Derivative of Laplace Transform

Theorem

Let $f: \R \to \R$ or $\R \to \C$ be a continuous function, differentiable on any closed interval $\closedint 0 a$.

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


Then, everywhere that $\dfrac \d {\d s} \laptrans f$ exists:

$\dfrac \d {\d s} \laptrans {\map f t} = -\laptrans {t \, \map f t}$


Proof


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\(\ds \frac \d {\d s} \laptrans {\map f t}\) \(=\) \(\ds \frac \d {\d s} \int_0^{\to +\infty} \map f t \, e^{-s t} \rd t\) Definition of Laplace Transform
\(\ds \) \(=\) \(\ds \int_0^{\to +\infty} \map {\frac {\partial} {\partial s} } {\map f t \, e^{-s t} } \rd t\) Definite Integral of Partial Derivative
\(\ds \) \(=\) \(\ds \int_0^{\to +\infty} \map f t \, \map {\frac {\partial} {\partial s} } {e^{-st} } \rd t\)
\(\ds \) \(=\) \(\ds -\int_0^{\to +\infty} t \, \map f t \, e^{-s t} \rd t\) Derivative of Exponential Function
\(\ds \) \(=\) \(\ds -\laptrans {t \, \map f t}\) Definition of Laplace Transform

$\blacksquare$


Examples

Example 1

$\ds \int_0^\infty t e^{-2 t} \cos t \rd t = \dfrac 3 {25}$


Also see


Sources

  • 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Appendix $\text A$: Table of General Properties of Laplace Transforms: $8.$
  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 32$: Table of General Properties of Laplace Transforms: $32.10$
  • 2009: William E. Boyce and Richard C. DiPrima: Elementary Differential Equations and Boundary Value Problems (9th ed.): $\S 6.1$
  • 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: $33.10$
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