Laplace Transform of Second Derivative

Theorem

Let $f: \R \to \R$ or $\R \to \C$ be a continuous function on any interval of the form $0 \le t \le a$.

Let $f$ be twice differentiable.

Let $f'$ be continuous and $f' '$ piecewise continuous with one-sided limits on said intervals.

Let $f$ and $f'$ be of exponential order.

Let $\laptrans f$ denote the Laplace transform of $f$.


Then $\laptrans {f' '}$ exists for $\map \Re s > a$, and:

$\laptrans {\map {f' '} t} = s^2 \laptrans {\map f t} - s \, \map f 0 - \map {f'} 0$


Proof


This article, or a section of it, needs explaining.
In particular: Why does $\laptrans {f' '}$ exist? It needs to be proven
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Explain}} from the code.


\(\ds \laptrans {\map {f' '} t}\) \(=\) \(\ds s \laptrans {\map {f'} t} - \map {f'} 0\) Laplace Transform of Derivative
\(\ds \) \(=\) \(\ds s \paren {s \laptrans {\map f t} - \map f 0} - \map {f'} 0\) Laplace Transform of Derivative
\(\ds \) \(=\) \(\ds s^2 \laptrans {\map f t} - s \, \map f 0 - \map {f'} 0\)

$\blacksquare$


Also see


Sources

  • 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Some Important Properties of Laplace Transforms: $5$. Laplace transform of derivatives: Theorem $1 \text{-} 9$
  • 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Solved Problems: Laplace Transform of Derivative: $14$
  • 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Appendix $\text A$: Table of General Properties of Laplace Transforms: $6.$
  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 32$: Table of General Properties of Laplace Transforms: $32.8$
  • 2009: William E. Boyce and Richard C. DiPrima: Elementary Differential Equations and Boundary Value Problems (9th ed.): $\S 6.2$
  • 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 33$: Laplace Transforms: Table of General Properties of Laplace Transforms: $33.8$
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.