Laplace Transform of Exponential times Sine
Jump to navigation
Jump to search
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 |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 32$: Table of Special Laplace Transforms: $32.34$