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 $s$.

Proof

 $\ds \map {\laptrans {e^{b t} \sin a t} } s$ $=$ $\ds \map {\laptrans {\sin a t} } {s - b}$ Laplace Transform of Exponential times Function $\ds$ $=$ $\ds \frac a {\paren {s - b}^2 + a^2}$ Laplace Transform of Sine

$\blacksquare$