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
\(\ds \laptrans {\cosh {a t} }\) | \(=\) | \(\ds \int_0^{\to +\infty} e^{-s t} \cosh {a t} \rd t\) | Definition of Laplace Transform | |||||||||||
\(\ds \) | \(=\) | \(\ds \intlimits {\frac {e^{-s t} \paren {-s \cosh a t + a \sinh a t} } {\paren {-s}^2 - a^2} } {t \mathop = 0} {t \mathop \to +\infty}\) | Primitive of $e^{a x} \cosh b x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 0 - \frac {-s \, \map \cosh {0 \times a} + a \, \map \sinh {0 \times a} } {s^2 - a^2}\) | Exponential Tends to Zero, Exponential of Zero | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {s \cosh 0 - a \sinh 0} {s^2 - a^2}\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac s {s^2 - a^2}\) | Hyperbolic Sine of Zero is Zero, Hyperbolic Cosine of Zero is One |
Proof 2
\(\ds \laptrans {\cosh at}\) | \(=\) | \(\ds \laptrans {\frac {e^{at} + e^{-at} } 2}\) | Definition of Hyperbolic Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \paren {\laptrans {e^{at} } + \laptrans {e^{-at} } }\) | Linear Combination of Laplace Transforms | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \paren {\frac 1 {s - a} + \frac 1 {s + a} }\) | Laplace Transform of Exponential | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \paren {\frac {s + a + s - a} {\paren {s - a} \paren {s + a} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac s {s^2 - a^2}\) |
$\blacksquare$
Proof 3
\(\ds \laptrans {\cosh a t}\) | \(=\) | \(\ds \laptrans {\frac {e^{a t} + e^{-a t} } 2}\) | 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 | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2 \paren {\frac {s + a + s - a} {\paren {s - a} \paren {s + a} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac s {s^2 - a^2}\) |
$\blacksquare$
Sources
- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Laplace Transforms of some Elementary Functions: $8$.
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 32$: Table of Special Laplace Transforms: $32.37$