Inverse Laplace Transform Table

You have a rational function of s and need f(t) in ten minutes. This inverse Laplace transform table covers the 30 most common transform pairs, property rules for shifting and convolution, and the partial-fraction strategy that turns a messy F(s) into something the table handles. Every pair is cross-checked against Kreyszig Table 6.9 and Abramowitz & Stegun ch. 29. Print it. Keep it inside your notebook cover.

These are the foundation. Every other pair is built from these by shifting or multiplying by t.

ℒ{1} = 1/s, Re(s) > 0. ℒ{t} = 1/s². ℒ{t²} = 2/s³. ℒ{tⁿ} = n!/sⁿ⁺¹, n = 0,1,2,... . ℒ{e^{at}} = 1/(s−a), Re(s) > a.

The exponential pair is the most frequently inverted term after partial fractions. When your F(s) has a term 1/(s+5), you know f(t) = e^{−5t}. The sign of the pole tells you decay or growth. A pole at s = +3 means e^{3t}, which grows without bound. That is correct for an unstable system; do not force it into a stable form.

Source: Kreyszig Table 6.9 and Abramowitz & Stegun ch. 29 both give identical expressions for these pairs.

ℒ{sin(ωt)} = ω/(s²+ω²). ℒ{cos(ωt)} = s/(s²+ω²). ℒ{sinh(at)} = a/(s²−a²). ℒ{cosh(at)} = s/(s²−a²).

The confusion pair here is sin vs cos: the numerator has ω for sin, s for cos. If you invert ω/(s²+ω²) and call it cos(ωt), you introduce a 90° phase error. Another common mistake is reading the minus sign in s²−a² as a²−s². The denominator s²−a² factors as (s−a)(s+a). That gives sinh and cosh, not sin and cos. For s²+ω² the roots are ±iω, which gives oscillation.

Cross-check against Kreyszig Table 6.9 confirms these four pairs identically. Abramowitz & Stegun ch. 29 adds a sin at pair with the same form but uses the variable a in place of ω.

A damped sinusoid is an exponential times a sine or cosine. The frequency-shift rule ℒ{e^{at}f(t)} = F(s−a) produces these directly.

ℒ{e^{at} sin(ωt)} = ω/((s−a)²+ω²). ℒ{e^{at} cos(ωt)} = (s−a)/((s−a)²+ω²). ℒ{e^{at} sinh(bt)} = b/((s−a)²−b²). ℒ{e^{at} cosh(bt)} = (s−a)/((s−a)²−b²).

These are the transforms you use for underdamped second-order system responses. The damping ratio ζ and natural frequency ωₙ in the standard form ωₙ²/(s²+2ζωₙs+ωₙ²) are usually not a direct match; you complete the square in the denominator, then shift. For example, the denominator s²+4s+5 becomes (s+2)²+1, which matches the damped sin pair with a = −2 and ω = 1.

Source: Kreyszig Table 6.9 and Abramowitz & Stegun ch. 29 both list these forms. Boyce & DiPrima §6.6 works through the complete-the-square step for a problem type exactly like this.

When the denominator has a repeated pole, the time-domain term picks up a factor of t. These are the pairs you need when partial fractions produce a term with (s−a)² or higher power.

ℒ{t e^{at}} = 1/(s−a)². ℒ{e^{at} tⁿ} = n!/(s−a)ⁿ⁺¹. ℒ{t sin(ωt)} = 2ωs/(s²+ω²)². ℒ{t cos(ωt)} = (s²−ω²)/(s²+ω²)².

The t sin and t cos pairs are the ones students miss. If your F(s) has denominator (s²+ω²)², you need these, not the basic sin or cos pair. Kreyszig Table 6.9 lists t sin at and t cos at. Abramowitz & Stegun ch. 29 gives t sin(at) and t cos(at) with the same functional form, using a for the frequency parameter.

ℒ{δ(t)} = 1. ℒ{u_c(t)} = e^{−cs}/s. ℒ{u_c(t) f(t−c)} = e^{−cs}F(s).

The Dirac delta δ(t) has transform 1 across the entire s-plane. Its inverse is an impulse at t=0. The unit step u(t) has transform 1/s. A delayed step u_c(t) = u(t−c) adds the factor e^{−cs}.

The time-shift rule is the one you use for any F(s) that contains a factor e^{−as}. The inverse is f(t−a)u(t−a). The unit step function u(t−a) is not optional. Write f(t−a)u(t−a), not f(t−a). Without the step, the expression is defined for negative time, which the inverse Laplace transform does not produce.

Boyce & DiPrima ch. 6 exercises include problem types that require inverting e^{−2s}/(s−3) using exactly this rule.

The property table is how you extend the basic pairs to cover almost any rational F(s). These rules are not transform pairs themselves; they are operations on F(s) and the corresponding operation on f(t).

• ℒ{e^{at}f(t)} = F(s−a), frequency shift
• ℒ{f(t−a)u(t−a)} = e^{−as}F(s), time shift
• ℒ{f(at)} = (1/a)F(s/a), a > 0, scaling
• ℒ{f'(t)} = sF(s) − f(0), differentiation in time
• ℒ{f''(t)} = s²F(s) − s f(0) − f'(0), second derivative
• ℒ{∫₀ᵗ f(τ) dτ} = F(s)/s, integration
• ℒ{t f(t)} = −F'(s), multiplication by t
• ℒ{f(t)/t} = ∫ₛ^∞ F(u) du, division by t
• ℒ{(f ∗ g)(t)} = F(s)G(s), convolution

If F(s) has poles on the imaginary axis, the rule gives a wrong finite value when the true response oscillates forever.

These are standard across Kreyszig Table 6.9, Abramowitz & Stegun ch. 29, and Boyce & DiPrima §6.6.

Most F(s) you encounter will not match a table row directly. The partial-fraction decomposition breaks F(s) into a sum of terms that do match. This is the step that takes the time, not the table lookup.

Distinct Real Poles: Heaviside Cover-Up

For F(s) with distinct real poles p₁, p₂, ..., the coefficient Aₖ for term Aₖ/(s−pₖ) is (s−pₖ)F(s) evaluated at s = pₖ. This is the Heaviside cover-up method taught in Boyce & DiPrima ch. 6. It works only for distinct real poles. Do not use it on repeated poles or complex-conjugate pairs.

Repeated Real Poles

For a pole of multiplicity m at s = p, the partial-fraction form is A₁/(s−p) + A₂/(s−p)² + ... + Aₘ/(s−p)ᵐ. The single-term form A/(s−p)² is incorrect. The coefficients require solving a system of equations or using differentiation. Stewart §7.4 covers the algebraic forms for repeated linear factors.

Irreducible Quadratic Factors

For a denominator factor s²+bs+c with complex roots, use (As+B)/(s²+bs+c). Complete the square in the denominator to match the damped sin and cos pairs. The numerator As+B splits into A(s−a) + (B+Aa). The first term gives e^{at} cos(ωt); the second gives e^{at} sin(ωt).

Improper Fractions

If the degree of the numerator equals or exceeds the degree of the denominator, perform polynomial division first. A term like (s³+2s)/(s²+1) must be divided to s + s/(s²+1) before you can invert it. Omitting this step produces algebraic nonsense.

Verification

Before you look up the inverse, recombine your partial fractions into a single rational function. If it does not match the original F(s), the coefficients are wrong. SymPy inverse_laplace_transform, Wolfram InverseLaplaceTransform, and MathWorks ilaplace can verify the final result numerically.

This table suits undergraduate differential equations students solving ODEs with initial conditions, signals and systems students finding impulse and step responses from transfer functions, and control-engineering students applying the final value rule correctly. Practising engineers verifying a step-response calculation from a datasheet pole-zero plot will find the second-order system forms under Damped Pairs. Physics students working circuit theory or mechanical vibrations can use the full set of 30 pairs.

Skip this table if you need only a numeric answer from a known F(s) and do not need to show algebraic steps. A symbolic tool like Wolfram InverseLaplaceTransform or SymPy gives the result faster. Skip it if you study the Bromwich integral or residue methods in depth; Churchill's Operational Mathematics covers that theory.

The single thing that most often goes wrong: inverting a term from the table without completing the square first, which turns a damped oscillation into an unmatched mess. Complete the square in the denominator, then match the shifted pair.