# Laplace Transform of Hyperbolic Sine/Proof 3

## 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 $\ds$ $=$ $\ds \int_0^{\to +\infty} e^{-s t} \paren {\frac {e^{a t} - e^{-a t} } 2} \rd t$ 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 $\ds$ $=$ $\ds \frac 1 2 \paren {\frac {s + a - s + a} {\paren {s - a} \paren {s + a} } }$ $\ds$ $=$ $\ds \frac a {s^2 - a^2}$

$\blacksquare$