Function of Exponential Order of Scalar Multiple
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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.
$\blacksquare$