Linear Combination of Functions of Exponential Order

Theorem

Let $f, g: \R \to \F$ be functions, where $\F \in \set {\R, \C}$.

Let $\lambda, \mu$ be complex numbers.

Suppose $f$ is of exponential order $a$ and $g$ is of exponential order $b$.

Then $\map {\paren {\lambda f + \mu g} } t = \lambda \, \map f t + \mu \, \map g t$ is of exponential order $\max \set {a, b}$.

Proof

Follows from:

• Scalar Multiple of Function of Exponential Order
• Sum of Functions of Exponential Order

$\blacksquare$