Laplace Transform

A learning series from introduction to advanced applications

About this series

The Laplace transform is a powerful tool that converts a time-domain function into a function on the complex $s$-plane. It turns differential equations into algebraic equations and is widely applied in control theory and circuit analysis.

This series starts from the definition of the Laplace transform and proceeds step by step through basic properties, inverse transforms, and applications to differential equations and transfer functions.

Definition of the Laplace transform

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

Learning by level

Introduction

University year 1 level

  • What is the Laplace transform
  • Relationship with the exponential function
  • Computing basic transforms
  • How to use a transform table

Basic

University year 2 level

  • Properties of the Laplace transform
  • Transforms of derivatives and integrals
  • Convolution theorem
  • Partial fraction decomposition
  • Inverse Laplace transform

Intermediate

University year 2-3 level

  • Solving differential equations
  • Transfer functions
  • Poles and zeros on the $s$-plane
  • Stability analysis

Advanced

University year 4 to graduate level

  • Inverse transform via complex integration
  • Applications of the residue theorem
  • Laplace transform of distributions
  • Extension to several variables

Learning path

Introduction Definition & basics Basic Properties & inversion Intermediate ODE applications Advanced Complex-analytic methods Introduction: definition, exponential function, basic transform table Basic: linearity, differentiation, convolution, inverse transform Intermediate: ODE solutions, transfer functions, poles and stability Advanced: complex integration, residues, distribution theory

Related topics

References