Higher Order Derivatives of Laplace Transform
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 $\dfrac {\partial f}{\partial s}$, the partial derivative of $f$ with respect to $s$, exist and be continuous on said intervals.
Let $\laptrans f = F$ denote the Laplace transform of $f$.
Then, everywhere that $\laptrans f$ exists and is $n$ times differentiable:
- $\dfrac {\d^n} {\d s^n} \laptrans {\map f t} = \paren {-1}^n \laptrans {t^n \, \map f t}$
Proof
The proof proceeds by induction on $n$, the order of the derivative of $\laptrans f$.
The case $n = 0$ is verified as follows:
\(\ds \frac {\d^0} {\d s^0} \laptrans {\map f t}\) | \(=\) | \(\ds \laptrans {\map f t}\) | Definition of Zeroth Derivative | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-1}^0 \laptrans {t^0 \, \map f t}\) | Definition of Zeroth Power |
Basis for the Induction
The case $n = 1$ is demonstrated in Derivative of Laplace Transform:
- $\dfrac \d {\d s} \laptrans {\map f t} = -\laptrans {t \, \map f t}$
This is the basis for the induction.
Induction Hypothesis
Fix $n \in \N$ with $n \ge 0$.
Assume:
- $\dfrac {\d^n} {\d s^n} \laptrans {\map f t} = \paren {-1}^n \laptrans {t^n \, \map f t}$
This is our induction hypothesis.
Induction Step
This is our induction step:
\(\ds \frac {\d^{n + 1} } {\d s^{n + 1} } \laptrans {\map f t}\) | \(=\) | \(\ds \frac {\d}{\d s} \frac {\d^n} {\d s^n} \laptrans {\map f t}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\d}{\d s} \paren {\paren {-1}^n \laptrans {t^n \, \map f t} }\) | Induction Hypothesis | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-1}^n \paren {-\laptrans {t \times t^n \, \map f t} }\) | Derivative of Laplace Transform | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-1}^{n + 1} \laptrans {t^{n + 1} \, \map f t}\) | simplification |
The result follows by the Principle of Mathematical Induction.
$\blacksquare$
Examples
Example $1$
- $\laptrans {t^2 e^{2 t} } = \dfrac 2 {\paren {s - 2}^3}$
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: $7$. Multiplication by $t^n$: Theorem $1 \text{-} 12$
- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Solved Problems: Multiplication by Powers of $t$: $19$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 32$: Table of General Properties of Laplace Transforms: $32.12$