Function of Exponential Order of Scalar Multiple

Theorem

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

Let $\lambda$ be a real constant.

Let $\map f t$ be of exponential order $a$.

Then the function defined by $t \mapsto \map f {\lambda t}$ is of exponential order $a\lambda$.

Proof

\(\ds \size {\map f t}\)

\(<\)

\(\ds K e^{a t}\)

Definition of Exponential Order to Real Index

\(\ds \leadsto \ \ \)

\(\ds \size {\map f {\lambda t} }\)

\(<\)

\(\ds K e^{a \lambda t}\)

replacing $t$ with $\lambda t$

The result follows by the definition of exponential order.

$\blacksquare$