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 $\hointr 0 \to$.

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Then $f$ is said to be of exponential order, denoted $f \in \EE$, 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 $f \in \EE_a$, 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

• Definition:Big-O Notation

• 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: $\S 3.1$