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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 complex numbers.

Definition: Real Part

The real part of \(f\) is the [[../Real Analysis/Real-Valued Functions|real-valued function]] \(\operatorname{Re} f: X \to \mathbb{R}\) which, for each \(x \in X\), returns the [[index|real part]] of \(f(x)\).

\[ \operatorname{Re} f (x) \overset{\text{def}}{=} \operatorname{Re}(f(x)) \qquad \forall x \in X \]

Definition: Imaginary Part

The imaginary part of \(f\) is the [[../Real Analysis/Real-Valued Functions|real-valued function]] \(\operatorname{Im} f: X \to \mathbb{R}\) which, for each \(x \in X\), returns the [[index|imaginary part]] of \(f(x)\).

\[ \operatorname{Im} f (x) \overset{\text{def}}{=} \operatorname{Im}(f(x)) \qquad \forall x \in X \]

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
\[ (f+g)(x) \overset{\text{def}}{=} f(x) + g(x) \qquad \forall x \in \mathcal{D}_f \cap \mathcal{D}_g \]
  • The product of \(f\) and \(g\) is the function \(fg: \mathcal{D}_f \cap \mathcal{D}_g \to \mathbb{C}\) defined as
\[ (fg)(x) \overset{\text{def}}{=} f(x) \cdot g(x) \qquad \forall x \in \mathcal{D}_f \cap \mathcal{D}_g \]
  • The quotient of \(f\) and \(g\) is the function \(fg: \mathcal{D}_f \cap \mathcal{D}_g \to \mathbb{C}\) defined as
\[ f/g (x) \overset{\text{def}}{=} \frac{f(x)}{g(x)} \qquad \forall x \in \mathcal{D}_f \cap \mathcal{D}_g \]