Laplace Transform

Laplace Transform Definition

ℒ{f(t)} = F(s) = ∫₀^∞ e^(-st)f(t)dt

Laplace Transform of 1

ℒ{1} = 1/s, s > 0

Laplace Transform of e^(at)

ℒ{e^(at)} = 1/(s-a), s > a

Laplace Transform of tⁿ

ℒ{tⁿ} = n!/s^(n+1), s > 0

Laplace Transform of sin(kt)

ℒ{sin(kt)} = k/(s²+k²), s > 0

Laplace Transform of cos(kt)

ℒ{cos(kt)} = s/(s²+k²), s > 0

Linearity Property

ℒ{af(t)+bg(t)} = aℒ{f(t)} + bℒ{g(t)}

Notation for Laplace Transform

If ℒ{f(t)} = F(s), then F(s) is called the Laplace transform of f(t)

First Derivative Property

ℒ{f'(t)} = sF(s) - f(0)

Second Derivative Property

ℒ{f''(t)} = s²F(s) - sf(0) - f'(0)

Third Derivative Property

ℒ{f'''(t)} = s³F(s) - s²f(0) - sf'(0) - f''(0)

nth Derivative Property

ℒ{f⁽ⁿ⁾(t)} = sⁿF(s) - sⁿ⁻¹f(0) - sⁿ⁻²f'(0) - ··· - f⁽ⁿ⁻¹⁾(0)

Pattern for Derivative Transforms

Start with sⁿF(s), then subtract descending powers of s multiplied by successive initial conditions

Exponential Shifting Theorem

ℒ{e^(at)f(t)} = F(s-a)

How to Apply the Exponential Shifting Theorem

Replace every s in F(s) with (s-a)

Region of Convergence for Exponential Shift

If F(s) exists for s > c, then ℒ{e^(at)f(t)} exists for s > a+c

Laplace Transform of e^(at)sin(kt)

k/((s-a)²+k²)

Laplace Transform of e^(at)cos(kt)

(s-a)/((s-a)²+k²)

Most Common Laplace Pairs to Memorize

1 ↔ 1/s, e^(at) ↔ 1/(s-a), tⁿ ↔ n!/s^(n+1), sin(kt) ↔ k/(s²+k²), cos(kt) ↔ s/(s²+k²)

Why Laplace Transforms Are Useful

They convert differential equations into algebraic equations in the variable s

What Happens to Initial Conditions in Laplace Transforms

They appear automatically through the derivative formulas

Common Mistake with Derivative Rule

F(s) is the Laplace transform of f(t), not an antiderivative of f(t)

Variable in the Time Domain

t

Variable in the Laplace Domain

s