Laplace Transform of Hyperbolic Sine

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Theorem

Let $\sinh t$ be the hyperbolic sine, where $t$ is real.

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


Then:

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

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


Proof 1

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

$\blacksquare$


Proof 2

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

$\blacksquare$


Proof 3

\(\displaystyle \laptrans {\sinh a t}\) \(=\) \(\displaystyle \laptrans {\frac {e^{a t} - e^{-a t} } 2}\) Definition of Hyperbolic Sine
\(\displaystyle \) \(=\) \(\displaystyle \int_0^{\to +\infty} e^{-s t} \paren {\frac {e^{a t} - e^{-a t} } 2} \rd t\) 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
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 2 \paren {\frac {s + a - s + a} {\paren {s - a} \paren {s + a} } }\)
\(\displaystyle \) \(=\) \(\displaystyle \frac a {s^2 - a^2}\)

$\blacksquare$


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