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Laplace Transform

Searchable table of common transform pairs and properties.

Laplace Transform Pairs

Common time-domain functions and their s-domain transforms.

Table of Laplace transform pairs
f(t) F(s) = ℒ{f(t)} Rule
No transform pairs match your search.
Reference table of common Laplace transform pairs.

About this topic

Laplace Transform Definition Convert time-domain functions into the s-domain.

The Laplace transform converts a function f(t) of time into a function F(s) of a complex variable s. This turns differential equations into algebraic ones.

Key formulas & rules

  • •Formula: F(s) = ∫₀^∞ f(t)·e^(−st) dt
  • •Converts calculus → algebra
  • •s is a complex frequency variable
  • •Only defined for t ≥ 0 (one-sided)
  • •Inverse: f(t) = (1/2πi) ∫ F(s)·e^(st) ds
  • •Denoted by ℒ{f(t)} = F(s)

Examples

Input Output Note
ℒ{1} 1/s Constant
ℒ{e^(2t)} 1/(s−2) Exponential
ℒ{sin(3t)} 3/(s²+9) Sinusoid
ℒ{t} 1/s² Ramp

Real-world uses

Circuit analysis Control systems Signal processing
Linearity Property Laplace transform is a linear operator.

Because integration is linear, the Laplace transform is linear: the transform of a sum is the sum of transforms, and constants can be pulled out.

Key formulas & rules

  • •ℒ{a·f(t) + b·g(t)} = a·F(s) + b·G(s)
  • •Constants factor out: ℒ{c·f(t)} = c·F(s)
  • •Enables solving linear ODEs
  • •Basis for superposition principle
  • •Works with infinite series
  • •Linear but not multiplicative

Examples

Input Output Note
ℒ{3 + 2t} 3/s + 2/s² Apply to each
ℒ{5·sin(2t)} 10/(s²+4) Pull out 5
ℒ{e^t + e^(2t)} 1/(s−1) + 1/(s−2) Split sum
ℒ{2·cos(t)} 2s/(s²+1) Pull out 2

Real-world uses

Circuit superposition ODE solution methods Linear system analysis
Derivative Property Differentiation in time becomes multiplication by s.

The most powerful property — it turns differential equations into algebraic equations by mapping derivatives in t to simple multiplications by s.

Key formulas & rules

  • •ℒ{f′(t)} = s·F(s) − f(0)
  • •ℒ{f″(t)} = s²F(s) − s·f(0) − f′(0)
  • •ℒ{f⁽ⁿ⁾} = sⁿF(s) − Σ s^(n−k−1)·f⁽ᵏ⁾(0)
  • •Initial conditions become constants
  • •Simplifies ODEs to algebra
  • •Inverse: multiplication by s → differentiation

Examples

Input Output Note
ℒ{f′} with f(0)=2 sF(s) − 2 First derivative
ℒ{f″} with f(0)=0 s²F(s) − 1 Second derivative
ℒ{t·f(t)} −dF/ds Multiplication by t
ℒ{f′′′} s³F(s) Third derivative

Real-world uses

Solving ODEs Circuit transients Mechanical vibrations
Shift Theorems Time and frequency shifts relate to exponentials in s.

Shifting a function in time multiplies its transform by an exponential in s. Multiplying a function by an exponential shifts the s-domain.

Key formulas & rules

  • •Time shift: ℒ{f(t−a)·u(t−a)} = e^(−as)·F(s)
  • •Freq shift: ℒ{e^(at)·f(t)} = F(s−a)
  • •Step function: ℒ{u(t−a)} = e^(−as)/s
  • •Delay property in signals
  • •Used to model turn-on switches
  • •Combined with convolution for LTI systems

Examples

Input Output Note
ℒ{u(t−3)} e^(−3s)/s Delayed step
ℒ{e^(−2t)·sin(t)} 1/((s+2)²+1) Freq shift
ℒ{sin(t−π/2)·u(t−π/2)} e^(−πs/2)/(s²+1) Delayed sine
ℒ{e^(3t)·t} 1/(s−3)² Shifted ramp

Real-world uses

Delayed responses Signal modulation Feedback systems
Convolution Theorem Multiplication in s-domain = convolution in time.

The convolution of two functions in time domain corresponds to multiplication of their Laplace transforms. This is the foundation of LTI system analysis.

Key formulas & rules

  • •ℒ{f * g} = F(s) · G(s)
  • •Convolution: (f*g)(t) = ∫₀ᵗ f(τ)g(t−τ)dτ
  • •Commutative: f * g = g * f
  • •Associative: (f*g)*h = f*(g*h)
  • •LTI response = convolution of input & impulse response
  • •Used to compute inverse transforms

Examples

Input Output Note
h(t)*x(t) for LTI Y(s) = H(s)·X(s) System output
ℒ{1 * 1} 1/s² = ℒ{t} Convolution → t
ℒ{e^(−t) * e^(−2t)} 1/((s+1)(s+2)) Sum property
ℒ{sin(t) * cos(t)} s/(s²+1)² Product

Real-world uses

Filter design Image processing Control systems

Modes & formulas

1

Pairs Table

• Search a reference library of common Laplace transform pairs. • Enter a time-domain function or s-domain expression in the search bar. • Copy any transform pair to your clipboard.
ℒ{f(t)} = F(s) = ∫₀^∞ f(t)·e^(−st) dt

The transform converts multiplication in the time domain to convolution in the s-domain, and differentiation to multiplication by s.

2

Properties & Theorems

• View key properties: linearity, derivative, time-shift, frequency-shift, convolution. • Derivative rule: ℒ{f′} = sF(s) − f(0). • Convolution rule: ℒ{f * g} = F(s)·G(s).
ℒ{f′(t)} = s·F(s) − f(0)

These properties are what make the Laplace transform so powerful: they turn calculus into algebra. Initial conditions appear naturally as constants.

3

Worked Examples

• See real examples of how to apply the transforms. • Each example links the time-domain input to its s-domain output. • Use these as templates for solving your own ODEs.
ℒ{e^(at)} = 1 / (s − a)

Common transformations: constants → 1/s, ramps → 1/s², exponentials → 1/(s−a), sines → ω/(s²+ω²), and cosines → s/(s²+ω²).

How to use it

Search the Pairs Table for a specific function, read the Properties tab to understand the rules, or browse the Examples tab to see the transforms in action.

Frequently asked questions

What is the inverse Laplace transform?
The inverse transform f(t) = (1/2πi) ∫ F(s)e^(st) ds converts back from the s-domain. In practice, tables and partial fractions are used.
Why is the Laplace transform one-sided?
The integral starts at t = 0, which is natural for causal systems (systems that are at rest before time zero).
What is the Laplace transform?
An integral transform that converts a function of time (t) into a function of complex frequency (s). It turns differential equations into algebraic equations.
What does the "s" in the s-domain mean?
It is a complex variable, s = σ + jω, where σ is the damping factor and ω is the angular frequency.
Why is the Laplace transform useful?
Because it simplifies solving linear differential equations by replacing calculus (derivatives and integrals) with algebra (multiplication and division).
What is the difference between Laplace and Fourier transforms?
The Laplace transform evaluates a signal from t=0 to infinity (one-sided) and includes a damping factor, allowing it to handle signals that grow exponentially. The Fourier transform evaluates over all time and strictly handles steady-state frequencies.

Related calculators

Complex Numbers Add, multiply, divide complex numbers in rectangular and polar form. Vector Math Dot product, cross product, magnitude, and angle between vectors. Matrix Calculator Calculate determinant, transpose, addition, and multiplication.
Read the full Laplace Transform guide — examples & 18 FAQs