Advanced Laplace Heaviside Calculator

Model delayed signals with step-based transform tools. Compare polynomial, exponential, sine, and cosine cases quickly. See formulas, values, plots, and exports in one place.

Calculator Inputs

Choose a delayed function family, enter parameters, and calculate the corresponding Laplace transform.

Supported families use the second shifting theorem and generate a sample table and graph for the delayed function.

Example Data Table

These examples show typical delayed inputs and their resulting transforms.

Case Input Function Transform Result
Constant step 3u(t-2) 3e-2s/s
Delayed quadratic 2(t-1)2u(t-1) 4e-s/s3
Delayed exponential 5e0.5(t-4)u(t-4) 5e-4s/(s-0.5)
Delayed sine 4sin(3(t-2))u(t-2) 12e-2s/(s2+9)

Formula Used

The page is based on the second shifting theorem for Laplace transforms:

L{u(t-a)g(t-a)} = e-asG(s)

Supported base transforms

  • L{A} = A/s
  • L{Atn} = A n! / sn+1
  • L{Aebt} = A / (s-b)
  • L{A sin(wt)} = Aw / (s2 + w2)
  • L{A cos(wt)} = As / (s2 + w2)

How delay changes the answer

Once the time-domain function is rewritten as u(t-a)g(t-a), the transform of g(t) is multiplied by e-as.

That exponential term is the hallmark of a Heaviside delay in the Laplace domain.

How to Use This Calculator

  1. Select the delayed function family that matches your problem.
  2. Enter amplitude and the shift value a.
  3. Fill the extra parameter needed for the selected family.
  4. Enter an s value to get a numeric transform evaluation.
  5. Set the plotting range and sampling step.
  6. Click Calculate Transform to show results above the form.
  7. Review the symbolic transform, sample table, and Plotly graph.
  8. Use the CSV or PDF buttons to export the output.

Frequently Asked Questions

1) What does this calculator compute?

It evaluates delayed functions written with a unit step and returns the matching Laplace transform for the selected family. It also samples the time-domain signal, plots it, and offers exports.

2) What is the Heaviside step function?

The Heaviside step, u(t-a), turns a function on at t=a. Before the shift point the output is zero; after it, the delayed expression becomes active.

3) Why does a delay create e-as in the transform?

Because the second shifting theorem states L{u(t-a)g(t-a)} = e-asG(s). The delay in time becomes exponential damping in the transform domain.

4) Should the shift always be nonnegative?

This version is intended for causal delays, so nonnegative shifts are most meaningful. Using a nonnegative shift keeps the plotted signal and formulas aligned with standard engineering and maths examples.

5) Why can the evaluated transform be undefined?

The numeric value becomes undefined at poles, such as s=b for delayed exponentials or s=0 for constant and polynomial modes. The symbolic formula still remains useful.

6) What does the graph show?

The chart shows the sampled delayed signal over time, plus the unit step indicator. It helps confirm when the function activates and how its shape changes after the shift.

7) Which function family should I choose?

Use polynomial for ramps and powers, exponential for growth or decay, sine for oscillations, cosine for phase-shifted oscillations, and constant for step-amplitude cases.

8) Can I use the exported files in coursework or reports?

Yes. The CSV file is convenient for spreadsheets, while the PDF captures the summary, formulas, and sample table for assignments, notes, or documentation.

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Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.