

$$(f+g)(x)\stackrel{\text{def}}{=}f(x)+g(x)\forall x\in {\mathcal{D}}_f\cap {\mathcal{D}}_g$$

• The product of $f$ and $g$ is the function $fg:{\mathcal{D}}_f\cap {\mathcal{D}}_g\to \mathbb{C}$ defined as

$$(fg)(x)\stackrel{\text{def}}{=}f(x)\cdot g(x)\forall x\in {\mathcal{D}}_f\cap {\mathcal{D}}_g$$

• The ratio of $f$ and $g$ is the function $fg:{\mathcal{D}}_f\cap {\mathcal{D}}_g\to \mathbb{C}$ defined as

$$f/g(x)\stackrel{\text{def}}{=}\frac{f(x)}{g(x)}\forall x\in {\mathcal{D}}_f\cap {\mathcal{D}}_g$$