Laplace Transform of Exponential times Sine

Theorem

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

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 $s$.

$\ds \map {\laptrans {e^{b t} \sin a t} } s$

$=$

$\ds \map {\laptrans {\sin a t} } {s - b}$

$\ds $

$=$

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

$\blacksquare$

can also be seen presented in the form:

$\laptrans {\dfrac {e^{b t} \sin a t} a} = \dfrac 1 {\paren {s - b}^2 + a^2}$

1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Appendix $\text B$: Table of Special Laplace Transforms: $10.$
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 32$: Table of Special Laplace Transforms: $32.34$
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: $33.34.$