Library / Foundations: Maths / FMA-49
Foundations: Maths
Laplace transform of a cosine
Laplace transform of cos ωt for t ≥ 0, evaluated at a real value of s.
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- Foundations: Maths
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- F(s)
The Laplace transform of cos ωt for t ≥ 0 is s/(s² + ω²). It has the same poles as the sine pair but a zero at s = 0. Using it, a sinusoid that starts at a nonzero value can be transformed. The value here is evaluated at a real s.
Laplace transform of cos ωt for t ≥ 0, evaluated at a real value of s. It is the partner of the sine pair and is needed to include a nonzero starting value of a sinusoid.
Calculator
- Frequency in Hz, ω/2π0.4775 Hz
- Period, 2π/ω2.094 s
- Transform of the sine with the same ω and s (companion pair)0.2308
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- At s = 2 and ω = 3 rad/s the transform of cos ωt is 0.1538. The sine pair at the same s is 0.2308.
- The cosine starts at 1 and the sine at 0, which is why the two transforms have s and ω in the numerator.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| ω Angular frequency | 0.1 to 50 rad/s | 7.987e-4 | 0.4988 | 0.498 | +10% | −12.7 % |
| s Laplace variable s (real value) | 0.1 to 20 1/s | 0.0111 | 0.1661 | 0.155 | +10% | +3.32 % |
Sweep of Laplace variable s (real value)
| Laplace variable s (real value) (1/s) | Transform at s |
|---|---|
| 0.1 | 0.0111 |
| 2.587 | 0.1649 |
| 5.075 | 0.146 |
| 7.562 | 0.1143 |
| 10.05 | 0.09136 |
| 12.54 | 0.07544 |
| 15.02 | 0.064 |
| 17.51 | 0.05547 |
| 20 | 0.0489 |
Sweep one variable
Variables and how each one contributes
Transform at s
What the equation gives you. Everything else feeds it.
Laplace variable s (real value)
The Laplace variable s, shown here as a real value in 1/s. In general s is complex. Raises the value at small s and lowers it at large s, because s is in both the numerator and the denominator. Raising it by 0.5 (to 2.5) moves the result by +6.56 %, from 0.1538 to 0.1639.
Angular frequency
The angular frequency ω of the cosine, in rad/s. Lowers the value, because ω only appears in the denominator. Raising it by 0.5 rad/s (to 3.5 rad/s) moves the result by −20 %, from 0.1538 to 0.1231.
Worked example with the values above
- Square both: s² = 4 and ω² = 9, so the denominator is 13.
- The numerator is s = 2.
- F(s) = 2 / 13 = 0.1538.
Assumptions and limits
- The function is cos ωt for t ≥ 0 and zero before.
- s is in the region of convergence (real part greater than 0).
Common mistakes
- Using the cosine pair for sine, which has ω in the numerator.
- Ignoring the initial value of the cosine, which is 1, not 0.
Statistics and uncertainty
The pair is exact; as for the sine pair, the uncertain quantity in practice is the frequency ω.
Where it is used
- Transforming a cosine forcing term or a cosine response.
- Deriving the response of an undamped oscillator.
Maths you need
- Laplace transform: turns a function of time into a function of s
- Causal function: one that is zero before time zero