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Laplace Transform

Source lecture(s): PHY621 Lec4

Intuition

Turn differential equations into algebra by trading derivatives for multiplication by \(s\).

Formal Definition

\[\mathcal{L}\{f(t)\}=\hat{f}(s)=\int_0^\infty f(t)\,e^{-st}\,dt\]

Mathematical Formulation

First derivative: \(\mathcal{L}\{f'\}=s\hat{f}-f(0)\) Second derivative: \(\mathcal{L}\{f''\}=s^2\hat{f}-sf(0)-f'(0)\)

Derivation

Integrate by parts: \(\int_0^\infty f'(t)e^{-st}\,dt = [-f(t)e^{-st}]_0^\infty + s\int_0^\infty f(t)e^{-st}\,dt\).

Worked Example

For \(f(t)=e^{at}\), \(\hat{f}(s)=\frac{1}{s-a}\).

Common Mistakes

  • Ignoring initial conditions when transforming derivatives.
  • Confusing bilateral vs unilateral transforms.

Quiz

Q1. What is the Laplace transform of a step function?

Answer

\(1/s\).