Integral transforms, operational calculus Laplace transform: definition and convergence Concept
Laplace transform
Summary
The Laplace transform is an integral transform. It sends a function of a real variable to the function of a complex variable . It is named after Pierre-Simon Laplace. See the definition.
Where this leads
15 pages build on this one.
- Heaviside expansion theorem
- Operational calculus
- Paley–Wiener theorem for Volterra resolvents
- Euler–Lotka equation
- Sumudu transform
- Iterated Laplace transform equals Stieltjes transform
- Starred transform
- Laguerre series expansion of the Laplace transform
- Bromwich justification of the Heaviside calculus
- Laplace transform of distributions
- Volterra equations with completely monotone kernels
- Resolvent as Laplace transform of a semigroup
- Multidimensional Laplace transform
- Laplace transform method for convolution equations
- Bernstein's theorem on monotone functions
Further reading
- Web https://en.wikipedia.org/wiki/Laplace_transform, lead section
About this page
| Tag | 0V49JEZW06 |
|---|---|
| URI | m2://0V49JEZW06 |
| Kind | concept |
| Writing | W1 stated |
| MSC | 44A10 |
| Same as |
Markdown · JSON · Edit on GitHub
What this page needs for W2
- edge instance_of in: at least two examples (0 of 2)
- edge not_instance_of in: at least one non-example (0 of 1)
The writing levels say how complete the text of a page is. They do not measure its truth.
What links here
Definitions
Results about this
- Abelian theorems for integral transforms
- Analyticity of the Laplace transform in its half-plane
- Final value theorem
- Watson's lemma
- Frequency shift theorem
- Integration in the transform variable
- Iterated Laplace transform equals Stieltjes transform
- Initial value theorem
- Time shift theorem
- Laplace transform tends to zero at infinity
- Tauberian theorems for integral transforms
- Scaling rule for the Laplace transform
- Linearity of the Laplace transform
- Lerch's uniqueness theorem
- Differentiation in the transform variable
- Laplace transform of an integral
Used by
- Heaviside expansion theorem
- Operational calculus
- Paley–Wiener theorem for Volterra resolvents
- Euler–Lotka equation
- Sumudu transform
- Iterated Laplace transform equals Stieltjes transform
- Starred transform
- Laguerre series expansion of the Laplace transform
- Bromwich justification of the Heaviside calculus
- Laplace transform of distributions
- Resolvent as Laplace transform of a semigroup
- Multidimensional Laplace transform
- Laplace transform method for convolution equations
- Bernstein's theorem on monotone functions
Examples
- Laplace transform of the Dirac delta
- Laplace transform of the error function
- Laplace transforms of sine and cosine
- Laplace transform of a power function
- Laplace transform method for the heat equation
- Laplace transform of the Heaviside step function
- Laplace transform of the exponential function
- Laplace transform of a periodic function