Mathematical equation: Laplace transformation

Source: Internet
Author: User

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

Contact Us

The content source of this page is from Internet, which doesn't represent Alibaba Cloud's opinion; products and services mentioned on that page don't have any relationship with Alibaba Cloud. If the content of the page makes you feel confusing, please write us an email, we will handle the problem within 5 days after receiving your email.

If you find any instances of plagiarism from the community, please send an email to: info-contact@alibabacloud.com and provide relevant evidence. A staff member will contact you within 5 working days.

A Free Trial That Lets You Build Big!

Start building with 50+ products and up to 12 months usage for Elastic Compute Service

  • Sales Support

    1 on 1 presale consultation

  • After-Sales Support

    24/7 Technical Support 6 Free Tickets per Quarter Faster Response

  • Alibaba Cloud offers highly flexible support services tailored to meet your exact needs.