"라플라스 변환"의 두 판 사이의 차이
Pythagoras0 (토론 | 기여) 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) |
Pythagoras0 (토론 | 기여) (→메타데이터) |
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(같은 사용자의 중간 판 9개는 보이지 않습니다) | |||
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==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/ | + | * 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 | + | * 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 | + | * 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>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 | + | <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, |
− | <math>\int_0^{\infty} f(t) \,dt</math>이 존재하고, | + | <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)!} | + | <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> | ||
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<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> | ||
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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, | + | * 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]] | + | * [[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:<math>\int_{0}^{\infty} f(e^{-t}) e^{-st}\,dt</math> |
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==역사== | ==역사== | ||
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* 라플라스 | * 라플라스 | ||
* 헤비사이드 | * 헤비사이드 | ||
− | * [[ | + | * [[수학사 연표]] |
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==메모== | ==메모== | ||
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* [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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==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 | ||
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− | ==사전 | + | ==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] | ||
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* [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)] | ||
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− | == | + | == 노트 == |
− | + | ===말뭉치=== | |
− | * http:// | + | # 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> |
+ | # When one says "the Laplace transform" without qualification, the unilateral or one-sided transform is usually intended.<ref name="ref_9d24b516" /> | ||
+ | # 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" /> | ||
+ | # Two integrable functions have the same Laplace transform only if they differ on a set of Lebesgue measure zero.<ref name="ref_9d24b516" /> | ||
+ | # 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> | ||
+ | # 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" /> | ||
+ | # Laplace transforms can also be used to solve IVP’s that we can’t use any previous method on.<ref name="ref_ceb72eb9" /> | ||
+ | # 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" /> | ||
+ | # 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> | ||
+ | # 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" /> | ||
+ | # A table of several important one-sided Laplace transforms is given below.<ref name="ref_46cf2dbe" /> | ||
+ | # The Laplace transform has many important properties.<ref name="ref_46cf2dbe" /> | ||
+ | # 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> | ||
+ | # 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> | ||
+ | # 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> | ||
+ | # The Laplace transform generates nonperiodic solutions.<ref name="ref_320625da" /> | ||
+ | # 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" /> | ||
+ | # In section 8.3.7 we will use the Laplace transform for solving a PDE.<ref name="ref_320625da" /> | ||
+ | # 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> | ||
+ | # Fourier transforms are often used to solve boundary value problems, Laplace transforms are often used to solve initial condition problems.<ref name="ref_8dd69bb3" /> | ||
+ | # Also, the Laplace transform succinctly captures input/output behavior or systems described by linear ODEs.<ref name="ref_8dd69bb3" /> | ||
+ | # 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> | ||
+ | # 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> | ||
+ | # 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> | ||
+ | # 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" /> | ||
+ | # Therefore, it has a Laplace transform.<ref name="ref_23424b92" /> | ||
+ | # This example introduces the idea of the inverse Laplace transform operator,, L −1.<ref name="ref_23424b92" /> | ||
+ | # 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> | ||
+ | # 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> | ||
+ | # The Laplace transform turns out to be a very efficient method to solve certain ODE problems.<ref name="ref_a1844089" /> | ||
+ | # The Laplace transform also has applications in the analysis of electrical circuits, NMR spectroscopy, signal processing, and elsewhere.<ref name="ref_a1844089" /> | ||
+ | # 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" /> | ||
+ | # 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> | ||
+ | # Many kinds of transformations already exist but Laplace transforms and Fourier transforms are the most well known.<ref name="ref_b29d8c7e" /> | ||
+ | # The Laplace transforms is usually used to simplify a differential equation into a simple and solvable algebra problem.<ref name="ref_b29d8c7e" /> | ||
+ | # There is always a table that is available to the engineer that contains information on the Laplace transforms.<ref name="ref_b29d8c7e" /> | ||
+ | # Compute the Laplace transform of exp(-a*t) .<ref name="ref_96a16540">[https://www.mathworks.com/help/symbolic/laplace.html Laplace transform]</ref> | ||
+ | # 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> | ||
+ | # Proof To prove this theorem we just use the definition of the Laplace transform and integration by parts.<ref name="ref_88945305" /> | ||
+ | # 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" /> | ||
+ | # Instead we will see that the method of Laplace Transforms tackles the entire problem with one fell swoop.<ref name="ref_88945305" /> | ||
+ | # 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> | ||
+ | # 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> | ||
+ | # 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> | ||
+ | # 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" /> | ||
+ | # However, before we investigate these properties, let us compute several Laplace transforms.<ref name="ref_8d80af84" /> | ||
+ | # The Laplace transform of a function does not always exist, even for functions that are infinitely differentiable.<ref name="ref_8d80af84" /> | ||
+ | # 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> | ||
+ | # 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" /> | ||
+ | # 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> | ||
+ | # 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" /> | ||
+ | # Then the function is defined by and is called the Laplace transform of .<ref name="ref_cfbdd7c1" /> | ||
+ | # 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" /> | ||
+ | # 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> | ||
+ | # 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" /> | ||
+ | # 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" /> | ||
+ | # 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.
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'}]