# Definition:Exponential Order

## Definition

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

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 $\hointr 0 \to$.

Then $f$ is said to be of **exponential order**, denoted $f \in \mathcal E$, if and only if it is of exponential order $a$ for some $a > 0$.

### Exponential Order $a$

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**.

## Also see

- Results about
**Exponential Order**can be found here.

## Sources

- 2009: William E. Boyce and Richard C. DiPrima:
*Elementary Differential Equations and Boundary Value Problems*(9th ed.): $\S 6.1$ - 2003: Anders Vretblad:
*Fourier Analysis and its Applications*: $\S 3.1$