Laplace Transform of t·e^(at)
The Laplace transform of t·e^(at) is 1/(s − a)². See the table rule behind it, then edit the function to transform anything else.
L{t·e^(at)} = 1/(s − a)². Multiplying by t raises the power of the shifted pole: 1/(s − a)².
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{t·e^(at)} = 1/(s − a)²
Worked example with numbers: L{t·e^(2t)} = 1/(s - 2)^2
- L{t·e^(2t)} = 1/(s - 2)^2 Table rule: L{t^1e^(at)} = 1!/(s − a)^2
Multiplying by t raises the power of the shifted pole: 1/(s − a)².
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Frequently asked questions
What is the Laplace transform of t·e^(at)? ▾
L{t·e^(at)} = 1/(s − a)². Multiplying by t raises the power of the shifted pole: 1/(s − a)².
Which rule gives the transform of t·e^(at)? ▾
L{t^1e^(at)} = 1!/(s − a)^2 applied to t·e^(2t).
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 t·e^(at) is 1/(s − a)². See the table rule behind it, then edit the function to transform anything else.