Laplace Transform of Exponential times Sine

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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

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

$\blacksquare$


Sources