Laplace Transform of Exponential times Function/Examples/Example 1

Examples of Use of Laplace Transform of Exponential times Function

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

$\laptrans {e^{-t} \cos 2 t} = \dfrac {s + 1} {s^2 + 2 s + 5}$

Proof

\(\ds \laptrans {e^{-t} \cos 2 t}\)

\(=\)

\(\ds \dfrac {\paren {s - \paren {-1} } } {\paren {s - \paren {-1} }^2 + 2^2}\)

Laplace Transform of Exponential times Function, Laplace Transform of Cosine

\(\ds \)

\(=\)

\(\ds \dfrac {s + 1} {s^2 + 2 s + 5}\)

simplification

$\blacksquare$

Sources

• 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Some Important Properties of Laplace Transforms: $2$. First translation or shifting property