But what is a Laplace Transform? (3Blue1Brown Laplace series)
by Grant Sanderson · 3Blue1Brown
Animated 35-minute explanation of the integral definition, why exponentials are the building blocks, analytic continuation, and how poles of F(s) reveal a signal's exponential components. Part of a trilogy with a prelude on complex exponents and a follow-up on solving ODEs. Supplies the 'why' that table-driven courses skip: what s means, why poles matter, why differentiation becomes multiplication by s.
More resources on Laplace Transforms
Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
Introduction to Differential Equations
Learn differential equations: explore methods for solving, modeling, and analyzing them. Taught by the Khan Academy Team.
The Laplace Transform: Theory and Applications
Springer undergraduate text developing the transform rigorously: existence and basic principles, applications to ODEs, integral and difference equations, the Dirac delta via Riemann-Stieltjes integrals, a complex variable chapter, the complex inversion formula, and PDEs, with many worked examples. Next step after the computational toolkit: proves what the tables assume and covers the Bromwich inversion integral. More accessible and modern than Widder.
A Student's Guide to Laplace Transforms
Short conceptual text explaining what the transform does, its convergence region, and its relationship to Fourier and Z-transforms, then applying it to ODEs, PDEs, and engineering systems. Fills the gap between intuition videos and theory texts for self-learners who want the meaning of s and the region of convergence, not just a table. Author-produced podcasts and interactive solutions to every problem are free on the book's website.
Notes on Diffy Qs, Chapter 6: The Laplace Transform
Free open textbook chapter covering the transform and its inverse, transforms of derivatives and solving ODEs, convolution, the Dirac delta and impulse response, and applying the transform to PDEs, with exercises and answers. A real textbook treatment with proofs-light but careful reasoning and graded exercises, between Paul's Notes (procedural) and Schiff (theory). Also available as a print paperback.
Paul's Online Notes: Laplace Transforms
Best free source of procedural fluency: partial-fraction inversions and step/delta IVPs worked line by line, which is where students actually get stuck. Ten-section chapter with worked examples: definition, computing transforms and inverses, step functions, solving IVPs, nonconstant-coefficient IVPs, discontinuous forcing, the Dirac delta, convolution integrals, and a printable transform table. Practice problems with solutions accompany each section.