Laplace Transform of Hyperbolic Cosine/Proof 3

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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 $\map \Re s > a$.


Proof

\(\ds \laptrans {\sinh a t}\) \(=\) \(\ds \laptrans {\frac {e^{a t} + e^{-a t} } 2}\) Definition of Hyperbolic Cosine
\(\ds \) \(=\) \(\ds \int_0^{\to +\infty} e^{-s t} \paren {\frac {e^{a t} + e^{-a t} } 2} \rd t\) Definition of Laplace Transform
\(\ds \) \(=\) \(\ds \dfrac 1 2 \int_0^{\to +\infty} e^{-s t} e^{a t} \rd t + \dfrac 1 2 \int_0^{\to +\infty} e^{-s t} e^{-a t} \rd t\) Linear Combination of Laplace Transforms
\(\ds \) \(=\) \(\ds \dfrac 1 2 \laptrans {e^{a t} } + \dfrac 1 2 \laptrans {e^{-a t} }\) Definition of Laplace Transform
\(\ds \) \(=\) \(\ds \frac 1 2 \paren {\frac 1 {s - a} + \frac 1 {s + a} }\) Laplace Transform of Exponential
\(\ds \) \(=\) \(\ds \frac 1 2 \paren {\frac {s + a + s - a} {\paren {s - a} \paren {s + a} } }\)
\(\ds \) \(=\) \(\ds \frac s {s^2 - a^2}\)

$\blacksquare$


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