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Reference Materials for Mathematics

Remkes Kooistra

Section 4.9 Laplace Transforms

Subsection 4.9.1 Definitions

A function \(f(t)\) defined on \((0, \infty)\) is said to be of exponential type \(\alpha\) if there exists a constant \(M\) such that for sufficiently large \(t\text{,}\) \(f(t) \lt M e^{\alpha t}\text{.}\) If \(f\) is of exponential type \(\alpha\) then the Laplace transform of \(f\) is a function of \(s\) with domain \((\alpha, \infty)\) defined as:
\begin{equation*} \calL\{f(t)\}(s) = \int_0^\infty f(t) e^{-ts} dt\text{.} \end{equation*}
If \(f\) and \(g\) are two integrable function on \((0, \infty)\text{,}\) then the convolution of \(f\) and \(g\) is a function on \((0, \infty)\) defined as:
\begin{equation*} f \star g (t) = \int_{0}^t f(\tau) g(t - \tau) d\tau\text{.} \end{equation*}

Subsection 4.9.2 Rules

For these rules, \(F(s)\) is the Laplace transform of \(f(t)\text{,}\) and \(G(s)\) is the Laplace transform of \(g(t)\text{.}\)
\begin{align*} \amp \text{Function } \amp \amp \text{ Laplace Transform}\\ \amp \alpha f(t) \amp \amp \alpha F(s)\\ \amp f(t) + g(t) \amp \amp F(s) + G(s)\\ \amp f(t) \star g(t) \amp \amp F(s) G(s)\\ \amp f(\alpha t) \amp \amp \frac{1}{\alpha} F\left( \frac{s}{\alpha} \right)\\ \amp \frac{df}{dt} \amp \amp sF(s) - f(0) \\ \amp \frac{d^n f}{dt^n} \amp \amp s^n F(s) - s^{n-1} f(0) - s^{n-2} f^\prime(0) - \ldots - s f^{(n-2)}(0) - f^{(n-1)}(0)\\ \amp \int_0^t f(\tau) dt \amp \amp \frac{F(s)}{s} \\ \amp t f(t) \amp \amp -\frac{dF}{ds}\\ \amp t^n f(t) \amp \amp (-1)^n \frac{d^n F}{ds^n} \\ \amp \frac{f(t)}{t} \amp \amp \int_s^\infty F(\tau ) d\tau\\ \amp e^{\alpha t} f(t) \amp \amp F(s-\alpha)\\ \amp u_\alpha(t) f(t-\alpha) \amp \amp e^{-\alpha s}F(s)\\ \amp f(t) \text{ with period } T \amp \amp \frac{1}{1-e^{-sT}} \int_0^T f(t) e^{-st} dt. \end{align*}

Subsection 4.9.3 Some Common Laplace Transforms

\begin{align*} \amp \text{Function} \amp \amp \text{Laplace Transform} \\ \amp 1 \amp \amp \frac{1}{s} \\ \amp t^n \amp \amp \frac{n!}{s^{n+1}} \\ \amp e^{\alpha t} \amp \amp \frac{1}{s-\alpha } \\ \amp \ln t \amp \amp \frac{-1}{s} (\ln s + \gamma) \\ \amp \sin \beta t \amp \amp \frac{\beta}{s^2 + \beta^2} \\ \amp \cos \beta t \amp \amp \frac{s}{s^2 + \beta^2} \\ \amp \sinh \beta t \amp \amp \frac{\beta}{s^2 - \beta^2} \\ \amp \cosh \beta \amp \amp \frac{s}{s^2 - \beta^2} \\ \amp e^{-\alpha t} \sin \beta t \amp \amp \frac{\beta}{(s+\alpha)^2 + \beta^2}\\ \amp e^{-\alpha t} \cos \beta t \amp \amp \frac{(s + \alpha)}{(s+\alpha)^2 + \beta^2}\\ \amp \delta_a(t) \amp \amp e^{-as} \\ \amp u_a(t) \amp \amp \frac{e^{-as}}{s} \end{align*}
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