Definition:Exponential Order/Real Index
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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 $\hointr 0 \to$.
Establish whether it is "finite subinterval" that is needed here, or what we have already defined as "Definition:Finite Subdivision". Also work out whether we can replace all the above with a link to Definition:Piecewise Continuous Function with One-Sided Limits .
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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: $\S 3.1$
