Definition:Exponential Order/Real Index

Definition
Let $f \left({t}\right): \R \to \R$ or $\R \to \C$ be a function.

Let $f$ be continuous on the real interval $\left [{0 \,.\,.\, \to} \right)$, except possibly for some finite number of discontinuities of the first kind in every finite subinterval of $\left [{0 \,.\,.\, \to} \right)$.

Let $\left \vert \, \cdot \, \right \vert$ be the absolute value if $f$ is real-valued, or the modulus if $f$ is complex-valued.

Let $e^{a t}$ be the exponential, where $a \in \R$ is constant.

Then $f\left({t}\right)$ is said to be of exponential order $a$, denoted $f \in \mathcal{E}_a$, there exist strictly positive real constants $M, K$ such that:


 * $\forall t \ge M: \left\vert {f \left({t}\right)} \right \vert < K e^{a t}$