Laplace Transform Table
Find common Laplace transform pairs, inverse Laplace transforms, and useful properties in one searchable table. Browse functions, check convergence conditions, and copy formulas with clear worked examples.
Definition · One-sided Laplace transform
This is a unilateral table for t ≥ 0. For ordinary functions, the lower limit is the usual zero; 0− specifies full inclusion of an impulse at zero. Derivative properties use preinitial values f(0−), with jumps interpreted distributionally. Ordinary right-sided derivatives without an impulse instead use f(0+). Do not mix these conventions.
Table of Common Laplace Transforms
32 of 32 transform pairs
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| Function / category | f(t) | F(s) | ROC / Conditions |
|---|---|---|---|
| ConstantConstants and Powers | 1 | Re(s) > 0 | |
No parameters. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Integrate exp(−st) on [0,∞). | |||
| Integer powersConstants and Powers | tⁿ | Re(s) > 0; n ≥ 1 integer | |
n! is factorial. The constant is the n = 0 case. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Repeated integration by parts. | |||
| Fractional powersConstants and Powers | tᵖ | Re(s) > 0; p > −1 real, noninteger | |
Principal power of s on the right half-plane; Γ is the gamma function. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Substitute v = st in the gamma integral and analytically continue within Re(s) > 0. | |||
| ExponentialExponential Functions | exp(at) | Re(s) > Re(a); a complex | |
s and a may be complex. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Integrate exp(−(s − a)t). | |||
| Exponentially weighted powerExponential Functions | tⁿ exp(at) | Re(s) > Re(a); n ≥ 1 integer | |
a complex. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Apply the s-shift to the integer-power pair. | |||
| SineTrigonometric Functions | sin(bt) | Re(s) > 0; b > 0 real | |
b is angular frequency. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Euler sine identity and exponential integrals. | |||
| CosineTrigonometric Functions | cos(bt) | Re(s) > 0; b > 0 real | |
b is angular frequency. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Euler cosine identity and exponential integrals. | |||
| Exponentially weighted sineExponential Functions | exp(at) sin(bt) | Re(s) > Re(a); b > 0 real | |
a complex. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Apply the exponential shift to sine. | |||
| Exponentially weighted cosineExponential Functions | exp(at) cos(bt) | Re(s) > Re(a); b > 0 real | |
a complex. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Apply the exponential shift to cosine. | |||
| Hyperbolic sineHyperbolic Functions | sinh(bt) | Re(s) > |b|; b ≠ 0 real | |
sinh(bt) = [exp(bt) − exp(−bt)]/2. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Take the difference of the two exponential integrals. | |||
| Hyperbolic cosineHyperbolic Functions | cosh(bt) | Re(s) > |b|; b real | |
cosh(bt) = [exp(bt) + exp(−bt)]/2. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Take the sum of the two exponential integrals. | |||
| Delayed unit stepUnit Step and Impulse | u(t − τ) | Re(s) > 0; τ ≥ 0 | |
u(t) = 1 for t > 0 and 0 for t < 0. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Integrate exp(−st) from τ to infinity. | |||
| Dirac impulseUnit Step and Impulse | δ(t) | 1 | All finite s; distribution |
The 0− convention includes the full impulse weight at zero. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Sifting identity at t = 0. | |||
| Delayed Dirac impulseUnit Step and Impulse | δ(t − τ) | exp(−τs) | All finite s; τ > 0; distribution |
δ is a distribution, not an ordinary pulse. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Sifting at t = τ. | |||
| Impulse derivativeUnit Step and Impulse | δ′(t) | s | All finite s; distribution |
Derivative of the impulse in the full-weight 0− convention. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: ⟨δ′, exp(−st)⟩ = s. | |||
| Delayed rampShifted Functions | (t − τ) u(t − τ) | Re(s) > 0; τ > 0 | |
The ramp begins at τ. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Time-delay rule applied to t. | |||
| Delayed exponentialShifted Functions | exp(a(t − τ)) u(t − τ) | Re(s) > Re(a); τ > 0 | |
a complex. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Substitute v = t − τ. | |||
| Finite rectangular pulseAdditional Common Pairs | u(t) − u(t − T) | All finite s; T > 0; value at s = 0 is T | |
Finite support; the apparent singularity is removable. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Integrate on [0,T], retaining the limit at zero. | |||
| Time-weighted sineTrigonometric Functions | t sin(bt) | Re(s) > 0; b > 0 real | |
Useful for repeated quadratic factors. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Differentiate b/(s² + b²) with respect to s and negate. | |||
| Time-weighted cosineTrigonometric Functions | t cos(bt) | Re(s) > 0; b > 0 real | |
Useful for repeated quadratic factors. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Negate the derivative of s/(s² + b²). | |||
| Squared sineTrigonometric Functions | sin²(bt) | Re(s) > 0; b > 0 real | |
sin²(bt) = [1 − cos(2bt)]/2. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Apply the double-angle identity and combine rational terms. | |||
| Squared cosineTrigonometric Functions | cos²(bt) | Re(s) > 0; b > 0 real | |
cos²(bt) = [1 + cos(2bt)]/2. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Apply the double-angle identity and combine terms. | |||
| Sine divided by timeAdditional Common Pairs | atan(b/s) | Re(s) > 0; b > 0 real | |
At t = 0 use limit b. Analytic branch continued from real s > 0. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Differentiate with respect to b, integrate the cosine transform, then use the zero value at b = 0. | |||
| One minus cosine divided by timeAdditional Common Pairs | Re(s) > 0; b > 0 real | ||
Analytic logarithm continued from positive real s; time limit at zero is 0. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Differentiate with respect to b, giving the sine transform, then integrate from 0 to b. | |||
| LogarithmAdditional Common Pairs | log(t) | Re(s) > 0 | |
γ is Euler’s constant; principal log on Re(s) > 0; locally integrable at t = 0. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Differentiate Γ(p + 1)/s^(p+1) at p = 0. | |||
| Bessel function J₀Additional Common Pairs | J₀(bt) | Re(s) > 0; b > 0 real | |
J₀ is the Bessel function of the first kind. Branch tends to 1/s as positive real s → ∞. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Use J₀(bt) = (1/π)∫₀^π cos(bt sin θ)dθ and integrate in t. | |||
| Bessel function J₁Additional Common Pairs | J₁(bt) | [1 − s/√(s² + b²)]/b | Re(s) > 0; b > 0 real |
Same square-root branch as J₀. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Use dJ₀(bt)/dt = −bJ₁(bt) and J₀(0) = 1. | |||
| Error functionAdditional Common Pairs | erf(b√t) | Re(s) > 0; b > 0 real | |
erf(z) = (2/√π)∫₀^z exp(−v²)dv; principal square root. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Differentiate erf(b√t): b exp(−b²t)/√(πt), then divide its transform by s. | |||
| Complementary error functionAdditional Common Pairs | erfc(b√t) | [1 − b/√(s + b²)]/s | Re(s) > −b²; b > 0 real; removable value at s = 0 is 1/(2b²) |
erfc(z) = 1 − erf(z). Square root has positive real part on this half-plane. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Subtract the erf transform from 1/s, then continue using erfc tail decay. | |||
| Difference of exponentialsExponential Functions | Re(s) > max(Re(a), Re(b)); a ≠ b | ||
a,b complex. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Subtract two exponential transforms and simplify; also their causal convolution. | |||
| Sine convolved with sineConvolution | Re(s) > 0; b > 0 real | ||
This is ∫₀ᵗ sin(bτ)sin(b(t − τ))dτ. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Multiply the two sine transforms, or combine the time-sine/cosine pairs. | |||
| Time-weighted hyperbolic sineHyperbolic Functions | t sinh(bt) | Re(s) > |b|; b ≠ 0 real | |
For negative b, both sides change sign. Reference for the pair or deriving property. Source normalizations are converted to the convention defined on this page. Mathematical check: Negate the s-derivative of b/(s² − b²). | |||
Inverse Laplace Transform Table
Select Inverse above to put the transform first and its corresponding time function second. Search works across both sides in either direction; all pair conditions still apply.
2/(s² + 4) ↔ sin(2t), Re(s) > 0, t ≥ 0.
Laplace Transform Properties
f ↔ F and g ↔ G use the unilateral convention. Initial-value terms are not optional. Sufficient convergence half-planes for properties may be smaller than an exact ROC after cancellation.
| Property | Time-domain operation | Transform-domain expression | Conditions |
|---|---|---|---|
| Linearity | A f(t) + B g(t) | A F(s) + B G(s) | At least the intersection of the original ROCs; cancellation may enlarge it. |
Linearity of the defining integral. Reference | |||
| Exponential shift | exp(at) f(t) | F(s − a) | ROC shifts by a: s − a must lie in the original ROC. |
Combine exp(at) with exp(−st). Reference | |||
| Time delay | u(t − τ) f(t − τ) | exp(−τs) F(s) | τ ≥ 0; causal extension; original ROC is retained for nonzero finite delay. |
Substitute v = t − τ. Reference | |||
| Gated function (not delayed argument) | u(t − τ) f(t) | exp(−τs) ℒ{f(t + τ)} | τ ≥ 0; use the ROC of the shifted function. Do not replace f(t) by f(t − τ). |
Substitute v = t − τ without changing the original function argument incorrectly. Reference | |||
| First derivative | f′(t) | sF(s) − f(0−) | 0− convention with distributional jumps; sufficient exponential order and initial limit. For an ordinary derivative over (0,∞), use f(0+) instead. Valid in a common convergence half-plane. |
Integration by parts including the jump at zero. Reference | |||
| Second derivative | f″(t) | s²F(s) − s f(0−) − f′(0−) | Same 0− convention; initial limits finite. Valid where all terms converge. |
Apply the first-derivative rule twice. Reference | |||
| Nth derivative | f⁽ⁿ⁾(t) | sⁿF(s) − Σ[k=0…n−1] s^(n−1−k) f⁽ᵏ⁾(0−) | n ≥ 1 integer; same initial-value convention and common convergence half-plane. |
Repeated integration by parts with initial terms retained. Reference | |||
| Multiplication by time | tⁿ f(t) | (−1)ⁿ F⁽ⁿ⁾(s) | n ≥ 1 integer; valid in the open ROC under standard differentiation hypotheses. |
Differentiate the kernel n times with respect to s. Reference | |||
| Integration | ∫₀⁻ᵗ f(τ)dτ | Causal accumulation; sufficient ROC Re(s) > max(σ,0) if Re(s) > σ is an original half-plane. Cancellation may enlarge it. | |
Convolve with the causal constant function. Reference | |||
| Causal convolution | (f ∗ g)(t) = ∫₀⁻ᵗ⁺ f(τ)g(t − τ)dτ | F(s)G(s) | Causal functions/distributions for which convolution exists; at least a shared convergence half-plane. |
Factor the double integral using Fubini under absolute convergence. Reference | |||
| Time scaling | f(at) | a > 0 real; s/a in the original ROC. Negative time scaling is not a unilateral rule. | |
Substitute v = at on the positive time axis. Reference | |||
| Periodic function | f(t + T) = f(t), t ≥ 0 | T > 0; locally integrable periodic f, Re(s) > 0 is sufficient. | |
Split into periods and sum a geometric series. Reference | |||
| Initial-value theorem | f(0+) | lim[s→+∞] sF(s) | Ordinary f with finite right limit, exponential order, and no impulse at zero; this is a limit identity, not an additional pair. |
Substitute v = st and use dominated convergence. Reference | |||
| Final-value theorem | lim[t→∞] f(t) | lim[s→0+] sF(s) | For rational F, all poles of sF must lie strictly in Re(s) < 0. A finite final limit must be justified; oscillations and unstable poles fail. |
Use the derivative rule and decay/stability hypotheses. Reference | |||
How to Use a Laplace Transform Table
- Find a time function f(t) or a transform F(s) using search.
- Check parameters and ROC; s is complex, not just a positive real number.
- Apply a listed property for shifts, derivatives or products corresponding to convolution.
- For inverse transforms, rearrange the expression or use partial fractions before matching.
For example, 1/[(s + 1)(s + 2)] = 1/(s + 1) − 1/(s + 2), so its causal inverse is exp(−t) − exp(−2t). The ROC is Re(s) > −1. A product F(s)G(s) corresponds to time-domain convolution, not ordinary multiplication.
Region of Convergence and Important Conditions
The ROC identifies where the defining integral converges, not merely where a rational formula has no pole. For exp(at), ℒ{exp(at)} = 1/(s − a) only in Re(s) > Re(a). For real nonzero sine or cosine frequencies, the open ROC is Re(s) > 0. The zero sine is a special zero-signal case with all finite s allowed.
All ordinary pairs describe the function for t ≥ 0, with a causal zero extension when interpreting it as a full-line signal. That extension is separate from preinitial values in a differential equation. Supply the actual initial state rather than assuming it is zero. A derivative across an initial jump includes an impulse; a classical derivative on (0,∞) does not.
u(t) denotes the unit step; δ(t) denotes the Dirac distribution. Γ(z) = ∫₀∞ vz−1 exp(−v)dv for Re(z) > 0. γ is Euler’s constant. J₀ and J₁ are Bessel functions of the first kind. Complex roots and logarithms use the branches stated in each row. ROC boundaries are not included unless explicitly stated. Finite pulses have entire transforms after removable singularities are filled.
Unilateral and bilateral Laplace transforms are not interchangeable without support assumptions. A Fourier transform can sometimes be obtained from s = iω, but only when convergence or a justified distributional limit permits it.
Step-by-Step Laplace Transform Examples
These examples use sample values, not an editable symbolic calculation.
Constant
ℒ{1} = ∫₀∞ exp(−st)dt = [−exp(−st)/s]₀∞ = 1/s
The boundary at infinity vanishes for Re(s) > 0.
Sine
sin(at) = [exp(iat) − exp(−iat)]/(2i)
ℒ{sin(at)} = [1/(s − ia) − 1/(s + ia)]/(2i) = a/(s² + a²)
For real a ≠ 0, Re(s) > 0. If a = 0, the time function and transform are both zero.
Inverse transform
F(s) = 2/(s² + 4) = 2/(s² + 2²)
Match b = 2 in the sine row: f(t) = sin(2t) for t ≥ 0, with Re(s) > 0.
Exponential shift
ℒ{exp(at)f(t)} = F(s − a)
For f(t) = t and a = 3, F(s) = 1/s² becomes 1/(s − 3)², with ROC Re(s) > 3 rather than Re(s) > 0.
Time delay
ℒ{u(t − a)f(t − a)} = exp(−as)F(s), a ≥ 0
A delayed ramp (t − 2)u(t − 2) has transform exp(−2s)/s², Re(s) > 0. Delaying the argument is essential; u(t − 2)t is a different function.
Frequently Asked Questions
What is a Laplace transform table?
It is a reference of time-domain functions and their Laplace-domain expressions, used in differential equations and systems analysis. A useful table also states parameter restrictions and convergence conditions.
How do I use a Laplace transform table?
Search for a matching function or expression, check parameters and the region of convergence, and apply the listed shift, derivative or convolution properties when necessary. Keep initial-value assumptions consistent.
Can I use a Laplace transform table in reverse?
Yes. Select Inverse, match F(s), and read f(t). Algebraic rearrangement or partial fractions may be needed before the expression matches a standard pair.
What is the Laplace transform of 1?
For this unilateral convention, ℒ{1} = 1/s with Re(s) > 0. The algebraic expression outside that half-plane is not the value of the convergent defining integral.
What is the Laplace transform of sin(at)?
For real a, ℒ{sin(at)} = a/(s² + a²) on Re(s) > 0. This is the exact open ROC for a ≠ 0; when a = 0 the signal is identically zero and its transform is zero for every s.
What is the difference between the Laplace and Fourier transforms?
Laplace uses a complex variable s = σ + iω and exponential weighting. Fourier describes frequency content along a real frequency axis. Substitution s = iω is justified only when the appropriate convergence or distributional conditions hold.
What does region of convergence mean?
The ROC is the set of complex s values for which the defining transform integral converges. An algebraically finite rational expression alone does not establish convergence.
Does a Laplace transform table solve any differential equation?
No. You still need an appropriate model, initial conditions, algebraic solution in s and an inverse transform. Partial fractions may help, but not every equation or transform has an elementary table-based solution.
Sources and verification
- MIT 18.031 — Laplace Transform Table (0− definition, initial values and ROC)
- NIST DLMF §1.14 — Integral Transforms
- Lamar University — Table of Laplace Transforms
The catalog has 32 distinct pair families and 14 properties, not 46 artificially labeled pairs. Each entry includes a mathematical derivation. Ordinary-function examples are independently checked using numerical integration; impulse identities use distributional checks.