Enter Window or Step Function Values
Choose a function type. Then enter coefficient, shift, interval, power, rate, and evaluation point.
Example Data Table
The table shows typical inputs and expected transform forms.
| Case | Function | Inputs | Transform Form |
|---|---|---|---|
| Delayed step | 4u(t - 3) | C = 4, a = 3 | 4e-3s / s |
| Window pulse | 5[u(t - 2) - u(t - 7)] | C = 5, a = 2, b = 7 | 5(e-2s - e-7s) / s |
| Shifted power | 2(t - 1)3u(t - 1) | C = 2, a = 1, n = 3 | 12e-s / s4 |
| Window power | 3t2[u(t - 1) - u(t - 4)] | C = 3, a = 1, b = 4, n = 2 | ∫143t2e-stdt |
Formula Used
The unit step is written as u(t - a). It is zero before a. It is one after a.
Delayed constant: L{C u(t - a)} = C e-as / s.
Delayed power: L{C(t - a)nu(t - a)} = C n! e-as / sn+1.
Rectangular window: L{C[u(t - a) - u(t - b)]} = C(e-as - e-bs) / s.
Finite window: L{f(t)[u(t - a) - u(t - b)]} = ∫ab f(t)e-stdt.
How to Use This Calculator
- Select the function type from the list.
- Enter coefficient C.
- Enter shift or start value a.
- Enter end value b for window functions.
- Enter power n for polynomial cases.
- Enter rate k for exponential cases.
- Enter the s value for numerical evaluation.
- Press the calculate button.
- Review the formula, value, table, and graph.
- Use CSV or PDF buttons to save the output.
Understanding Window and Step Laplace Transforms
Why Step Functions Matter
Step functions describe signals that begin at a fixed time. They are common in control systems, circuits, and differential equations. A unit step lets you turn a function on after a delay. This makes piecewise models easier to write. It also makes transform work cleaner. Instead of solving separate regions by hand, you can use a compact shifted expression.
What a Window Function Does
A window function turns a signal on and then turns it off. The expression u(t - a) - u(t - b) creates that interval. It equals one between a and b. It equals zero outside that range. This is useful for pulses, gates, short loads, and temporary inputs. The calculator handles rectangular, exponential, and power based windows.
Using the Shifting Rule
The shifting rule is the key idea. A delayed signal adds an exponential multiplier to its transform. The multiplier is e to the negative a s. It records the start delay. For powers of t minus a, the calculator uses factorial terms. These terms come from the standard transform of a power.
Reading the Result
The output includes the original function, transform formula, and numeric value at your chosen s. The graph shows the time signal. This helps you confirm the interval and shape. The table gives export friendly values. The CSV button saves spreadsheet data. The PDF button creates a simple report for notes, homework, or documentation.
FAQs
1. What is a unit step function?
A unit step function is zero before a chosen time. It becomes one after that time. It helps model delayed signals and switched inputs.
2. What is a window function?
A window function keeps a signal active only between two points. The expression u(t - a) - u(t - b) creates that finite interval.
3. Why does e^-as appear in the answer?
The factor e^-as appears because the signal is delayed by a. It is the time shifting effect in the Laplace transform.
4. Can I evaluate a finite pulse?
Yes. Choose the window constant mode. Enter the coefficient, start time, end time, and s value. The calculator returns the pulse transform.
5. What does s mean?
The variable s is the complex frequency variable. This calculator evaluates formulas using a real positive s value for practical numerical output.
6. Can I use powers in a window?
Yes. Select the window power mode. The calculator evaluates the finite integral for t raised to an integer power.
7. Why must b be greater than a?
The window starts at a and ends at b. If b is not greater than a, the interval has no positive length.
8. What can I export?
You can export the function, transform formula, input values, conditions, and numerical answer. CSV is for spreadsheets. PDF is for reports.