# Definition:Exponential Order/Real Index

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## Contents

## 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** $a$.

## Also see

## Sources

- 1965: Murray R. Spiegel:
*Theory and Problems of Laplace Transforms*... (previous) ... (next): Chapter $1$: The Laplace Transform: Functions of Exponential Order - 2009: William E. Boyce and Richard C. DiPrima:
*Elementary Differential Equations and Boundary Value Problems*(9th ed.): $\S 6.1$ - 2005: Anders Vretblad:
*Fourier Analysis and its Applications*: $\S 3.1$