# Definition:Exponential Order/Real Index

## Definition

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

Let $f$ be continuous on the real interval $\hointr 0 \to$, except possibly for some finite number of discontinuities of the first kind in every finite subinterval of $\hointr 0 \to$.

Let $\size {\, \cdot \,}$ be the absolute value if $f$ is real-valued, or the modulus if $f$ is complex-valued.

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

Then $\map f t$ is said to be of exponential order $a$, denoted $f \in \mathcal E_a$, if and only if there exist strictly positive real numbers $M, K$ such that:

$\forall t \ge M: \size {\map f t} < K e^{a t}$

## Also known as

Such a function is also known as being of exponential type $a$.