Laplace Transform Calculator
Use this Laplace Transform Calculator to enter values, adjust options, and review results in a compact responsive workspace.
Results are calculated automatically as you enter data.
Piecewise input
Use one interval per line, for example:
0 to 2: t
Derivation
Laplace transform
▼ See explanations and tips below ▼
Related Calculators
What Is a Laplace Transform?
A Laplace transform is a way to convert a function of time, usually written as \(f(t)\), into a function of a new variable, usually written as \(F(s)\). In many courses and applications, \(t\) represents time, while \(s\) represents the transform-domain variable.
The transform is defined by an improper integral:
$$
F(s)=\mathcal{L}\{f(t)\}=\int_0^\infty e^{-st}f(t)\,dt
$$
This formula says: multiply the time-domain function by the decaying exponential \(e^{-st}\), then integrate from \(0\) to infinity. When the integral converges, the result is a new expression in terms of \(s\).
The main value of the Laplace transform is that it often changes difficult time-domain problems into simpler algebraic problems. This is especially useful in differential equations, control systems, circuit analysis, signal processing, and engineering mathematics.
For example, derivatives in a differential equation can become polynomial expressions in \(s\). A forcing function that turns on at a certain time can be represented with a unit step function. A piecewise input can often be handled by integrating over the intervals where the function has different formulas.
Why Laplace Transforms Matter
Laplace transforms are useful because they connect two different views of the same problem:
- The time-domain view, where a function describes how something changes with \(t\).
- The s-domain view, where the transformed function can be easier to manipulate algebraically.
Students often meet Laplace transforms while solving differential equations with initial conditions. Engineers use them to analyze systems, circuits, transfer functions, and inputs that start, stop, jump, or change over time.
They are also practical because many common functions have standard transform rules. Instead of evaluating the defining integral every time, you can use a transform table and the linearity property to build the answer term by term.
Key Terms to Know
- Time-domain function: The original function, written as \(f(t)\).
- Transform-domain function: The transformed function, written as \(F(s)\).
- Transform pair: A matched pair such as \(f(t) \leftrightarrow F(s)\).
- Linearity: The property that lets you transform sums and constant multiples term by term.
- Unit step function: A function that is \(0\) before a chosen time and \(1\) from that time onward.
- Heaviside function: Another common name for the unit step function.
- Piecewise function: A function defined by different formulas on different intervals.
- Region of convergence: The set of \(s\) values for which the Laplace integral converges.
- Transform table: A list of standard functions and their Laplace transforms.
How Laplace Transforms Work
The definition of the Laplace transform is based on an integral, but many practical calculations use rules that come from that definition.
The most important rule is linearity:
$$
\mathcal{L}\{A f(t)+B g(t)\}
=
A\mathcal{L}\{f(t)\}+B\mathcal{L}\{g(t)\}
$$
where \(A\) and \(B\) are constants.
Linearity means that a function such as \(3t^2+4t+7\) can be transformed one term at a time. The transform of the whole sum is the sum of the transforms of the separate terms.
Common transform rules include:
| Time-domain function | Laplace transform |
|---|---|
| \(c\) | \(\frac{c}{s}\) |
| \(t\) | \(\frac{1}{s^2}\) |
| \(t^n\), where \(n\) is a nonnegative integer | \(\frac{n!}{s^{n+1}}\) |
| \(e^{at}\) | \(\frac{1}{s-a}\) |
| \(\sin(at)\) | \(\frac{a}{s^2+a^2}\) |
| \(\cos(at)\) | \(\frac{s}{s^2+a^2}\) |
| \(u(t-a)\) | \(\frac{e^{-as}}{s}\) |
These rules have convergence conditions, even when the condition is not written beside every formula. For example, the transform of \(e^{at}\) requires \(s>a\). In rigorous work, always check the region of convergence or the assumptions required by your course or textbook.
Unit Step and Piecewise Functions
The unit step function \(u(t-a)\) turns on at \(t=a\):
$$
u(t-a)=
\begin{cases}
0, & t
Its Laplace transform is:
$$
\mathcal{L}\{u(t-a)\}=\frac{e^{-as}}{s}
$$
Piecewise functions can also be transformed from the integral definition by splitting the integral over the relevant intervals.
For example, if a function equals \(1\) from \(t=a\) to \(t=b\) and is \(0\) elsewhere on that interval contribution, then:
$$
\int_a^b e^{-st}\,dt
=
\frac{e^{-as}-e^{-bs}}{s}
$$
If a function equals \(t\) from \(t=a\) to \(t=b\), then the interval contribution is:
$$
\int_a^b t e^{-st}\,dt
=
\frac{e^{-as}(as+1)}{s^2}
-
\frac{e^{-bs}(bs+1)}{s^2}
$$
When the upper endpoint is infinity, the upper-end term disappears if the integral converges.
Examples of Laplace Transforms in Practice
Example 1: Transforming a Polynomial
Suppose:
$$
f(t)=3t^2+4t+7
$$
Use linearity:
$$
\mathcal{L}\{3t^2+4t+7\}
=
3\mathcal{L}\{t^2\}
+
4\mathcal{L}\{t\}
+
7\mathcal{L}\{1\}
$$
Now apply the standard rules:
$$
\mathcal{L}\{t^2\}=\frac{2!}{s^3}=\frac{2}{s^3}
$$
$$
\mathcal{L}\{t\}=\frac{1}{s^2}
$$
$$
\mathcal{L}\{1\}=\frac{1}{s}
$$
So:
$$
F(s)=\frac{6}{s^3}+\frac{4}{s^2}+\frac{7}{s}
$$
This example shows the basic workflow: split the expression into supported terms, transform each term, then add the results.
Example 2: Transforming Sine and Cosine Terms
Suppose:
$$
f(t)=4\sin(2t)+3\cos(5t)
$$
Use the sine rule:
$$
\mathcal{L}\{\sin(at)\}=\frac{a}{s^2+a^2}
$$
For \(\sin(2t)\):
$$
\mathcal{L}\{\sin(2t)\}=\frac{2}{s^2+4}
$$
Multiplying by \(4\) gives:
$$
4\mathcal{L}\{\sin(2t)\}=\frac{8}{s^2+4}
$$
Use the cosine rule:
$$
\mathcal{L}\{\cos(at)\}=\frac{s}{s^2+a^2}
$$
For \(\cos(5t)\):
$$
\mathcal{L}\{\cos(5t)\}=\frac{s}{s^2+25}
$$
Multiplying by \(3\) gives:
$$
3\mathcal{L}\{\cos(5t)\}=\frac{3s}{s^2+25}
$$
Therefore:
$$
F(s)=\frac{8}{s^2+4}+\frac{3s}{s^2+25}
$$
Example 3: A Unit Step Input
Suppose:
$$
f(t)=2u(t-3)
$$
The function is \(0\) before \(t=3\) and \(2\) from \(t=3\) onward. Using the unit step rule:
$$
\mathcal{L}\{u(t-3)\}=\frac{e^{-3s}}{s}
$$
Multiplying by \(2\) gives:
$$
F(s)=\frac{2e^{-3s}}{s}
$$
The factor \(e^{-3s}\) signals a shift or delay in the time domain.
Example 4: A Simple Piecewise Function
Suppose a function is defined as:
$$
f(t)=
\begin{cases}
t, & 0\le t<2\\
1, & t\ge 2
\end{cases}
$$
Split the transform integral across the two intervals:
$$
F(s)=\int_0^2 t e^{-st}\,dt+\int_2^\infty e^{-st}\,dt
$$
For the interval where the function is \(t\):
$$
\int_0^2 t e^{-st}\,dt
=
\frac{1}{s^2}
-
\frac{e^{-2s}(2s+1)}{s^2}
$$
For the interval where the function is \(1\):
$$
\int_2^\infty e^{-st}\,dt
=
\frac{e^{-2s}}{s}
$$
So one valid term-by-term form is:
$$
F(s)=
\frac{1}{s^2}
-
\frac{e^{-2s}(2s+1)}{s^2}
+
\frac{e^{-2s}}{s}
$$
A symbolic algebra system may simplify this expression further, but the interval-by-interval form is often easier to connect back to the original piecewise definition.
How to Interpret the Result
The outputs \(\mathcal{L}\{f(t)\}\) and \(F(s)\) mean the same thing: the Laplace transform of the input function.
The original function tells you what was transformed. The transform-domain result tells you how that function appears in the \(s\) domain. The rules or steps help identify which standard transform entries were applied.
A result containing powers of \(s\), such as \(\frac{1}{s^2}\) or \(\frac{6}{s^3}\), often comes from polynomial terms in \(t\). A result containing \(s^2+a^2\) in the denominator often comes from sine or cosine terms. A result containing \(e^{-as}\) usually indicates a delayed or shifted feature, such as a unit step or a finite piecewise interval beginning at \(t=a\).
A validation message means no valid transform was produced for the entered form. In that case, the displayed result should not be treated as a mathematical answer.
Common Mistakes and Misconceptions
One common mistake is thinking that the Laplace transform changes products into products. Linearity applies to sums and constant multiples, but in general:
$$
\mathcal{L}\{f(t)g(t)\}
\ne
\mathcal{L}\{f(t)\}\mathcal{L}\{g(t)\}
$$
Another common mistake is ignoring convergence. A formula may be correct only for certain values of \(s\). For many beginner calculations this condition is handled in the background, but rigorous work should include it.
Other practical mistakes include:
-
Using
**instead of^for powers. - Entering negative or fractional powers such as \(t^{-1}\) or \(t^{0.5}\).
- Using symbolic parameters such as \(a\) in \(\sin(at)\) when a numeric value is expected.
-
Omitting explicit multiplication, for example typing
3tinstead of3*t. -
Entering symbolic parameters such as
exp(a*t); parameters must be numeric. - Filling both the standard and piecewise fields; choose exactly one input mode.
- Treating the displayed sufficient ROC as a proof for expressions outside the supported scope.
- Expecting every result to be algebraically simplified into one fraction.
When to Use Laplace Transforms
Use Laplace transforms when you need to:
- Convert common time-domain functions into \(s\)-domain expressions.
- Solve or check work in differential equations.
- Analyze systems with step inputs, delays, or piecewise forcing functions.
- Work with signals, circuits, transfer functions, or engineering models.
- Build a transform from standard table entries using linearity.
For quick educational checks, a rule-based calculator is helpful. For advanced symbolic work, proofs, inverse transforms, convolution, derivatives, arbitrary products, or full convergence analysis, use a complete symbolic math system or consult a textbook.
Limitations and Things to Keep in Mind
Laplace transforms are powerful, but not every function has a Laplace transform. The defining integral must converge for the values of \(s\) being considered. Functions that grow too quickly may fail to have a transform in the usual sense.
For piecewise functions, the exact value at a jump discontinuity usually does not affect the Laplace transform, because a single point does not change the value of the integral. The intervals and formulas on those intervals matter much more than the value assigned at the jump point.
For this calculator, keep these practical limits in mind:
- It is designed for a fixed set of common transform forms, not for arbitrary symbolic expressions.
- It supports exact numeric constants, nonnegative integer powers of \(t\), exponentials with numeric parameters, sine and cosine with numeric parameters, unit-step terms, sums and differences, and an exponential multiplied by a supported polynomial or trigonometric term.
- Negative powers, fractional powers, symbolic parameters, general products, quotients, convolutions, derivatives, and nested expressions are not handled as general symbolic transformations.
- Piecewise support is limited to interval bodies equal to \(0\), \(1\), or \(t\).
- Piecewise intervals must be ordered and non-overlapping. Gaps are interpreted as zero, and an infinite endpoint must occur only on the final interval.
- The output may not be simplified into a single rational expression.
- Decimal, scientific-notation, and fractional numeric coefficients are retained exactly in the symbolic result.
- The result displays a sufficient right-half-plane region of convergence for supported inputs; it is not a general convergence proof.
- Inputs are bounded to 500 characters, 300 parsed nodes, 40 expanded terms, 50 piecewise intervals, integer exponents no larger than 10,000, exact numeric coefficients no longer than 20,000 decimal digits, and exact result strings no longer than 50,000 characters.
- The calculator does not compute inverse Laplace transforms.
When a Laplace transform is part of a graded assignment, engineering analysis, safety-related calculation, or professional model, double-check the result and the convergence assumptions.
How to Use This Calculator
- Enter a supported time-domain expression using \(t\) as the variable.
-
Use
*for multiplication and^for powers, such as3*t^2 + 4*t + 7. -
Use supported forms such as constants,
t^n,exp(a*t),sin(a*t),cos(a*t), andu(t-a), with numeric values fora. -
For piecewise functions, enter one interval per line, such as
0 to 2: tor2 to inf: 1. - Use either the standard expression field or the piecewise field. If both contain input, the calculator asks you to clear one.
- Review the original function, the \(\mathcal{L}\{f(t)\}\) result, the \(F(s)\) result, the sufficient ROC, and the listed rules or steps.
- Use the clear action to reset the inputs if you want to start over.
Frequently Asked Questions
What does \(F(s)\) mean?
\(F(s)\) is the Laplace transform of \(f(t)\). It represents the same function information in the \(s\) domain rather than the time domain.
Why does the Laplace transform start at \(0\)?
The common one-sided Laplace transform integrates from \(0\) to infinity, which fits many initial-value problems and time-based applications. It focuses on the behavior of a system from the starting time onward.
Is the Laplace transform the same as the Fourier transform?
No. The two transforms are related, but they are not the same tool. The Laplace transform uses the factor \(e^{-st}\) and is especially useful for initial-value problems, differential equations, and systems with growth or decay conditions.
Why do unit step transforms contain \(e^{-as}\)?
The factor \(e^{-as}\) is associated with a delay or shift that begins at \(t=a\). In a unit step term, it indicates that the function turns on later rather than at \(t=0\).
Why does the calculator reject some mathematically valid expressions?
The calculator uses a limited set of supported transform rules. A function may have a valid Laplace transform mathematically but still fall outside the supported input forms.
Does the result include the region of convergence?
Yes, it reports a sufficient right-half-plane condition for supported expressions, such as \(\operatorname{Re}(s)>a\). For rigorous or unsupported work, determine the full region of convergence independently.
Can this calculator find inverse Laplace transforms?
No. It calculates forward Laplace transforms for supported input forms. Use an inverse Laplace transform resource or a symbolic math tool when you need to return from \(F(s)\) to \(f(t)\).
Sources and References
Books
- Jiří Lebl. Notes on Diffy Qs: Differential Equations for Engineers. Open educational web version. Sections 6.1, 6.2, and Appendix B. Accessed June 29, 2026. https://www.jirka.org/diffyqs/
- William F. Trench. Elementary Differential Equations with Boundary Value Problems. Trinity University / LibreTexts edition. Chapter 8, especially Sections 8.1, 8.4, and 8.8. Accessed June 29, 2026. https://math.libretexts.org/Bookshelves/DifferentialEquations/ElementaryDifferentialEquationswithBoundaryValueProblems(Trench))
- Russell Herman. A First Course in Differential Equations for Scientists and Engineers. LibreTexts edition. Chapter 5, especially Sections 5.1 and 5.4. Accessed June 29, 2026. https://math.libretexts.org/Bookshelves/DifferentialEquations/AFirstCourseinDifferentialEquationsforScientistsandEngineers_(Herman))
Online and Official Sources
- MIT OpenCourseWare. “Laplace Transform: Basics.” 18.03SC Differential Equations, Fall 2011. Accessed June 29, 2026. https://ocw.mit.edu/courses/18-03sc-differential-equations-fall-2011/pages/unit-iii-fourier-series-and-laplace-transform/laplace-transform-basics/
- MIT OpenCourseWare. “Unit Step and Unit Impulse Response.” 18.03SC Differential Equations, Fall 2011. Accessed June 29, 2026. https://ocw.mit.edu/courses/18-03sc-differential-equations-fall-2011/pages/unit-iii-fourier-series-and-laplace-transform/unit-step-and-unit-impulse-response/