Mathematical equation: Laplace transformation

Updated: 1 APR 2016

Laplace transform

Set function \ (f (t) \) is defined at \ (t>0\), integral

\ (F (s) =\int_0^{+\infty}f (t) e^{-st}dt \qquad (S\in \mathbb{c})

If convergence is within a domain of s, this mapping is called the Laplace transform, which is recorded as

\ (f (s) =\mathscr{l}[f (T)],\qquad f (t) =\mathscr{l}^{-1}[f (s)]\)

In fact, the Laplace transform of \ (f (t) \) is the Fourier transform (f (t) u (t) E^{-\beta T} (\beta>0) \).

Laplace Transformation Properties

1. Linear

2. Differential

\ (\mathscr{l}[f ' (t)]=s\mathscr{l}[f (t)]-f (0) \)

\ (\mathscr{l}[f^{(n)} (t)]=s^n\mathscr{l}[f (t)]-s^{n-1}f (0)-s^{n-2}f ' (0)-\cdots-f^{(n-1)} (0) \)

3. Integration

\ (\MATHSCR{L}\LEFT[\INT_0^TF (t) dt\right]=\dfrac{1}{s}\mathscr{l}[f (t)]\)

4. Displacement Properties

5. Delay Nature

6. Similar properties

7. Initial value theorem

8. The final value theorem

Laplace inverse transformation

The Fourier transform can be used to derive

\ (f (t) =\dfrac{1}{2\pi\mathrm{i}}\int_{\beta-\mathrm{i}\omega}^{\beta+\mathrm{i}\omega}f (s) e^{st}ds, t>0\)

Integral becomes Laplace inversion integral. This inversion integral can be calculated using the left number:

If \ (S_1, s_2, ..., s_n\) is the function \ (f (s) \) of all the singularities, and when \ (S \rightarrow \infty\) \ (f (s) \rightarrow 0\), then

\ (f (t) =\dfrac{1}{2\pi \mathrm{i}}\int_{\beta-\mathrm{i}\omega}^{\beta+\mathrm{i}\omega}f (s) e^{st}ds=\sum\limits _{k=1}^{n}\underset{s=s_k}{\operatorname{res}}[f (s) e^{st}]\)

Mathematical equation: Laplace transformation