Laplace Transform of e^(at)·sin(bt)
The Laplace transform of e^(at)·sin(bt) is b/((s − a)² + b²). See the table rule behind it, then edit the function to transform anything else.
L{e^(at)·sin(bt)} = b/((s − a)² + b²). First shifting theorem: replace s with s − a in the sine transform.
Use t as the variable. Examples: t^3, e^(-2t), sin(3t), t*cos(2t), 4t^2+3. Write δ(t) as delta(t).
The result
L{e^(at)·sin(bt)} = b/((s − a)² + b²)
Worked example with numbers: L{e^(2t)·sin(3t)} = 3/((s - 2)^2 + 9)
- L{e^(2t)·sin(3t)} = 3/((s - 2)^2 + 9) Table rule: L{e^(at)sin(bt)} = b/((s − a)² + b²)
First shifting theorem: replace s with s − a in the sine transform.
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Frequently asked questions
What is the Laplace transform of e^(at)·sin(bt)? ▾
L{e^(at)·sin(bt)} = b/((s − a)² + b²). First shifting theorem: replace s with s − a in the sine transform.
Which rule gives the transform of e^(at)·sin(bt)? ▾
L{e^(at)sin(bt)} = b/((s − a)² + b²) applied to e^(2t)·sin(3t).
How do I get back from F(s) to f(t)? ▾
Use the inverse Laplace transform. Split F(s) into partial fractions and invert each piece with the same table read backwards — the inverse Laplace transform calculator does this step by step.
About this Calculator
The Laplace transform of e^(at)·sin(bt) is b/((s − a)² + b²). See the table rule behind it, then edit the function to transform anything else.