# 라플라스 변환

## 개요

- 푸리에 변환의 변형
- 어떤 미분방정식들의 해를 대수적 조작을 통해 얻을 수 있게 해주는 변환
- 라플라스 변환을 미분방정식에 응용한 사람은 Oliver Heaviside http://en.wikipedia.org/wiki/Oliver_Heaviside 이다
- operational calculus 또는 Heaviside calculus 의 도구

## 정의

- 함수 \(f(t)\)에 대한 라플라스 변환을 다음과 같이 정의함\[F(s) = \mathcal{L} \left\{f(t)\right\}=\int_0^{\infty} e^{-st} f(t) \,dt\]

## 성질

- 함수 \(f\)에 대한 도함수의 라플라스 변환은 다음과 같다\[\mathcal{L}\left\{\frac{df}{dt}\right\} = s\cdot\mathcal{L} \left\{ f(t) \right\}-f(0)\]

(정리)

\(f\)가 유계이고, \(t\geq 0\)에서 조각적 연속(piecewise continuous)라 하자.

\(\mathfrak{R}(s)\geq 0\)에서 정의된 함수 \(F(s) = \mathcal{L} \left\{f(t)\right\}=\int_0^{\infty} e^{-st} f(t) \,dt\) 가 \(\mathfrak{R}(s)\geq 0\)에서 해석함수로 확장되면,

\(\int_0^{\infty} f(t) \,dt\)이 존재하고, \(F(0) = \int_0^{\infty} f(t) \,dt\)가 성립한다.

## 예

\(\left(\frac{t^ne^t}{n!}\right)'=\frac{t^{n-1}e^t}{(n-1)!}+\frac{t^ne^t}{n!}\) 로부터 \(\mathcal{L}\left\{\frac{t^{n-1}e^t}{(n-1)!}\right\} = (s-1)\cdot\mathcal{L} \left\{ \frac{t^ne^t}{n!}\right\}\)

\(\mathcal{L}\left\{e^t\right\} = \frac{1}{s-1}\)

\(\mathcal{L}\left\{t e^t\right\} = \frac{1}{(s-1)^2}\)

\(\mathcal{L}\left\{\frac{t^2 e^t}{2!}\right\} = \frac{1}{(s-1)^3}\)

\(\mathcal{L}\left\{\frac{t^3 e^t}{3!}\right\} = \frac{1}{(s-1)^4}\)

...

## 상수계수 미분방정식에의 응용

- \(y''(t)-2 y'(t)+y(t)=e^t\)
- 양변에 라플라스 변환을 취하면,\[s^2 Y(s)+Y(s)-2 (s Y(s)-1)-s+1=\frac{1}{s-1}\], 여기서 \(Y(s)=\mathcal{L} \left\{ f(t) \right\}\).
- \(Y(s)=\frac{1}{s-1}-\frac{2}{(s-1)^2}+\frac{1}{(s-1)^3}\)
- \(y(t)=e^t-2t e^t+\frac{t^2}{2}e^t\) 는 주어진 미분방정식의 해가 된다

## 멜린변환과의 관계

- 푸리에 변환 항목 참조\[\hat{f}(s)= \int_{0}^{\infty} f(x) x^{s}\frac{dx}{x}\]
- 멜린변환에서 \(x=e^{-t}\)로 변수를 치환하면, 라플라스 변환을 얻는다\[\int_{0}^{\infty} f(e^{-t}) e^{-st}\,dt\]

## 역사

- 오일러
- 라플라스
- 헤비사이드
- 수학사 연표

## 메모

## 관련된 항목들

## 매스매티카 파일 및 계산 리소스

## 사전 형태의 자료

- http://ko.wikipedia.org/wiki/라플라스_변환
- http://en.wikipedia.org/wiki/Laplace_transform
- http://en.wikipedia.org/wiki/Laplace–Stieltjes_transform
- http://en.wikipedia.org/wiki/Moment_(mathematics)

## 노트

### 말뭉치

- This limit emphasizes that any point mass located at 0 is entirely captured by the Laplace transform.
^{[1]} - When one says "the Laplace transform" without qualification, the unilateral or one-sided transform is usually intended.
^{[1]} - The Laplace transform can be alternatively defined as the bilateral Laplace transform, or two-sided Laplace transform, by extending the limits of integration to be the entire real axis.
^{[1]} - Two integrable functions have the same Laplace transform only if they differ on a set of Lebesgue measure zero.
^{[1]} - Laplace transforms and Fourier transforms are probably the main two kinds of transforms that are used.
^{[2]} - As we will see in later sections we can use Laplace transforms to reduce a differential equation to an algebra problem.
^{[2]} - Laplace transforms can also be used to solve IVP’s that we can’t use any previous method on.
^{[2]} - For “simple” differential equations such as those in the first few sections of the last chapter Laplace transforms will be more complicated than we need.
^{[2]} - The Laplace transform is particularly useful in solving linear ordinary differential equations such as those arising in the analysis of electronic circuits.
^{[3]} - The inverse Laplace transform is known as the Bromwich integral, sometimes known as the Fourier-Mellin integral (see also the related Duhamel's convolution principle).
^{[3]} - A table of several important one-sided Laplace transforms is given below.
^{[3]} - The Laplace transform has many important properties.
^{[3]} - The Laplace transform is used to quickly find solutions for differential equations and integrals.
^{[4]} - Like all transforms, the Laplace transform changes one signal into another according to some fixed set of rules or equations.
^{[5]} - In general, the Laplace transform is used for applications in the time-domain for t ≥ 0.
^{[6]} - The Laplace transform generates nonperiodic solutions.
^{[6]} - We will illustrate the usability of the Laplace transform in section 8.2.5 where we discuss an example using the Laplace transform to solve an ODE.
^{[6]} - In section 8.3.7 we will use the Laplace transform for solving a PDE.
^{[6]} - As mentioned in another answer, the Laplace transform is defined for a larger class of functions than the related Fourier transform.
^{[7]} - Fourier transforms are often used to solve boundary value problems, Laplace transforms are often used to solve initial condition problems.
^{[7]} - Also, the Laplace transform succinctly captures input/output behavior or systems described by linear ODEs.
^{[7]} - The Laplace transform 3{6 sinusoid: rst express f (t) = cos !
^{[8]} - The Laplace transform f(p), also denoted by L{F(t)} or Lap F(t), is defined by the integral involving the exponential parameter p in the kernel K = e−pt.
^{[9]} - Therefore, sin kx and cos kx each have a Laplace transform, since they are continuous and bounded functions.
^{[10]} - Furthermore, any function of the form e kx , as well as any polynomial, is continuous and, although unbounded, is of exponential order and therefore has a Laplace transform.
^{[10]} - Therefore, it has a Laplace transform.
^{[10]} - This example introduces the idea of the inverse Laplace transform operator,, L −1.
^{[10]} - For a function defined on , its Laplace transform is denoted as obtained by the following integral: where is real and is called the Laplace Transform Operator.
^{[11]} - In this chapter we will discuss the Laplace transform.
^{[12]} - The Laplace transform turns out to be a very efficient method to solve certain ODE problems.
^{[12]} - The Laplace transform also has applications in the analysis of electrical circuits, NMR spectroscopy, signal processing, and elsewhere.
^{[12]} - Finally, understanding the Laplace transform will also help with understanding the related Fourier transform, which, however, requires more understanding of complex numbers.
^{[12]} - In this article, we will be discussing Laplace transforms and how they are used to solve differential equations.
^{[13]} - Many kinds of transformations already exist but Laplace transforms and Fourier transforms are the most well known.
^{[13]} - The Laplace transforms is usually used to simplify a differential equation into a simple and solvable algebra problem.
^{[13]} - There is always a table that is available to the engineer that contains information on the Laplace transforms.
^{[13]} - Compute the Laplace transform of exp(-a*t) .
^{[14]} - Problems Now that we know how to find a Laplace transform, it is time to use it to solve differential equations.
^{[15]} - Proof To prove this theorem we just use the definition of the Laplace transform and integration by parts.
^{[15]} - What this tells us is that if we have a differential equation, then the Laplace transform will turn it into an algebraic equation.
^{[15]} - Instead we will see that the method of Laplace Transforms tackles the entire problem with one fell swoop.
^{[15]} - Laplace transforms convert a function f(t) in the time domain into function in the Laplace domain F(s).
^{[16]} - The domain of its Laplace transform depends on f and can vary from a function to a function.
^{[17]} - However, the Laplace transform , an integral transform, allows us to change a differential equation to an algebraic equation.
^{[18]} - We shall define the Laplace transform of a function \(f(t)\) by \begin{equation*} {\mathcal L}(f)(s)= F(s) = \int_0^\infty e^{-st} f(t) \, dt, \end{equation*} provided the integral converges.
^{[18]} - However, before we investigate these properties, let us compute several Laplace transforms.
^{[18]} - The Laplace transform of a function does not always exist, even for functions that are infinitely differentiable.
^{[18]} - Two weeks ago I did that with the Laplace transform, and now I feel like it finally makes sense to me.
^{[19]} - Wikipedia sketches this briefly, but I would never have discovered it there because I was looking for that generalization rather than looking for a way to understand the Laplace transform.
^{[19]} - In the first part, we give an explicit formula for the Laplace transform and verify that this formula satisfies properties (??).
^{[20]} - In the second part, we compute a table of Laplace transforms for a number of special functions including step functions and impulse functions.
^{[20]} - Then the function is defined by and is called the Laplace transform of .
^{[20]} - Note that the Laplace transform is an improper integral — which implies that some care must be taken when discussing its properties.
^{[20]} - The Laplace transform of a function f (t) is dened for those values of s at which the integral converges.
^{[21]} - Note that the Laplace transform of f (t) is a function of s. Hence the transform is sometimes denoted L{f (t)}(s), L{f }(s), or simply F (s).
^{[21]} - You can integrate by parts obtain the Laplace transform of f (t) = t: Integrate by parts n times to get L{tn} = tnest dt (cid:90) 0 = n! sn+1 , for s > 0, and n = 0, 1, 2, . . .
^{[21]} - As (x) generalizes the factorial, the Laplace transform (8) generalizes (5).
^{[21]} - CHAPTER 1 Laplace Transform Methods Laplace transform is a method frequently employed by engineers.
^{[22]} - By applying the Laplace transform, one can change an ordinary dif- ferential equation into an algebraic equation, as algebraic equation is generally easier to deal with.
^{[22]} - Our rst theorem states when Laplace transform can be performed, Theorem 1.1.
^{[22]} - The next result shows that Laplace transform is unique in the sense that dierent continuous functions will have dierent Laplace trans- form.
^{[22]} - The convolution theorem for Laplace transform is a useful tool for solving certain Laplace transforms.
^{[23]} - An extremely powerful tool that helps us to solve this kind of real world problems are the Laplace transforms.
^{[24]} - But first let us become familiar with the Laplace transform itself.
^{[24]} - You can say that we "Laplace transform f from the t-space into F inside of the s-space.
^{[24]} - We can actually use the linearity in order to find even more new Laplace transforms.
^{[24]}

### 소스

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}Laplace transform - ↑
^{2.0}^{2.1}^{2.2}^{2.3}Differential Equations - ↑
^{3.0}^{3.1}^{3.2}^{3.3}Laplace Transform -- from Wolfram MathWorld - ↑ Laplace transform table ( F(s) = L{ f(t) } )
- ↑ Laplace Transform- Definition, Properties, Formulas, Equation & Examples
- ↑
^{6.0}^{6.1}^{6.2}^{6.3}Laplace Transforms - an overview - ↑
^{7.0}^{7.1}^{7.2}What exactly is Laplace transform? - ↑ S. boyd
- ↑ Laplace transform | mathematics
- ↑
^{10.0}^{10.1}^{10.2}^{10.3}The Laplace Transform Operator - ↑ eFunda: Laplace Transforms
- ↑
^{12.0}^{12.1}^{12.2}^{12.3}6.1: The Laplace Transform - ↑
^{13.0}^{13.1}^{13.2}^{13.3}Laplace Transform Table, Formula, Examples & Properties - ↑ Laplace transform
- ↑
^{15.0}^{15.1}^{15.2}^{15.3}Using the Laplace Transform to Solve Initial Value Problems - ↑ Laplace Transforms
- ↑ The laplace transform
- ↑
^{18.0}^{18.1}^{18.2}^{18.3}The Laplace Transform - ↑
^{19.0}^{19.1}The n-Category Café - ↑
^{20.0}^{20.1}^{20.2}^{20.3}Laplace Transforms and Their Computation - ↑
^{21.0}^{21.1}^{21.2}^{21.3}1. the laplace transform of a function f (t) is - ↑
^{22.0}^{22.1}^{22.2}^{22.3}Chapter 1 - ↑ Brilliant Math & Science Wiki
- ↑
^{24.0}^{24.1}^{24.2}^{24.3}Laplace Transforms:

## 메타데이터

### 위키데이터

- ID : Q199691

### Spacy 패턴 목록

- [{'LOWER': 'laplace'}, {'LEMMA': 'transform'}]