Laplace Transform of Exponential times Cosine

Theorem

$\map {\laptrans {e^{b t} \cos a t} } s = \dfrac {s - b} {\paren {s - b}^2 + a^2}$

where:

$a$ and $b$ are real numbers

$s$ is a complex number with $\map \Re s > a + b$

$\laptrans f$ denotes the Laplace transform of $f$ evaluated at $s$.

Proof

\(\ds \map {\laptrans {e^{b t} \cos a t} } s\)

\(=\)

\(\ds \map {\laptrans {\cos a t} } {s - b}\)

Laplace Transform of Exponential times Function

\(\ds \)

\(=\)

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

Laplace Transform of Cosine

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 32$: Table of Special Laplace Transforms: $32.35$