Преобразованием Лапласа некоторой функции
называется интегральное преобразование вида:
Функция
называется оригиналом, функция
— изображением. Причём
является функцией комлексной переменной, т.е.
.
В качестве примера, найдём изображение
функции оригинала
.
Для этого нам необходимо воспользоваться приведённой выше формулой и
вычислить интеграл:
То, что функция
является изображением функции
записывается как
или
.
Важным свойством
преобразования Лапласа
является то, что если
, то
Указанное свойство активно используется при
решении дифференциальных уравнений
поскольку позволяет сводить последние к алгебраическим.
Наш онлайн калькулятор, построенный на основе системы Wolfram Alpha позволяет найти преобразование Лапласа практически любой, даже очень сложной функции.
Интегральное преобразование Лапласа онлайн, калькулятор изображения функций.
Теория функций комплексного переменного.
: x^a
: x^aмодуль x: abs(x)
- : Sqrt[x]
- : x^(1/n)
- : a^x
- : Log[a, x]
- : Log[x]
- : cos[x] или Cos[x]
: Sqrt[x]![sqrt[n]{x}](https://upload.wikimedia.org/math/5/e/4/5e4352778f3b156f05ef056f9793ec36.png)
: x^(1/n)
: a^x
: Log[a, x]
: Log[x]
: cos[x] или Cos[x]
: sin[x] или Sin[x]
: tan[x] или Tan[x]
: cot[x] или Cot[x]
: sec[x] или Sec[x]
: csc[x] или Csc[x]
: ArcCos[x]
: ArcSin[x]
: ArcTan[x]
: ArcCot[x]
: ArcSec[x]
: ArcCsc[x]
: cosh[x] или Cosh[x]
: sinh[x] или Sinh[x]
: tanh[x] или Tanh[x]
: coth[x] или Coth[x]
: sech[x] или Sech[x]
: csch[x] или Csch[е]
: ArcCosh[x]
: ArcSinh[x]
: ArcTanh[x]
: ArcCoth[x]
: ArcSech[x]
: ArcCsch[x]
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Number Line
Examples
-
laplace:e^{frac{t}{2}}
-
laplace:e^{-2t}sin^{2}(t)
-
laplace:8pi
-
laplace:g(t)=3sinh(2t)+3sin(2t)
-
inverse:laplace:frac{s}{s^{2}+4s+5}
-
inverse:laplace:frac{1}{x^{frac{3}{2}}}
-
inverse:laplace:frac{sqrt{pi}}{3x^{frac{3}{2}}}
-
inverse:laplace:frac{5}{4x^2+1}+frac{3}{x^3}-5frac{3}{2x}
- Show More
Description
Find the Laplace and inverse Laplace transforms of functions step-by-step
Frequently Asked Questions (FAQ)
-
How do you calculate the Laplace transform of a function?
- The Laplace transform of a function f(t) is given by: L(f(t)) = F(s) = ∫(f(t)e^-st)dt, where F(s) is the Laplace transform of f(t), s is the complex frequency variable, and t is the independent variable.
-
What is mean by Laplace equation?
- The Laplace equation is a second-order partial differential equation that describes the distribution of a scalar quantity in a two-dimensional or three-dimensional space. The Laplace equation is given by: ∇^2u(x,y,z) = 0, where u(x,y,z) is the scalar function and ∇^2 is the Laplace operator.
-
What kind of math is Laplace?
- Laplace transforms are a type of mathematical operation that is used to transform a function from the time domain to the frequency domain. They are a specific example of a class of mathematical operations called integral transforms.
-
Why is it called Laplace?
- The Laplace equation is named after the discoverer Pierre-Simon Laplace, a French mathematician and physicist who made significant contributions to the field of mathematics and physics in the 18th and 19th centuries.
-
What does the Laplace equation use for?
- The Laplace equations are used to describe the steady-state conduction heat transfer without any heat sources or sinks
- Show more
laplace-calculator
en
Related Symbolab blog posts
Advanced Math Solutions – Laplace Calculator, Laplace Transform
In previous posts, we talked about the four types of ODE — linear first order, separable, Bernoulli, and exact….
Read More
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19:26 найти оригинал функции по ее изображению |
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Калькулятор для оригиналов функций по изображению (калькулятор обратного преобразования Лапласа ) |
Категория: Теория функций комплексного переменного | Просмотров: 34918 | Добавил: Admin | Теги: тфкп | Рейтинг: 4.5/2 |
Laplace Transform Calculator is a free online tool that displays the transformation of the real variable function to the complex variable. BYJU’S online Laplace transform calculator tool makes the calculations faster and the integral change is displayed in a fraction of seconds.
How to Use the Laplace Transform Calculator?
The procedure to use the Laplace transform calculator is as follows:
Step 1: Enter the function, variable of function, transformation variable in the input field
Step 2: Click the button “Calculate” to get the integral transformation
Step 3: The result will be displayed in the new window
What is the Laplace Transform?
In Mathematics, the Laplace transform is an integral transformation, which transforms the real variable function “t” to the complex variable function. The main purpose of this transformation is to convert the ordinary differential equations into an algebraic equation that helps to solve the ordinary differential equations easily. Laplace transform has many applications in the field of Science and Engineering.
Standard Form
The standard form to represent the Laplace transform is as follows:
(begin{array}{l}F(s)=L(f(t))= int_{0}^{infty }e^{-st}f(t)dtend{array} )
Where
f(t) is defined for all real numbers t ≥ 0
“s” is the complex number frequency parameter
Frequently Asked Questions on Laplace Transform Calculator
Q1
What is the main purpose of the Laplace Transform?
The main purpose of the Laplace transform is to transform the Ordinary Differential Equations (ODEs) into algebraic equations. This conversion helps to solve the ODEs easily.
Q2
Mention the significance of the Laplace Transform
The Laplace transform gives useful techniques for determining certain types of differential equations when initial conditions are given, especially when the primary values are zero.
Q3
Write down the applications of the Laplace transform.
The applications of the Laplace transform are:
- Circuit Analysis
- Signal processing
- Communication Systems