Calculator
This tool applies the second shifting theorem to delayed functions written with a unit step term.
Formula Used
The calculator uses the second shifting theorem:
If f(t) = A · u(t-a) · g(t-a), then L{f(t)} = A · e^(-as) · G(s)
where G(s) = L{g(t)}
Supported base transforms:
- 1: G(s) = 1 / s
- t: G(s) = 1 / s²
- tⁿ: G(s) = n! / sⁿ⁺¹
- e^(bt): G(s) = 1 / (s - b)
- sin(ωt): G(s) = ω / (s² + ω²)
- cos(ωt): G(s) = s / (s² + ω²)
The delay contributes the exponential factor e-as, while the chosen function determines the remaining transform term.
How to Use This Calculator
- Enter the amplitude that multiplies the delayed signal.
- Enter the delay value a for the unit step u(t-a).
- Select the function that starts after the delay.
- Fill exponent, rate, or frequency inputs when needed.
- Optionally enter a numeric s value for direct evaluation.
- Press Calculate Transform to display the symbolic and numeric results above the form.
- Use the export buttons to save the result as CSV or PDF.
Example Data Table
| Amplitude A | Shift a | Base Function | Time-domain Form | Laplace Transform |
|---|---|---|---|---|
| 1 | 2 | 1 | u(t-2) | e-2s / s |
| 3 | 1 | (t-a) | 3u(t-1)(t-1) | 3e-s / s² |
| 2 | 4 | sin(5(t-a)) | 2u(t-4)sin(5(t-4)) | 10e-4s / (s² + 25) |
| 1.5 | 0.5 | e^(3(t-a)) | 1.5u(t-0.5)e^(3(t-0.5)) | 1.5e-0.5s / (s - 3) |
Frequently Asked Questions
1. What does the unit step do in this calculator?
It delays when the function becomes active. Before t = a, the expression is zero. After t = a, the chosen function starts and the transform gains an e-as factor.
2. Why is the result multiplied by e-as?
That factor comes from the second shifting theorem. Any delayed signal written as u(t-a)g(t-a) transforms into e-asG(s), where G(s) is the transform of the undelayed base function.
3. Can I evaluate the transform at a specific s value?
Yes. Enter a numeric s value and the calculator evaluates the symbolic transform directly. The value is meaningful only when the chosen s satisfies the relevant convergence condition.
4. What functions are supported?
The tool supports constants, linear ramps, powers, exponentials, sine functions, and cosine functions after the delay. These cover many common Laplace transform exercises in engineering and mathematics.
5. What is the difference between (t-a) and t?
Inside a delayed expression, the active function is written relative to the shift. So the correct form is usually g(t-a), not g(t). This keeps the second shifting theorem valid.
6. Why might a numeric result be invalid?
Some transforms converge only for certain s values. For example, ebt needs Re(s) > b. If your chosen s breaks that condition, the numeric evaluation may be undefined or not physically useful.
7. What does the amplitude change?
Amplitude scales the entire signal. In the transform, it multiplies the delayed expression directly, so both the symbolic result and any numeric evaluation increase or decrease proportionally.
8. When should I use CSV or PDF export?
Use CSV when you want tabular data for spreadsheets or reports. Use PDF when you want a clean summary of the current result for homework, notes, or printed study material.