Math 2.0

Integral transforms, operational calculus Laplace transform: operational rules Theorem

Differentiation in the transform variable

Builds on Laplace transformAbscissa of absolute convergenceHolomorphic function

Statement

Let f ⁣:[0,∞)→ℂf \colon [0, \infty) \to \C be integrable on each interval [0,T][0, T], and let FF be its Laplace transform. Let aa be a real number such that ∫0∞∣f(t)∣e−σt dt<∞\int_0^\infty \abs{f(t)} e^{-\sigma t} \, dt < \infty for every σ>a\sigma > a. Then FF is holomorphic on the half-plane Re⁡s>a\operatorname{Re} s > a, and for every integer n≥1n \ge 1 and every ss with Re⁡s>a\operatorname{Re} s > a, F(n)(s)=∫0∞(−t)nf(t) e−st dt=ℒ{(−t)nf(t)}(s).F^{(n)}(s) = \int_0^\infty (-t)^n f(t) \, e^{-s t} \, dt = \mathcal{L}\{(-t)^n f(t)\}(s).

Remarks

The hypothesis says that the Laplace integral converges absolutely to the right of aa; the least such aa is the abscissa of absolute convergence. For Re⁡s≥σ>σ′>a\operatorname{Re} s \ge \sigma > \sigma' > a, one has tne−σt≤Ce−σ′tt^n e^{-\sigma t} \le C e^{-\sigma' t}, so tn∣f(t)∣e−σtt^n \abs{f(t)} e^{-\sigma t} is integrable. This domination justifies differentiation under the integral sign. Example: from ℒ{ebt}(s)=1/(s−b)\mathcal{L}\{e^{b t}\}(s) = 1/(s - b) one gets ℒ{tebt}(s)=1/(s−b)2\mathcal{L}\{t e^{b t}\}(s) = 1/(s - b)^2. From ℒ{sin⁡ωt}(s)=ω/(s2+ω2)\mathcal{L}\{\sin \omega t\}(s) = \omega/(s^2 + \omega^2) one gets ℒ{tsin⁡ωt}(s)=2ωs/(s2+ω2)2\mathcal{L}\{t \sin \omega t\}(s) = 2 \omega s / (s^2 + \omega^2)^2.

Going deeper

For a student

This rule produces the transforms of tnebtt^n e^{b t}, tcos⁡ωtt \cos \omega t and similar functions, which appear as resonant solutions of linear equations. It also shows that a Laplace transform is infinitely differentiable, and in fact holomorphic, in its half-plane of convergence. The reverse operation is integration in the transform variable.

Reading path

To understand this page from the ground up, read these 15 pages in order. The foundations come first.

Show the path as a graph
Abscissa of absolute convergence, concept, no evidence levelAbscissa of absolute …conceptComplex numbers, object, no evidence levelComplex numbersobjectFunction, concept, no evidence levelFunctionconceptReal numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionLaplace transform, definition, no evidence levelLaplace transformdefinitionSupremum, definition, no evidence levelSupremumdefinitionCompleteness axiom, statement, no evidence levelCompleteness axiomstatementExistence of square roots of nonnegative reals, statement, no evidence levelExistence of square r…statementSquare root, definition, no evidence levelSquare rootdefinitionModulus of a complex number, definition, no evidence levelModulus of a complex …definitionComplex differentiability, definition, no evidence levelComplex differentiabi…definitionHolomorphic function, definition, no evidence levelHolomorphic functiondefinitionDifferentiation in the transform variable, statement, no evidence levelDifferentiation in th…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Abscissa of absolute convergence concept
  2. Complex numbers object
  3. Function concept
  4. Real numbers object
  5. Set bounded above definition
  6. Real and imaginary parts definition
  7. Laplace transform definition
  8. Supremum definition
  9. Completeness axiom statement
  10. Existence of square roots of nonnegative reals statement
  11. Square root definition
  12. Modulus of a complex number definition
  13. Complex differentiability definition
  14. Holomorphic function definition
  15. Differentiation in the transform variable statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Laplace_transform
About this page
TagVEPJN1NK3C
URIm2://VEPJN1NK3C
Kindstatement, theorem
WritingW0 stub
MSC44A10
StatusThis copy of the site has no build data. Run m2 status VEPJN1NK3C.
AboutLaplace transform
UsesLaplace transform, Abscissa of absolute convergence, Holomorphic function

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by