Laplace Transform of Hyperbolic Sine/Proof 2

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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

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

$\blacksquare$


Sources