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

\(\ds \map {\laptrans {e^{b t} \cos a t} } s\) \(=\) \(\ds \map {\laptrans {\cos a t} } {s - b}\) First Translation Property of Laplace Transforms
\(\ds \) \(=\) \(\ds \frac {s - b} {\paren {s - b}^2 + a^2}\) Laplace Transform of Cosine

$\blacksquare$


Sources

  • 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Appendix $\text B$: Table of Special Laplace Transforms: $11.$
  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 32$: Table of Special Laplace Transforms: $32.35$
  • 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 33$: Laplace Transforms: Table of Special Laplace Transforms: $33.35.$
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