"라플라스 변환"의 두 판 사이의 차이

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잔글 (찾아 바꾸기 – “==관련도서== * 도서내검색<br> ** http://books.google.com/books?q= ** http://book.daum.net/search/contentSearch.do?query= * 도서검색<br> ** http://books.google.com/books?q= ** http://book.daum.net/search/mainSearch.d)
 
(같은 사용자의 중간 판 9개는 보이지 않습니다)
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==이 항목의 스프링노트 원문주소==
 
 
* [[라플라스 변환]]
 
 
 
 
 
 
 
 
 
==개요==
 
==개요==
  
 
* [[푸리에 변환]]의 변형
 
* [[푸리에 변환]]의 변형
 
* 어떤 미분방정식들의 해를 대수적 조작을 통해 얻을 수 있게 해주는 변환
 
* 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
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* 라플라스 변환을 미분방정식에 응용한 사람은 Oliver Heaviside http://en.wikipedia.org/wiki/Oliver_Heaviside  이다
 
* operational calculus 또는 Heaviside calculus 의 도구
 
* operational calculus 또는 Heaviside calculus 의 도구
  
 
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==정의==
 
==정의==
  
*  함수 <math>f(t)</math>에 대한 라플라스 변환을 다음과 같이 정의함<br><math>F(s) = \mathcal{L} \left\{f(t)\right\}=\int_0^{\infty} e^{-st} f(t) \,dt</math><br>
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*  함수 <math>f(t)</math>에 대한 라플라스 변환을 다음과 같이 정의함:<math>F(s) = \mathcal{L} \left\{f(t)\right\}=\int_0^{\infty} e^{-st} f(t) \,dt</math>
  
 
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==성질==
 
==성질==
  
*  함수 <math>f</math>에 대한 도함수의 라플라스 변환은 다음과 같다<br><math>\mathcal{L}\left\{\frac{df}{dt}\right\} = s\cdot\mathcal{L} \left\{ f(t) \right\}-f(0)</math><br>
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*  함수 <math>f</math>에 대한 도함수의 라플라스 변환은 다음과 같다:<math>\mathcal{L}\left\{\frac{df}{dt}\right\} = s\cdot\mathcal{L} \left\{ f(t) \right\}-f(0)</math>
  
 
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(정리)
 
(정리)
  
<math>f</math>가 유계이고, <math>t\geq 0</math>에서 조각적 연속(piecewise continuous)라 하자.
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<math>f</math>가 유계이고, <math>t\geq 0</math>에서 조각적 연속(piecewise continuous)라 하자.
  
<math>\mathfrak{R}(s)\geq 0</math>에서 정의된 함수 <math>F(s) = \mathcal{L} \left\{f(t)\right\}=\int_0^{\infty} e^{-st} f(t) \,dt</math> 가 <math>\mathfrak{R}(s)\geq 0</math>에서 해석함수로 확장되면,
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<math>\mathfrak{R}(s)\geq 0</math>에서 정의된 함수 <math>F(s) = \mathcal{L} \left\{f(t)\right\}=\int_0^{\infty} e^{-st} f(t) \,dt</math> <math>\mathfrak{R}(s)\geq 0</math>에서 해석함수로 확장되면,
  
<math>\int_0^{\infty} f(t) \,dt</math>이 존재하고, <math>F(0) = \int_0^{\infty} f(t) \,dt</math>가 성립한다. 
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<math>\int_0^{\infty} f(t) \,dt</math>이 존재하고, <math>F(0) = \int_0^{\infty} f(t) \,dt</math>가 성립한다.  
  
 
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==예==
 
==예==
  
<math>\left(\frac{t^ne^t}{n!}\right)'=\frac{t^{n-1}e^t}{(n-1)!}\right+\frac{t^ne^t}{n!}\right</math> 로부터 <math>\mathcal{L}\left\{\frac{t^{n-1}e^t}{(n-1)!}\right\} = (s-1)\cdot\mathcal{L} \left\{ \frac{t^ne^t}{n!}\right\}</math>
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<math>\left(\frac{t^ne^t}{n!}\right)'=\frac{t^{n-1}e^t}{(n-1)!}+\frac{t^ne^t}{n!}</math> 로부터 <math>\mathcal{L}\left\{\frac{t^{n-1}e^t}{(n-1)!}\right\} = (s-1)\cdot\mathcal{L} \left\{ \frac{t^ne^t}{n!}\right\}</math>
  
 
<math>\mathcal{L}\left\{e^t\right\} = \frac{1}{s-1}</math>
 
<math>\mathcal{L}\left\{e^t\right\} = \frac{1}{s-1}</math>
52번째 줄: 44번째 줄:
 
<math>\mathcal{L}\left\{t e^t\right\} = \frac{1}{(s-1)^2}</math>
 
<math>\mathcal{L}\left\{t e^t\right\} = \frac{1}{(s-1)^2}</math>
  
 
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<math>\mathcal{L}\left\{\frac{t^2 e^t}{2!}\right\} = \frac{1}{(s-1)^3}</math>
 
<math>\mathcal{L}\left\{\frac{t^2 e^t}{2!}\right\} = \frac{1}{(s-1)^3}</math>
  
 
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<math>\mathcal{L}\left\{\frac{t^3 e^t}{3!}\right\} = \frac{1}{(s-1)^4}</math>
 
<math>\mathcal{L}\left\{\frac{t^3 e^t}{3!}\right\} = \frac{1}{(s-1)^4}</math>
62번째 줄: 54번째 줄:
 
...
 
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==상수계수 미분방정식에의 응용==
 
==상수계수 미분방정식에의 응용==
  
 
* <math>y''(t)-2 y'(t)+y(t)=e^t</math>
 
* <math>y''(t)-2 y'(t)+y(t)=e^t</math>
*  양변에 라플라스 변환을 취하면,<br><math>s^2 Y(s)+Y(s)-2 (s Y(s)-1)-s+1=\frac{1}{s-1}</math>, 여기서 <math>Y(s)=\mathcal{L} \left\{ f(t) \right\}</math>.<br>
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*  양변에 라플라스 변환을 취하면,:<math>s^2 Y(s)+Y(s)-2 (s Y(s)-1)-s+1=\frac{1}{s-1}</math>, 여기서 <math>Y(s)=\mathcal{L} \left\{ f(t) \right\}</math>.
 
* <math>Y(s)=\frac{1}{s-1}-\frac{2}{(s-1)^2}+\frac{1}{(s-1)^3}</math>
 
* <math>Y(s)=\frac{1}{s-1}-\frac{2}{(s-1)^2}+\frac{1}{(s-1)^3}</math>
 
* <math>y(t)=e^t-2t e^t+\frac{t^2}{2}e^t</math> 는 주어진 미분방정식의 해가 된다
 
* <math>y(t)=e^t-2t e^t+\frac{t^2}{2}e^t</math> 는 주어진 미분방정식의 해가 된다
  
 
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==멜린변환과의 관계==
 
==멜린변환과의 관계==
  
* [[푸리에 변환]] 항목 참조<br><math>\hat{f}(s)= \int_{0}^{\infty} f(x) x^{s}\frac{dx}{x}</math><br>
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* [[푸리에 변환]] 항목 참조:<math>\hat{f}(s)= \int_{0}^{\infty} f(x) x^{s}\frac{dx}{x}</math>
멜린변환에서 <math>x=e^{-t}</math>로 변수를 치환하면, 라플라스 변환을 얻는다<br><math>\int_{0}^{\infty} f(e^{-t}) e^{-st}\,dt</math><br>
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멜린변환에서 <math>x=e^{-t}</math>로 변수를 치환하면, 라플라스 변환을 얻는다:<math>\int_{0}^{\infty} f(e^{-t}) e^{-st}\,dt</math>
  
 
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==역사==
 
==역사==
93번째 줄: 85번째 줄:
 
* 라플라스
 
* 라플라스
 
* 헤비사이드
 
* 헤비사이드
* [[수학사연표 (역사)|수학사연표]]
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* [[수학사 연표]]
  
 
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==메모==
 
==메모==
103번째 줄: 95번째 줄:
 
* [http://www.math.ttu.edu/%7Eklong/Notebooks/LaplaceTransforms.nb.pdf http://www.math.ttu.edu/~klong/Notebooks/LaplaceTransforms.nb.pdf]
 
* [http://www.math.ttu.edu/%7Eklong/Notebooks/LaplaceTransforms.nb.pdf http://www.math.ttu.edu/~klong/Notebooks/LaplaceTransforms.nb.pdf]
  
 
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==관련된 항목들==
 
==관련된 항목들==
111번째 줄: 103번째 줄:
 
* [[푸리에 변환]]
 
* [[푸리에 변환]]
  
 
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==수학용어번역==
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* http://www.google.com/dictionary?langpair=en|ko&q=
 
* [http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr= 대한수학회 수학 학술 용어집]<br>
 
** http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
 
* [http://kms.or.kr/home/kor/board/bulletin_list_subject.asp?bulletinid=%7BD6048897-56F9-43D7-8BB6-50B362D1243A%7D&boardname=%BC%F6%C7%D0%BF%EB%BE%EE%C5%E4%B7%D0%B9%E6&globalmenu=7&localmenu=4 대한수학회 수학용어한글화 게시판]
 
 
 
 
 
 
 
 
 
  
 
==매스매티카 파일 및 계산 리소스==
 
==매스매티카 파일 및 계산 리소스==
  
 
* https://docs.google.com/file/d/0B8XXo8Tve1cxekVyZUNKR2RGY0U/edit
 
* https://docs.google.com/file/d/0B8XXo8Tve1cxekVyZUNKR2RGY0U/edit
* http://www.wolframalpha.com/input/?i=
 
* http://functions.wolfram.com/
 
* [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]
 
* [http://people.math.sfu.ca/%7Ecbm/aands/toc.htm Abramowitz and Stegun Handbook of mathematical functions]
 
* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
 
* [http://numbers.computation.free.fr/Constants/constants.html Numbers, constants and computation]
 
* [https://docs.google.com/open?id=0B8XXo8Tve1cxMWI0NzNjYWUtNmIwZi00YzhkLTkzNzQtMDMwYmVmYmIxNmIw 매스매티카 파일 목록]
 
  
 
 
  
 
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==사전 형태의 자료==
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==사전 형태의 자료==
  
 
* [http://ko.wikipedia.org/wiki/%EB%9D%BC%ED%94%8C%EB%9D%BC%EC%8A%A4_%EB%B3%80%ED%99%98 http://ko.wikipedia.org/wiki/라플라스_변환]
 
* [http://ko.wikipedia.org/wiki/%EB%9D%BC%ED%94%8C%EB%9D%BC%EC%8A%A4_%EB%B3%80%ED%99%98 http://ko.wikipedia.org/wiki/라플라스_변환]
147번째 줄: 121번째 줄:
 
* [http://en.wikipedia.org/wiki/Laplace%E2%80%93Stieltjes_transform http://en.wikipedia.org/wiki/Laplace–Stieltjes_transform]
 
* [http://en.wikipedia.org/wiki/Laplace%E2%80%93Stieltjes_transform http://en.wikipedia.org/wiki/Laplace–Stieltjes_transform]
 
* [http://en.wikipedia.org/wiki/Moment_%28mathematics%29 http://en.wikipedia.org/wiki/Moment_(mathematics)]
 
* [http://en.wikipedia.org/wiki/Moment_%28mathematics%29 http://en.wikipedia.org/wiki/Moment_(mathematics)]
* http://www.wolframalpha.com/input/?i=
 
* [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]
 
* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]<br>
 
** http://www.research.att.com/~njas/sequences/?q=
 
 
 
 
 
 
 
  
==관련논문==
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== 노트 ==
  
* http://www.jstor.org/action/doBasicSearch?Query=
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===말뭉치===
* http://dx.doi.org/
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# This limit emphasizes that any point mass located at 0 is entirely captured by the Laplace transform.<ref name="ref_9d24b516">[https://en.wikipedia.org/wiki/Laplace_transform Laplace transform]</ref>
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# When one says "the Laplace transform" without qualification, the unilateral or one-sided transform is usually intended.<ref name="ref_9d24b516" />
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# 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.<ref name="ref_9d24b516" />
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# Two integrable functions have the same Laplace transform only if they differ on a set of Lebesgue measure zero.<ref name="ref_9d24b516" />
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# Laplace transforms and Fourier transforms are probably the main two kinds of transforms that are used.<ref name="ref_ceb72eb9">[https://tutorial.math.lamar.edu/classes/de/LaplaceIntro.aspx Differential Equations]</ref>
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# As we will see in later sections we can use Laplace transforms to reduce a differential equation to an algebra problem.<ref name="ref_ceb72eb9" />
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# Laplace transforms can also be used to solve IVP’s that we can’t use any previous method on.<ref name="ref_ceb72eb9" />
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# 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.<ref name="ref_ceb72eb9" />
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# The Laplace transform is particularly useful in solving linear ordinary differential equations such as those arising in the analysis of electronic circuits.<ref name="ref_46cf2dbe">[https://mathworld.wolfram.com/LaplaceTransform.html Laplace Transform -- from Wolfram MathWorld]</ref>
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# 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).<ref name="ref_46cf2dbe" />
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# A table of several important one-sided Laplace transforms is given below.<ref name="ref_46cf2dbe" />
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# The Laplace transform has many important properties.<ref name="ref_46cf2dbe" />
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# The Laplace transform is used to quickly find solutions for differential equations and integrals.<ref name="ref_d2045873">[https://www.rapidtables.com/math/calculus/laplace_transform.html Laplace transform table ( F(s) = L{ f(t) } )]</ref>
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# Like all transforms, the Laplace transform changes one signal into another according to some fixed set of rules or equations.<ref name="ref_feb07027">[https://byjus.com/maths/laplace-transform/ Laplace Transform- Definition, Properties, Formulas, Equation & Examples]</ref>
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# In general, the Laplace transform is used for applications in the time-domain for t ≥ 0.<ref name="ref_320625da">[https://www.sciencedirect.com/topics/engineering/laplace-transforms Laplace Transforms - an overview]</ref>
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# The Laplace transform generates nonperiodic solutions.<ref name="ref_320625da" />
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# 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.<ref name="ref_320625da" />
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# In section 8.3.7 we will use the Laplace transform for solving a PDE.<ref name="ref_320625da" />
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# As mentioned in another answer, the Laplace transform is defined for a larger class of functions than the related Fourier transform.<ref name="ref_8dd69bb3">[https://math.stackexchange.com/questions/181160/what-exactly-is-laplace-transform/181261 What exactly is Laplace transform?]</ref>
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# Fourier transforms are often used to solve boundary value problems, Laplace transforms are often used to solve initial condition problems.<ref name="ref_8dd69bb3" />
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# Also, the Laplace transform succinctly captures input/output behavior or systems described by linear ODEs.<ref name="ref_8dd69bb3" />
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# The Laplace transform 3{6 sinusoid: rst express f (t) = cos !<ref name="ref_49d34905">[https://web.stanford.edu/~boyd/ee102/laplace.pdf S. boyd]</ref>
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# 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.<ref name="ref_86be309d">[https://www.britannica.com/science/Laplace-transform Laplace transform | mathematics]</ref>
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# Therefore, sin kx and cos kx each have a Laplace transform, since they are continuous and bounded functions.<ref name="ref_23424b92">[https://www.cliffsnotes.com/study-guides/differential-equations/the-laplace-transform/the-laplace-transform-operator The Laplace Transform Operator]</ref>
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# 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.<ref name="ref_23424b92" />
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# Therefore, it has a Laplace transform.<ref name="ref_23424b92" />
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# This example introduces the idea of the inverse Laplace transform operator,, L −1.<ref name="ref_23424b92" />
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# 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.<ref name="ref_2249ba02">[https://www.efunda.com/math/laplace_transform/index.cfm eFunda: Laplace Transforms]</ref>
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# In this chapter we will discuss the Laplace transform.<ref name="ref_a1844089">[https://math.libretexts.org/Bookshelves/Differential_Equations/Book%3A_Differential_Equations_for_Engineers_(Lebl)/6%3A_The_Laplace_Transform/6.1%3A_The_Laplace_Transform 6.1: The Laplace Transform]</ref>
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# The Laplace transform turns out to be a very efficient method to solve certain ODE problems.<ref name="ref_a1844089" />
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# The Laplace transform also has applications in the analysis of electrical circuits, NMR spectroscopy, signal processing, and elsewhere.<ref name="ref_a1844089" />
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# Finally, understanding the Laplace transform will also help with understanding the related Fourier transform, which, however, requires more understanding of complex numbers.<ref name="ref_a1844089" />
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# In this article, we will be discussing Laplace transforms and how they are used to solve differential equations.<ref name="ref_b29d8c7e">[https://www.electrical4u.com/laplace-transformation/ Laplace Transform Table, Formula, Examples & Properties]</ref>
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# Many kinds of transformations already exist but Laplace transforms and Fourier transforms are the most well known.<ref name="ref_b29d8c7e" />
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# The Laplace transforms is usually used to simplify a differential equation into a simple and solvable algebra problem.<ref name="ref_b29d8c7e" />
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# There is always a table that is available to the engineer that contains information on the Laplace transforms.<ref name="ref_b29d8c7e" />
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# Compute the Laplace transform of exp(-a*t) .<ref name="ref_96a16540">[https://www.mathworks.com/help/symbolic/laplace.html Laplace transform]</ref>
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# Problems Now that we know how to find a Laplace transform, it is time to use it to solve differential equations.<ref name="ref_88945305">[https://ltcconline.net/greenl/courses/204/PowerLaplace/initialValueProblems.htm Using the Laplace Transform to Solve Initial Value Problems]</ref>
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# Proof To prove this theorem we just use the definition of the Laplace transform and integration by parts.<ref name="ref_88945305" />
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# What this tells us is that if we have a differential equation, then the Laplace transform will turn it into an algebraic equation.<ref name="ref_88945305" />
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# Instead we will see that the method of Laplace Transforms tackles the entire problem with one fell swoop.<ref name="ref_88945305" />
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# Laplace transforms convert a function f(t) in the time domain into function in the Laplace domain F(s).<ref name="ref_a90e72a7">[https://apmonitor.com/pdc/index.php/Main/LaplaceTransforms Laplace Transforms]</ref>
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# The domain of its Laplace transform depends on f and can vary from a function to a function.<ref name="ref_8a6ceafe">[https://web.iit.edu/sites/web/files/departments/academic-affairs/academic-resource-center/pdfs/LaplaceTransformIIT.pdf The laplace transform]</ref>
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# However, the Laplace transform , an integral transform, allows us to change a differential equation to an algebraic equation.<ref name="ref_8d80af84">[http://faculty.sfasu.edu/judsontw/ode/html-20180819/laplace01.html The Laplace Transform]</ref>
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# 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.<ref name="ref_8d80af84" />
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# However, before we investigate these properties, let us compute several Laplace transforms.<ref name="ref_8d80af84" />
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# The Laplace transform of a function does not always exist, even for functions that are infinitely differentiable.<ref name="ref_8d80af84" />
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# Two weeks ago I did that with the Laplace transform, and now I feel like it finally makes sense to me.<ref name="ref_ecba2e70">[https://golem.ph.utexas.edu/category/2019/07/what_is_the_laplace_transform.html The n-Category Café]</ref>
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# 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.<ref name="ref_ecba2e70" />
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# In the first part, we give an explicit formula for the Laplace transform and verify that this formula satisfies properties (??).<ref name="ref_cfbdd7c1">[https://ximera.osu.edu/laode/textbook/laplaceTransforms/laplaceTransformsAndTheirComputation Laplace Transforms and Their Computation]</ref>
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# In the second part, we compute a table of Laplace transforms for a number of special functions including step functions and impulse functions.<ref name="ref_cfbdd7c1" />
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# Then the function is defined by and is called the Laplace transform of .<ref name="ref_cfbdd7c1" />
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# Note that the Laplace transform is an improper integral — which implies that some care must be taken when discussing its properties.<ref name="ref_cfbdd7c1" />
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# The Laplace transform of a function f (t) is dened for those values of s at which the integral converges.<ref name="ref_6ee9e577">[https://www.math.unl.edu/~scohn1/EngRevf08/laplace1.pdf 1. the laplace transform of a function f (t) is]</ref>
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# 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).<ref name="ref_6ee9e577" />
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# 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, . . .<ref name="ref_6ee9e577" />
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# As (x) generalizes the factorial, the Laplace transform (8) generalizes (5).<ref name="ref_6ee9e577" />
 +
# CHAPTER 1 Laplace Transform Methods Laplace transform is a method frequently employed by engineers.<ref name="ref_e2e77648">[https://www.unf.edu/~mzhan/chapter6.pdf Chapter 1]</ref>
 +
# 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.<ref name="ref_e2e77648" />
 +
# Our rst theorem states when Laplace transform can be performed, Theorem 1.1.<ref name="ref_e2e77648" />
 +
# The next result shows that Laplace transform is unique in the sense that dierent continuous functions will have dierent Laplace trans- form.<ref name="ref_e2e77648" />
 +
# The convolution theorem for Laplace transform is a useful tool for solving certain Laplace transforms.<ref name="ref_363a51e8">[https://brilliant.org/wiki/laplace-transform/ Brilliant Math & Science Wiki]</ref>
 +
# An extremely powerful tool that helps us to solve this kind of real world problems are the Laplace transforms.<ref name="ref_99548916">[https://www.math.fsu.edu/~fusaro/EngMath/Ch5/SCOLT.html Laplace Transforms:]</ref>
 +
# But first let us become familiar with the Laplace transform itself.<ref name="ref_99548916" />
 +
# You can say that we "Laplace transform f from the t-space into F inside of the s-space.<ref name="ref_99548916" />
 +
# We can actually use the linearity in order to find even more new Laplace transforms.<ref name="ref_99548916" />
 +
===소스===
 +
<references />
  
 
+
== 메타데이터 ==
  
 
+
===위키데이터===
 +
* ID :  [https://www.wikidata.org/wiki/Q199691 Q199691]
 +
===Spacy 패턴 목록===
 +
* [{'LOWER': 'laplace'}, {'LEMMA': 'transform'}]

2021년 2월 26일 (금) 02:55 기준 최신판

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.



\(\left(\frac{t^ne^t}{n!}\right)'=\frac{t^{n-1}e^t}{(n-1)!}+\frac{t^ne^t}{n!}\) 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



메모



관련된 항목들




매스매티카 파일 및 계산 리소스



사전 형태의 자료

노트

말뭉치

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

소스

메타데이터

위키데이터

Spacy 패턴 목록

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