Laplace Transforms
Cheat sheet.

The key rules, formulas and reminders from all 5 topics, gathered into reference cards.

Key formulas, conditions and traps · Read down each column.

Laplace Transforms and Their Domain

Core rule

\mathcal L[y'']=s^2Y-sy(0)-y'(0).

Watch for

Derivative formulas require suitable regularity and growth. Keep initial terms until after substituting the stated values. Losing them replaces the intended initial-value problem with a different one.

Laplace Inversion and Initial-Value Problems

Core rule

Solve for $Y(s)$ with the initial terms included, then match numerator and denominator to a transform pair.

Watch for

A shift in $s$ produces an exponential in time. A factor $e^{-as}$ produces a time delay instead; these are different transform rules. Repeated poles require terms for each power in partial fractions.

Delayed Inputs and Convolution

Core rule

\mathcal L[H(t-a)f(t-a)]=e^{-as}F(s).

Watch for

An ordinary step in forcing usually leaves the state continuous while changing its derivative. An impulse can cause a jump instead. Separate the forcing’s discontinuities from the response’s continuity requirements.

Impulse Inputs and Jump Conditions

Core rule

Integrate across the impulse to obtain the jump; preserve area when approximating delta inputs.

Watch for

If the leading coefficient is a mass $m$ , the velocity jump is $J/m$ . A finite pulse approximation must preserve impulse area when its width shrinks. Its peak height alone does not determine the limiting effect.

Transfer Functions, Poles and Response

Core rule

G(s)=Y(s)/U(s)\quad\text{with zero initial conditions}.

Watch for

Frequency response uses $G(i\omega)$ after stable transients have decayed. A pole-zero cancellation can hide an unstable internal mode, so transfer behavior alone should not be treated as a complete state-stability proof.