Laplace Transforms: every key term you need (+ practice quiz)
25 flashcard terms for Differential Equations Topic 7, written to match the course framework. Study them here, then drill them as interactive flashcards, or test yourself with the 15-question quiz — free, no account needed.
An integral operator sending a function of time to a function of a complex parameter by integrating the function against a decaying exponential over the positive half line. It converts calculus problems into algebra.
Exponential order
The growth restriction that a function is eventually bounded by a constant times an exponential. Together with piecewise continuity it guarantees the defining integral converges for large enough values of the parameter.
Piecewise continuity requirement
The condition that the function has only finitely many jump discontinuities on any bounded interval, with finite one-sided limits. Jumps are allowed, which is why the transform handles switched inputs so gracefully.
Linearity of the transform
The transform of a sum is the sum of the transforms and constants pass through. This is what allows a complicated forcing term to be transformed piece by piece.
Transform of a derivative
The transform of the derivative equals the parameter times the transform minus the initial value. Repeating the rule for the second derivative brings in the initial derivative as well, which is why initial data enter automatically.
Why initial conditions are built in
Because the derivative rule consumes the initial values, the transformed equation already encodes them, so no arbitrary constants ever appear and no final fitting step is required.
Inverse Laplace transform
The operation recovering a time function from its transform. In practice it is carried out by algebraic rearrangement into recognisable pieces rather than by evaluating the complex contour integral.
Uniqueness of the inverse
Two piecewise continuous functions of exponential order with the same transform agree wherever both are continuous. Values at isolated jump points are not determined, which is harmless for modelling.
Partial fraction decomposition for inversion
Splitting a rational transform into simple pieces with linear or irreducible quadratic denominators so that each piece matches a standard entry. Repeated factors require successive powers in the decomposition.
Completing the square for inversion
Rewriting an irreducible quadratic denominator as a shifted square so the result matches a damped sine or cosine entry. Numerators must be adjusted to match the shift as well as the denominator.
First shifting theorem
Multiplying a time function by an exponential shifts its transform in the parameter. This single rule explains why every damped oscillation entry is the undamped entry with a translated argument.
Unit step function
The function that is zero before a chosen switch-on time and one afterwards. It is the basic building block for writing any piecewise-defined forcing as a single algebraic expression.
Second shifting theorem
A time function delayed and switched on by a unit step has transform equal to the undelayed transform multiplied by an exponential in the parameter. Rewriting the forcing in terms of the shifted variable is the crucial preparatory step.
Piecewise forcing assembly
Expressing a forcing function defined by cases as a sum of unit steps multiplying the difference between successive formulas. Each step switches one formula off and the next one on at the right instant.
Dirac impulse
An idealised instantaneous input of unit total effect, treated as the limit of tall narrow pulses. Its transform is an exponential in the parameter, or simply one when the impulse arrives at the origin.
Impulse response
The output of a system at rest driven by a unit impulse. Its transform is the transfer function itself, which makes it the fingerprint that determines the response to every other input.
Transfer function
The ratio of the transform of the output to the transform of the input for a system starting from rest. For a constant-coefficient equation it is the reciprocal of the characteristic polynomial evaluated at the parameter.
Convolution integral
An operation combining two time functions by integrating one against a time-reversed and shifted copy of the other. It is commutative and it is what multiplication of transforms corresponds to.
Convolution theorem
The transform of a convolution is the product of the individual transforms. It lets any forced response be written as the impulse response convolved with the input, without partial fractions.
Transform of an integral
Integrating a function from zero divides its transform by the parameter. It follows from the convolution theorem with the constant function one and is useful for circuits carrying charge terms.
Derivative of the transform rule
Multiplying a time function by the time variable corresponds to negating the derivative of its transform. Repeating it handles polynomial multipliers and is how power entries are generated.
Transform of a periodic function
For a function repeating with a fixed period, the transform equals the integral over one period divided by one minus an exponential in the period. It avoids integrating over the infinite tail.
Initial and final value reasoning
Limits of the parameter times the transform at large and small parameter recover the starting and long-run values of the time function, provided the relevant limits exist. It gives quick checks without inverting.
When to prefer the transform method
For discontinuous, switched, impulsive or periodic forcing, and for problems where initial data at the origin are given. For smooth forcing with general initial points the classical methods are usually quicker.
Common transform pitfalls
Failing to rewrite a delayed forcing in terms of the shifted time variable before applying the delay rule, and forgetting that the transform is defined only for nonnegative time so nothing before the origin is represented.