Laplace Transform of Hyperbolic Cosine/Proof 2

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

\(\ds \laptrans {\cosh at}\)

\(=\)

\(\ds \laptrans {\frac {e^{at} + e^{-at} } 2}\)

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

\(\ds \)

\(=\)

\(\ds \frac 1 2 \paren {\frac {s + a + s - a} {\paren {s - a} \paren {s + a} } }\)

\(\ds \)

\(=\)

\(\ds \frac s {s^2 - a^2}\)

$\blacksquare$

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: $3 \ \text {(b)}$