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
- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Solved Problems: Laplace Transforms of some Elementary Functions: $3 \ \text {(a)}$