What is the unilateral Laplace transform?

It maps a suitable function $f(t)$, usually defined for $t\geq0$, to $F(s)=\int_0^\infty f(t)e^{-st}\,dt$.

What do $f(t)$ and $F(s)$ usually represent?

$f(t)$ is the time-domain function, and $F(s)$ is its Laplace-domain representation as a function of complex $s$.

What do the real and imaginary parts of $s$ represent?

Writing $s=\sigma+i\omega$, $\sigma$ is the real part and controls exponential damping or growth, while $\omega$ is the imaginary component associated with oscillation.

How does the transform simplify differential equations?

It changes derivatives into algebraic expressions involving powers of $s$ and initial values, allowing the transformed equation to be solved algebraically.

Why is an inverse transform needed?

The inverse Laplace transform converts a solution expressed in the s-domain back into the original time-domain function.

Differential equationsLesson 1 of 2 in the course Laplace transform and partial differential equations

Laplace Transforms: Convergence, Transform Rules, and Applications

5 flashcards · 5 quiz questions · 5 match pairs · 4 fill-in sentences · English

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Summary

The Laplace transform changes a function of time into a function of a complex variable. This change of viewpoint can turn difficult calculus problems into algebraic ones.

A time-domain function is commonly written as $f(t)$ , where $t \geq 0$ . Its Laplace transform is written as $F(s)$ , where $s$ is a complex variable. The transform is defined by

$F(s)=\mathcal{L}\{f(t)\}=\int_0^\infty f(t)e^{-st}\,dt.$

All flashcards

What is the unilateral Laplace transform?: It maps a suitable function $f(t)$ , usually defined for $t\geq0$ , to $F(s)=\int_0^\infty f(t)e^{-st}\,dt$ .
What do $f(t)$ and $F(s)$ usually represent?: $f(t)$ is the time-domain function, and $F(s)$ is its Laplace-domain representation as a function of complex $s$ .
What do the real and imaginary parts of $s$ represent?: Writing $s=\sigma+i\omega$ , $\sigma$ is the real part and controls exponential damping or growth, while $\omega$ is the imaginary component associated with oscillation.
How does the transform simplify differential equations?: It changes derivatives into algebraic expressions involving powers of $s$ and initial values, allowing the transformed equation to be solved algebraically.
Why is an inverse transform needed?: The inverse Laplace transform converts a solution expressed in the s-domain back into the original time-domain function.

Quiz questions

Over what interval does the bilateral Laplace transform integrate?Answer: Over the entire real axis, from $-\infty$ to $\infty$ .
For the unilateral Laplace transform, what is $\mathcal{L}\{f'(t)\}$ ?Answer: $sF(s)-f(0^-)$
What does the region of convergence (ROC) of a Laplace transform specify?Answer: The values of $s$ for which the defining integral converges.
For an LTI system, what does convolution of the input with the impulse response become in the Laplace domain?Answer: Multiplication of the input and impulse-response transforms.
Under the applicable pole conditions, which expression gives the final value of a function from its Laplace transform?Answer: $\lim_{s\to 0}sF(s)$

Based on the Wikipedia article Laplace transform, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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