Laplace Transforms and Their Domain

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The idea

The one-sided Laplace transform packages a function on $t\ge0$ into $F(s)=\int_0^\infty e^{-st}f(t)dt$ . Exponential weighting makes many growing functions integrable for sufficiently large real $s$ . Linearity turns sums of inputs into sums of transforms.

Visual guide

VISUAL GUIDEThe exponential weight can overcome growth

s = 3: e⁻ᵗ
s = 2: constant 1

For f(t) = e²ᵗ, the transform integrand is e⁻⁽ˢ⁻²⁾ᵗ. At s = 3 it decays and integrates to 1; at s = 2 it stays at 1 and has infinite area. The convergence condition s > 2 matters.

Method and assumptions

For piecewise continuous functions of exponential order, the transform exists in a right half-plane. Useful pairs include $1\leftrightarrow1/s$ , $e^{at}\leftrightarrow1/(s-a)$ , $\sin bt\leftrightarrow b/(s^2+b^2)$ and $\cos bt\leftrightarrow s/(s^2+b^2)$ . Integration by parts gives $\mathcal L[y']=sY-y(0)$ .

Worked example: an exponential

$\mathcal L[e^{2t}]=\int_0^\infty e^{-(s-2)t}dt=1/(s-2)$ for $s>2$ . The expression alone does not record the convergence condition; the integral diverges when $s\le2$ .

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

If Y(s)=ℒ{y(t)}, what is ℒ{y′(t)}?

Y(s)/s

sY(s)−y(0)

sY(s)

Hint 1 · Find a starting point

Integration by parts leaves a boundary term.

Hint 2 · Take the next step

The initial value appears with a minus sign.

Show the reasoning

Answer: sY(s)−y(0)

ℒ{y′}=sY−y(0), under the transform’s existence conditions. Omitting y(0) changes an initial-value problem.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

Worked example: a second derivative

Apply the derivative rule twice: $\mathcal L[y'']=s^2Y-sy(0)-y'(0)$ . For $y=\cos t$ , this is $s^3/(s^2+1)-s=-s/(s^2+1)$ , matching the transform of $-\cos t$ .

Interpreting the result

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.

Practice

Find the transform of $3+2e^{-t}$ .
Find the transform of $t$ .
What is $\mathcal L[y']$ if $y(0)=4$ ?

Show worked solutions

$3/s+2/(s+1)$ for $s>0$ .
$1/s^2$ for $s>0$ , by integration by parts.
$sY-4$ .

Further study

MIT OpenCourseWare: Differential Equations provides a full university course with additional lectures and exercises.

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