Laplace Transform of Hyperbolic Cosine

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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 1

\(\displaystyle \laptrans {\cosh {a t} }\) \(=\) \(\displaystyle \int_0^{\to +\infty} e^{-s t} \cosh {a t} \rd t\) Definition of Laplace Transform
\(\displaystyle \) \(=\) \(\displaystyle \intlimits {\frac {e^{-s t} \paren {-s \cosh a t + a \sinh a t} } {\paren {-s}^2 - a^2} } {t \mathop = 0} {t \mathop \to +\infty}\) Primitive of $e^{a x} \cosh b x$
\(\displaystyle \) \(=\) \(\displaystyle 0 - \frac {-s \, \map \cosh {0 \times a} + a \, \map \sinh {0 \times a} } {s^2 - a^2}\) Exponential Tends to Zero, Exponential of Zero
\(\displaystyle \) \(=\) \(\displaystyle \frac {s \cosh 0 - a \sinh 0} {s^2 - a^2}\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle \frac s {s^2 - a^2}\) Hyperbolic Sine of Zero is Zero, Hyperbolic Cosine of Zero is One


Proof 2

\(\displaystyle \laptrans {\cosh at}\) \(=\) \(\displaystyle \laptrans {\frac {e^{at} + e^{-at} } 2}\) Definition of Hyperbolic Cosine
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 2 \paren {\laptrans {e^{at} } + \laptrans {e^{-at} } }\) Linear Combination of Laplace Transforms
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 2 \paren {\frac 1 {s - a} + \frac 1 {s + a} }\) Laplace Transform of Exponential
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 2 \paren {\frac {s + a + s - a} {\paren {s - a} \paren {s + a} } }\)
\(\displaystyle \) \(=\) \(\displaystyle \frac s {s^2 - a^2}\)

$\blacksquare$


Proof 3

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

$\blacksquare$


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