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Math101learn.math101.caStep Functions
A precise guide to unit-step representations, delayed inputs, piecewise functions, and Laplace transforms.
Precise definition
The Heaviside unit step $u(t-a)$ is 0 for $t<a$ and 1 for $t>a$; its value at $t=a$ depends on convention and usually does not affect ordinary Laplace integrals. It switches a term on at time $a$. A delayed copy is $u(t-a)f(t-a)$.
Notation and mathematical language
The second shifting theorem is $\mathcal L\{u(t-a)f(t-a)\}=e^{-as}F(s)$ for $a\ge0$. A piecewise change from $f_1$ to $f_2$ at $a$ can be written $f_1(t)+u(t-a)[f_2(t)-f_1(t)]$; rewrite the bracket in powers of $t-a$ before transforming by the delay theorem.
Conceptual picture
A step representation separates a baseline from changes. Each switch contributes only after its activation time. Delayed notation preserves the shape of a response while moving its start and enforcing causality.
Conditions and key results
Ordinary step forcing may make a solution's derivative jump while the solution remains continuous, depending on equation order. Differentiating steps leads to impulses in distribution theory and should not be treated as an ordinary derivative at the jump. Transform rules assume causal $t\ge0$ functions.
A reliable strategy
- Write the original piecewise intervals and identify each change time and jump amount.
- Express the baseline plus unit-step corrections, then test the formula on every interval.
- For Laplace work, rewrite switched terms as functions of $t-a$ and apply the delay theorem.
- Invert or solve, then re-expand piecewise to verify values before and after each switch.
Fully worked example
Interpretation and application
Step inputs model switches, dosing, loads, and control commands. Real devices switch over a finite transition time, so a step is an idealization; the solution is exact for that ideal input, not a claim of physically instantaneous change.
Common mistakes
Verification and reasonableness
- Evaluate the step formula at a sample time in each piece.
- Take the inverse transform and compare with the original piecewise definition.
- Check continuity or jump sizes explicitly at switch times according to the model.
Practice
- What is $u(t-4)$ for $t<4$?
- Transform $u(t-3)(t-3)$.
- Write a jump from 2 to 7 at $t=5$.
Answers and brief solutions
- $0$.
- $e^{-3s}/s^2$.
- $2+5u(t-5)$.
Further deduction
Multiple switches add linearly. A rectangular pulse of height $A$ active from $a$ to $b$ is $A[u(t-a)-u(t-b)]$, with transform $A(e^{-as}-e^{-bs})/s$. This representation makes duration and amplitude separate, and evaluating before $a$, between $a,b$, and after $b$ verifies the signs.
When a first-order equation has bounded step forcing, integrating across a tiny interval around the switch shows that the state remains continuous even though its derivative may jump. In a second-order mechanical equation with a step force, displacement and velocity usually remain continuous. An impulse can instead create a velocity jump. These conclusions follow from the equation order and input type, not from the drawing convention for $u(0)$.
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
What multiplier appears in the transform of $u(t-6)f(t-6)$?
- A delay of $a$ multiplies the transform by $e^{-as}$.
- With $a=6$, the factor is $e^{-6s}$.
End of lesson
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