Calculator inputs
This tool applies the second shifting theorem to expressions of the form u(t - a)f(t - a).
Formula used
The calculator uses the second shifting theorem for Laplace transforms:
Common base transforms used here are:
| Base function f(t) | Base transform F(s) |
|---|---|
| A | A / s |
| Atn | A·n! / sn+1 |
| Aebt | A / (s - b) |
| A sin(bt) | Ab / (s2 + b2) |
| A cos(bt) | As / (s2 + b2) |
How to use this calculator
- Enter the delay time a where the unit step becomes active.
- Select the delayed function type you want to transform.
- Set the amplitude and any needed coefficient or polynomial power.
- Choose one evaluation point s for a numeric transform check.
- Set the graph range and number of sample points.
- Press the calculate button to display formulas, values, and the plot above the form.
- Review the region of convergence before interpreting numeric values.
- Use CSV or PDF export if you want a record of the computed results.
Example data table
| Case | Delayed expression | Laplace transform | ROC |
|---|---|---|---|
| Constant | u(t - 2)·3 | 3e-2s/s | Re(s) > 0 |
| Polynomial | u(t - 1)·4(t - 1)2 | 8e-s/s3 | Re(s) > 0 |
| Exponential | u(t - 3)·2e1.5(t - 3) | 2e-3s/(s - 1.5) | Re(s) > 1.5 |
| Sine | u(t - 2)·5sin(3(t - 2)) | 15e-2s/(s2 + 9) | Re(s) > 0 |
| Cosine | u(t - 0.5)·6cos(4(t - 0.5)) | 6se-0.5s/(s2 + 16) | Re(s) > 0 |
FAQs
1) What does this calculator transform?
It transforms delayed functions written as u(t - a)f(t - a). That form activates the function at time a and then applies the second shifting theorem directly.
2) Why does the result contain e-as?
The exponential factor appears because delaying a time-domain signal by a units multiplies its Laplace transform by e-as. This is the core idea behind the second shifting theorem.
3) What is the region of convergence?
The region of convergence states where the transform is valid in the s-domain. It depends on the underlying base function, not only on the delay.
4) Why are some graph values missing?
Missing values usually mean the sample lies outside the region of convergence or too close to a pole. The calculator skips those points to avoid misleading output.
5) What does the coefficient field mean?
The coefficient changes meaning with the chosen function. It is the growth rate for exponentials and the angular frequency for sine or cosine models.
6) Can I use a negative step time?
This version expects a nonnegative delay because that is the most common step-function form in applied Laplace problems. Negative values are blocked to keep interpretation consistent.
7) Are the outputs symbolic or numeric?
Both are provided. You get a symbolic transform expression, a numeric evaluation at one selected s value, a graph, and a computed sample table.
8) What does the CSV or PDF export contain?
CSV export includes the graph data points. PDF export summarizes the chosen inputs, transform formula, region of convergence, and a preview table of sampled values.