Exponential is of Exponential Order Real Part of Index

Theorem

Let $\map f t = e^{\psi t}$ be the complex exponential function, where $t \in \R, \psi \in \C$.

Let $a = \map \Re \psi$.

Then $e^{\psi t}$ is of exponential order $a$.

Proof

\(\ds \forall t \ge 1: \, \)

\(\ds \size {e^{\psi t} }\)

\(=\)

\(\ds e^{a t}\)

Modulus of Exponential is Exponential of Real Part

\(\ds \)

\(<\)

\(\ds 2 e^{a t}\)

Exponential of Real Number is Strictly Positive

The result follows from the definition of exponential order with $M = 1$, $K = 2$, and $a = a$.

$\blacksquare$