Laplace Transform of a Heaviside Function Calculator

Calculator inputs

Choose a delayed base function, then compute its transformed expression and sample value.

Amplitude A

Delay a

Base function

Polynomial order n

Secondary parameter

Sample s value

Graph end time

Example data table

Input function	Base transform	Shifted Laplace transform
2u(t - 3)	L{1} = 1/s	2e^-3s/s
u(t - 1)(t - 1)²	L{t²} = 2/s³	2e^-s/s³
4u(t - 2)sin(5(t - 2))	L{sin(5t)} = 5/(s² + 25)	20e^-2s/(s² + 25)
3u(t - 4)e^{2(t - 4)}	L{e^2t} = 1/(s - 2)	3e^-4s/(s - 2)

Formula used

Second shifting theorem
If L{f(t)} = F(s), then L{u(t-a)f(t-a)} = e^-asF(s).

Common base transforms
L{1} = 1/s
L{tⁿ} = n!/sⁿ⁺¹
L{e^bt} = 1/(s-b)
L{sin(ωt)} = ω/(s²+ω²)
L{cos(ωt)} = s/(s²+ω²)

How this calculator applies the rule
First, it transforms the base function. Next, it multiplies by e^-as. Finally, it reports the shifted expression, convergence region, sample value, and graph of the delayed time function.

How to use this calculator

Enter the amplitude A.
Set the delay a for the unit step.
Select the base function type.
Enter a polynomial order or secondary parameter when needed.
Choose a positive sample value for s.
Set the graph end time.
Press the calculate button.
Review the transform, ROC, value table, and graph.
Use the export buttons to save CSV or PDF output.

Frequently asked questions

1) What does the Heaviside function do here?

It turns the base function on at time t = a. Before that instant, the function equals zero. After that instant, the delayed expression becomes active.

2) Why does the answer include e^-as?

That factor comes from the second shifting theorem. A time delay in the original function becomes multiplication by e^-as in the Laplace domain.

3) Can I use zero delay?

Yes. When a = 0, the unit step starts immediately. The shift factor becomes one, so the result reduces to the ordinary Laplace transform of the base function.

4) What values of s are valid?

That depends on the base function. Constants, powers, sine, and cosine need Re(s) > 0. Exponentials need Re(s) > b.

5) What does polynomial order mean?

It is the power on the delayed term. For example, order 3 means the calculator uses (t-a)³ inside the shifted function.

6) Are sine and cosine also delayed?

Yes. The calculator uses sin(ω(t-a)) or cos(ω(t-a)) after the switching time. That keeps the theorem application consistent.

7) What does the graph show?

It plots the time-domain function, not the transform. You see the zero region before the delay and the active waveform after the delay.

8) Can I export the computed result?

Yes. Use the CSV button for tabular output and the PDF button for a clean summary document containing the main settings and final result.