Laplace transform
Laplace transforms are a powerful mathematical tool used to transform differential equations (usually in the time domain) into algebraic equations (in the frequency or complex domain), which simplifies their analysis and solution. They are widely used in engineering, physics, and applied mathematics, especially in the analysis of linear systems and control processes.
Definition
The Laplace transform of a function \(f(t)\), defined for \(t \geq 0\), is:
$$\mathcal{L}\left\lbrace f(t) \right\rbrace = F(s) = \int_{0}^{\infty} s^{-st} f(t) dt $$
where:
- \(f(t)\) is a real or complex function of time \(t\).
- \(s\) is a complex variable: \(s = \sigma + i\omega\).
Main Properties
- Linearity:
- Derivatives:
- Integration:
- Displacement in time, \(u(t - a)\) is the unit step function.
- Convolution Theorem
- Inverse transforms: allow returning from the \(s\)-domain to the time domain.
$$\begin{split}\mathcal{L}\{af(t) + bg(t)\} &= a \mathcal{L}\{f(t)\} + b \mathcal{L}\{g(t)\} \\ &= a F(s) + b G(s) \end{split}$$
$$\begin{split}\mathcal{L}\left\{ \frac{d^n f(t)}{dt^n} \right\} &= s^n \mathcal{L}\{f(t)\} - s^{n-1}f(0) - s^{n-2}f'(0) - \\ &\cdots - f^{(n-1)}(0) \end{split}$$
$$\mathcal{L}\left\{ \int_0^t f(\tau) d\tau \right\} = \frac{1}{s} \mathcal{L}\{f(t)\}$$
$$\mathcal{L}\{f(t-a)u(t-a)\} = e^{-as} \mathcal{L}\{f(t)\}, $$
$$\mathcal{L}\left\{ \left( f \ast g \right) \left( t \right) \right\} = F\left( s \right)G\left( s \right)$$
Basic Example
Laplace transform of a unit step function:
$$\mathcal{L} \left\{ 1 \right\} = \int_{0}^{\infty} e^{-st} dt = \frac{1}{s}, \quad \frak{R}\left( s \right)>0$$
Inverse Laplace Transform Formula
The inverse Laplace transform, denoted as \(\mathcal{L}^{-1}\), is used to recover the original function \(f(t)\) from its transform \(F(s)\). A simple example of the inverse transform is:
$$\mathcal{L}^{-1} \left\{ \frac{1}{s} \right\} = 1$$
The general formula for the inverse Laplace transform is not unique, and methods such as partial fraction decomposition or inverse transform tables can be used to obtain \(f(t)\).
Table
| f(t) | F(s) |
|---|---|
| $$1$$ | $$\frac{1}{s}$$ |
| $$t$$ | $$\frac{1}{s^{2}}$$ |
| $$t^{n}, n = 1,2,3,\ldots$$ | $$\frac{n!}{s^{n + 1}}$$ |
| $$t^{n-\frac{1}{2}}, n = 1,2,3,\ldots$$ | $$\frac{1 \cdot 3 \cdot 5 \cdots (2n - 1)\sqrt{\pi}}{2^{n}s^{n + \frac{1}{2}}}$$ |
| $$t^{\alpha},\alpha > -1$$ | $$\frac{\Gamma(\alpha + 1)}{s^{\alpha + 1}}$$ |
| $$\sqrt{t}$$ | $$\frac{\sqrt{\pi}}{2s^{\frac{3}{2}}}$$ |
| $$e^{at}$$ | $$\frac{1}{s - a}$$ |
| $$t^{n}e^{at}, n = 1,2,3,\ldots$$ | $$\frac{n!}{(s - a)^{n + 1}}$$ |
| $$\sin{\omega t}$$ | $$\frac{\omega}{s^{2} + \omega^{2}}$$ |
| $$\cos{\omega t}$$ | $$\frac{s}{s^{2} + \omega^{2}}$$ |
| $$\sinh{\omega t}$$ | $$\frac{\omega}{s^{2} - \omega^{2}}$$ |
| $$\cosh{\omega t}$$ | $$\frac{s}{s^{2} - \omega^{2}}$$ |
| $$e^{at}\sin{\omega t}$$ | $$\frac{\omega}{(s - a)^{2} + \omega^{2}}$$ |
| $$e^{at}\cos{\omega t}$$ | $$\frac{s - a}{(s - a)^{2} + \omega^{2}}$$ |
| $$e^{at}\sinh{\omega t}$$ | $$\frac{\omega}{(s - a)^{2} - \omega^{2}}$$ |
| $$e^{at}\cosh{\omega t}$$ | $$\frac{s - a}{(s - a)^{2} - \omega^{2}}$$ |
| $$t \sin{\omega t}$$ | $$\frac{2\omega s}{(s^{2} + \omega^{2})^{2}}$$ |
| $$t\cos{\omega t}$$ | $$\frac{s^{2} - \omega^{2}}{(s^{2} + \omega^{2})^{2}}$$ |
| $$\sin{\omega t} - \omega t \cos{\omega t}$$ | $$\frac{2\omega^{3}}{(s^{2} + \omega^{2})^{2}}$$ |
| $$\sin{\omega t} + \omega t \cos{\omega t}$$ | $$\frac{2\omega s^{2}}{(s^{2} + \omega^{2})^{2}}$$ |
| $$\cos \omega t - \omega t \sin \omega t$$ | $$\frac{s\left( s^{2} - \omega^{2} \right)}{\left( s^{2} + \omega^{2} \right)^{2}}$$ |
| $$\cos \omega t + \omega t \sin \omega t$$ | $$\frac{s \left( s^{2} + 3 \omega^{2} \right)}{\left( s^{2} + \omega^{2} \right)^{2}}$$ |
| $$\sin (\omega t + \phi)$$ | $$\frac{s \sin (\phi) + \omega \cos (\phi)}{s^{2} + \omega^{2}}$$ |
| $$\cos (\omega t + \phi)$$ | $$\frac{s \sin (\phi) - \omega \cos (\phi)}{s^{2} + \omega^{2}}$$ |
| $$\frac{1}{a - b}(e^{at} - e^{bt})$$ | $$\frac{1}{(s - a)(s - b)}$$ |
| $$\frac{1}{a - b}(ae^{at} - be^{bt})$$ | $$\frac{s}{(s - a)(s - b)}$$ |
| $$\frac{1}{a^{2}}(1 - \cos{at})$$ | $$\frac{1}{s(s^{2} + a^{2})}$$ |
| $$\frac{1}{a^{3}}(at - \sin{at})$$ | $$\frac{1}{s^{2}(s^{2} + a^{2})}$$ |
| $$f(t) + g(t)$$ | $$F(s) + G(s)$$ |
| $$cf(t)$$ | $$cF(s)$$ |
| $$f^{\prime}(t)$$ | $$sF(s) - f(0)$$ |
| $$f^{\prime\prime}(t)$$ | $$s^{2}F(s) - sf(0) - f^{\prime}(0)$$ |
| $$f^{(n)}(t)$$ | $$s^{n}F(s) - s^{n - 1}f(0) - \ldots - f^{(n - 2)}(0)$$ |
| $$e^{at}f(t)$$ | $$F(s - a)$$ |
| $$t^{n}f(t)$$ | $$(-1)^{n}\frac{d^{n}}{ds^{n}}F(s)$$ |
| $$\delta(t - c)$$ | $$e^{-ts}$$ |
| $$U_{a}(t) = U(t - a)$$ | $$\frac{e^{-as}}{s}$$ |
| $$U_{a}(t)f(t - a)$$ | $$e^{-ts}F(s)$$ |
| $$f \cdot g = \int_{0}^{t}f(t - \sigma)g(\sigma)d\sigma$$ | $$F(s)G(s)$$ |
| $$f(t + T) = f(t)$$ | $$\frac{\int_{0}^{T}e^{-at}f(t)dt}{1 - e^{-sT}}$$ |
| $$\int_{0}^{t}f(\sigma)d\sigma$$ | $$\frac{1}{s}F(s)$$ |
| $$\frac{f(t)}{t}$$ | $$\int_{s}^{\infty}F(\sigma)d\sigma$$ |
| $$f(at)$$ | $$\frac{1}{a}F(\frac{s}{a})$$ |
See also
Fourier transform