# 라플라스 변환

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## 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$\int_{0}^{\infty} f(e^{-t}) e^{-st}\,dt$

## 노트

### 말뭉치

1. This limit emphasizes that any point mass located at 0 is entirely captured by the Laplace transform.
2. When one says "the Laplace transform" without qualification, the unilateral or one-sided transform is usually intended.
3. 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.
4. Two integrable functions have the same Laplace transform only if they differ on a set of Lebesgue measure zero.
5. Laplace transforms and Fourier transforms are probably the main two kinds of transforms that are used.
6. As we will see in later sections we can use Laplace transforms to reduce a differential equation to an algebra problem.
7. Laplace transforms can also be used to solve IVP’s that we can’t use any previous method on.
8. 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.
9. The Laplace transform is particularly useful in solving linear ordinary differential equations such as those arising in the analysis of electronic circuits.
10. 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).
11. A table of several important one-sided Laplace transforms is given below.
12. The Laplace transform has many important properties.
13. The Laplace transform is used to quickly find solutions for differential equations and integrals.
14. Like all transforms, the Laplace transform changes one signal into another according to some fixed set of rules or equations.
15. In general, the Laplace transform is used for applications in the time-domain for t ≥ 0.
16. The Laplace transform generates nonperiodic solutions.
17. 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.
18. In section 8.3.7 we will use the Laplace transform for solving a PDE.
19. As mentioned in another answer, the Laplace transform is defined for a larger class of functions than the related Fourier transform.
20. Fourier transforms are often used to solve boundary value problems, Laplace transforms are often used to solve initial condition problems.
21. Also, the Laplace transform succinctly captures input/output behavior or systems described by linear ODEs.
22. The Laplace transform 3{6 sinusoid: rst express f (t) = cos !
23. 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.
24. Therefore, sin kx and cos kx each have a Laplace transform, since they are continuous and bounded functions.
25. 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.
26. Therefore, it has a Laplace transform.
27. This example introduces the idea of the inverse Laplace transform operator,, L −1.
28. 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.
29. In this chapter we will discuss the Laplace transform.
30. The Laplace transform turns out to be a very efficient method to solve certain ODE problems.
31. The Laplace transform also has applications in the analysis of electrical circuits, NMR spectroscopy, signal processing, and elsewhere.
32. Finally, understanding the Laplace transform will also help with understanding the related Fourier transform, which, however, requires more understanding of complex numbers.
33. In this article, we will be discussing Laplace transforms and how they are used to solve differential equations.
34. Many kinds of transformations already exist but Laplace transforms and Fourier transforms are the most well known.
35. The Laplace transforms is usually used to simplify a differential equation into a simple and solvable algebra problem.
36. There is always a table that is available to the engineer that contains information on the Laplace transforms.
37. Compute the Laplace transform of exp(-a*t) .
38. Problems Now that we know how to find a Laplace transform, it is time to use it to solve differential equations.
39. Proof To prove this theorem we just use the definition of the Laplace transform and integration by parts.
40. What this tells us is that if we have a differential equation, then the Laplace transform will turn it into an algebraic equation.
41. Instead we will see that the method of Laplace Transforms tackles the entire problem with one fell swoop.
42. Laplace transforms convert a function f(t) in the time domain into function in the Laplace domain F(s).
43. The domain of its Laplace transform depends on f and can vary from a function to a function.
44. However, the Laplace transform , an integral transform, allows us to change a differential equation to an algebraic equation.
45. 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.
46. However, before we investigate these properties, let us compute several Laplace transforms.
47. The Laplace transform of a function does not always exist, even for functions that are infinitely differentiable.
48. Two weeks ago I did that with the Laplace transform, and now I feel like it finally makes sense to me.
49. 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.
50. In the first part, we give an explicit formula for the Laplace transform and verify that this formula satisfies properties (??).
51. In the second part, we compute a table of Laplace transforms for a number of special functions including step functions and impulse functions.
52. Then the function is defined by and is called the Laplace transform of .
53. Note that the Laplace transform is an improper integral — which implies that some care must be taken when discussing its properties.
54. The Laplace transform of a function f (t) is dened for those values of s at which the integral converges.
55. 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).
56. 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, . . .
57. As (x) generalizes the factorial, the Laplace transform (8) generalizes (5).
58. CHAPTER 1 Laplace Transform Methods Laplace transform is a method frequently employed by engineers.
59. 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.
60. Our rst theorem states when Laplace transform can be performed, Theorem 1.1.
61. The next result shows that Laplace transform is unique in the sense that dierent continuous functions will have dierent Laplace trans- form.
62. The convolution theorem for Laplace transform is a useful tool for solving certain Laplace transforms.
63. An extremely powerful tool that helps us to solve this kind of real world problems are the Laplace transforms.
64. But first let us become familiar with the Laplace transform itself.
65. You can say that we "Laplace transform f from the t-space into F inside of the s-space.
66. We can actually use the linearity in order to find even more new Laplace transforms.

## 메타데이터

### Spacy 패턴 목록

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