Laplace Transform of Hyperbolic Cosine/Proof 1

Theorem

Let $\cosh t$ be the hyperbolic cosine, where $t$ is real.

Let $\laptrans f$ denote the Laplace transform of the real function $f$.

Then:

$\laptrans {\cosh a t} = \dfrac s {s^2 - a^2}$

where $a \in \R_{>0}$ is constant, and $\map \Re s > a$.

$\ds \laptrans {\cosh {a t} }$

$=$

$\ds \int_0^{\to +\infty} e^{-s t} \cosh {a t} \rd t$

$\ds $

$=$

$\ds \intlimits {\frac {e^{-s t} \paren {-s \cosh a t + a \sinh a t} } {\paren {-s}^2 - a^2} } {t \mathop = 0} {t \mathop \to +\infty}$

$\ds $

$=$

$\ds 0 - \frac {-s \, \map \cosh {0 \times a} + a \, \map \sinh {0 \times a} } {s^2 - a^2}$

$\ds $

$=$

$\ds \frac {s \cosh 0 - a \sinh 0} {s^2 - a^2}$

simplifying

$\ds $

$=$

$\ds \frac s {s^2 - a^2}$