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Topic: Laplace transform

	Laplace Transform
		The Laplace transform is a close relative of the Fourier transform.
		In addition, the Laplace transform is useful in determining solutions of partial differential equations -- particularly where the time or spatial domains are semi-infinite, e.g., heat flow in an infinite rod where the temperature at one end is known.
		For the Laplace transform, the appropriate convolution f
	www.math.duke.edu /education/ccp/materials/engin/laplace/laplace1.html (339 words)

	PlanetMath: Laplace transform
		Notice the Laplace transform is a linear transformation.
		The most popular usage of the Laplace transform is to solve initial value problems by taking the Laplace transform of both sides of an ordinary differential equation.
		This is version 19 of Laplace transform, born on 2003-06-11, modified 2007-03-01.
	planetmath.org /encyclopedia/LaplaceTransform.html (148 words)

	Laplace transform
		The Laplace transform can also be used to solve differential equations and is used extensively in electrical engineering.
		Laplace transform of a function with period p
		The Laplace transform is closely related to the Fourier transform and the z-transform.
	www.xasa.com /wiki/en/wikipedia/l/la/laplace_transform.html (369 words)

	eFunda: Laplace Transforms
		The Laplace transform is a powerful tool formulated to solve a wide variety of initial-value problems.
		The strategy is to transform the difficult differential equations into simple algebra problems where solutions can be easily obtained.
		One then applies the Inverse Laplace transform to retrieve the solutions of the original problems.
	www.efunda.com /math/laplace_transform/index.cfm (94 words)

	The Laplace Transform
		have a Fourier transform and that the integral
		A sufficient condition for the existence of the Laplace transform is that
		Inverting the Laplace transform is usually accomplished with the aid of a table of known Laplace transforms and the technique of partial fraction expansion.
	math.fullerton.edu /mathews/c2003/LaplaceTransformMod.html (493 words)

	RMP Lecture Notes
		The last point is the biggest single reason the Laplace transform is valued -- it transforms linear differential equations into algebraic equations, which many people find easier to solve.
		You can always use the definition to find the Laplace transform of a function, but that usually is more trouble than it is worth.
		The major reason people bother with Laplace transforms is that they can make it easier to obtain analytical solutions of many linear ordinary differential equations.
	www.cbu.edu /~rprice/lectures/laplace.html (766 words)

	Laplace Transform
		The Fourier transform has an inverse transformation, and the transformation between time and frequency domains is equally convenient both ways, involving a simple integral along the real axes.
		The Laplace transform f(s) of a time function F(t) is an analytic function of the complex variable s, except at its singularities.
		Note how the contribution to the inverse transform came from the neighbourhoods of the pole and the branch cut, which is typical.
	www.du.edu /~jcalvert/math/laplace.htm (2163 words)

	The Laplace Transform
		Laplace was a French mathematician, astronomer, and physicist who applied the Newtonian theory of gravitation to the solar system (an important problem of his day).
		The Laplace Transform is widely used in engineering applications (mechanical and electronic), especially where the driving force is discontinuous.
		The main idea behind the Laplace Transformation is that we can solve an equation (or system of equations) containing differential and integral terms by transforming the equation in "t-space" to one in "s-space".
	www.intmath.com /Laplace-transformation/Intro.php (256 words)

	Laplace Transform Analysis
		We see that the Laplace transform can be viewed as a generalization of the Fourier transform from the real line (a simple frequency axis) to the entire complex plane.
		An advantage of the Laplace transform is the ability to transform signals which have no Fourier transform.
		Thus, the Laplace transform can be seen as the Fourier transform of an exponentially windowed input signal.
	ccrma.stanford.edu /~jos/filters/Laplace_Transform_Analysis.html (232 words)

	Laplace Transforms and Circuit Analysis - Recommended Books Page
		In the words of Dr. Math "The Laplace transform is an aid in solving differential equations of a continuous-time system, and the z transform performs the same task using difference equations for a discrete-time system, e.g.
		There are tables of z transforms used in the same manner as the Laplace transforms for finding the inverse functions, and there are many other analogies.
		Laplace Transforms and Circuit Analysis page is geared to bring some interesting books to your attention.
	www.a-ten.com /art/laplace.htm (420 words)

	Laplace Transform in Comp.DSP (Site not responding. Last check: 2007-11-06)
		Laplace to Z transform for second order lag.
		I have a problem with verifying the results for the conversion of a Laplace transform to a z transform.
		The Laplace transform is in a table and is: (b-a)/((s+a)(s+b)) The z transform*...
	www.dsprelated.com /comp.dsp/keyword/Laplace_Transform.php (878 words)

	Lapace Transforms
		Similarly, the application of Laplace Transforms to the analysis of systems which can be described by linear, ordinary time differential equations overcomes some of the complexities encountered in the time-domain solution of such equations.
		Laplace Transforms are used to convert time domain relationships to a set of equations expressed in terms of the Laplace operator 's'.
		The necessary operations are carried out and the laplace transforms obtained in terms of s are then inverted from the s domain to the time (t) domain.
	www.roymech.co.uk /Related/Control/Laplace_Transforms.html (669 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-11-06)
		The Laplace transform is a simple way of converting functions in one domain to functions of another domain.
		With the Laplace transform, we can convert this to a function of frequency, which is cos(w*t) ----L{}-----> w / (s^2 + w^2) This is useful for a very simple reason: it makes solving differential equations much easier.
		Since the Laplace transform of a derivative becomes a multiple of the domain variable, the Laplace transform turns a complicated n-th order differential equation to a corresponding nth degree polynomial.
	www.mathforum.com /library/drmath/view/52122.html (348 words)

	Laplace Transform Example #1
		The overall Laplace Transform is the sum of the individual terms, making use of the linearity property.
		Using the defining integral for the Laplace Transform or using the Tables of Laplace Transforms from the textbook, the individual transforms and the total transform for this system are shown below.
		The region of convergence for the Laplace Transform is all of the s-plane to the right of Re[s] = -0.1.
	ece.gmu.edu /~gbeale/ece_220/laplace_01.html (689 words)

	6.2 Properties of the Laplace transform
		The properties of the Laplace transform enable us to find Laplace transforms without having to compute them directly from he definition.
		Determine the Laplace transform of the causal function f(x)=x.
		Determine the Laplace transform of the function f(x) illustrated in Figure 6.1.
	www.math.ut.ee /~toomas_l/harmonic_analysis/Fourier/node36.html (445 words)

	Laplace Transform Example #2
		The inverse Laplace Transform of the transfer function H(s) is the impulse response h(t).
		Since we are only considering the 1-sided Laplace Transform, the inverse process is unique without worrying about the Region of Convergence, and h(t) is assumed to be 0 for t < 0.
		The magnitude curve has a slope of -20 db/decade of frequency for all positive frequencies, and the phase is constant at -90 degrees for all positive frequencies.
	ece.gmu.edu /~gbeale/ece_220/laplace_02.html (1249 words)

	m07 - The Laplace Transform
		The Laplace transform can be thought of as a generalization of the Fourier transform in order to include a larger class of functions, to allow the use of complex variable theory, to solve initial value differential equations, and to give a tool for input-output description of linear systems.
		This definition is often called the bilateral Laplace transform to distinguish it from the unilateral transform (ULT) which is defined with zero as the lower limit of the forward transform integral (Equation 1).
		It does not reduce to the Fourier transform for signals that exist for negative time, but if the negative time part of a signal can be neglected, the unilateral transform will converge for a much larger class of function that the bilateral transform will.
	cnx.org /content/m13876/latest (479 words)

	475LaplaceTransformExamples.nb
		Since the Laplace transform is linear, it follows that the inverse laplace transform is linear, i.e.
		Notice that the x-terms are not affected by the transform.
		If we apply the Laplace transform to an IVBVP in x and t, all time derivatives are turned into multiplication, leaving an ODE in x.
	www.bsu.edu /web/mkarls/475LaplaceTransformExamples (770 words)

	The Laplace Transform (Site not responding. Last check: 2007-11-06)
		The Laplace transform is one of the most useful tools we'll see in this course for solving differential equations with initial conditions.
		One of the greatest accomplishments of the Laplace transform is that it provides us a relatively easy way to solve the types of equations that come up in real-world applications where functions aren't always continuous.
		The Laplace transform is an effective computational tool for solving linear equations with constant coefficients in the presence of initial conditions.
	math.ucsd.edu /~math20d/Fall/Lab8F/Lab8F.html (1374 words)

	SAM/SRI Laplace transform (Site not responding. Last check: 2007-11-06)
		Conditions for the Existence of a Laplace Transform of f(t)=F(s)
		Conditions for the Existence of an Inverse Laplace Transform of F(t)=f(s)
		Laplace image is usually given as a fraction of two polynomials
	dce.felk.cvut.cz /sari/index.php?ID=ltransform-en (140 words)

	The Laplace transform
		This is a tool and it is indispensable as most of linear system dynamics are described in a mapped space that can only be understood when the main theorems of the Laplace transform are known.
		The correspondences of the Laplace transform are given in tabular form to be simply used for the forward and back transformation.
		Special focus is put on the solution of differential equations using the Laplace transform and on special signals, e.g.
	virtual.cvut.cz /dynlabmodules/syscontrol/node6.html (181 words)

	Laplace transform 2 CC Poles (Site not responding. Last check: 2007-11-06)
		The above diagram shows a surface plot of the magnitude of the Laplace transform function F(s) when F(s) has two complex conjugate poles in the complex s plane.
		The plot is characterised by its "flatness" over most of the s plane except at the poles where it rises to infinity.
		The red line shows the magnitude of F(s) along the imaginary s axis and corresponds to the frequency response of a system which has Laplace Transform F(s).
	cnyack.homestead.com /files/alaplace/laptr2.htm (442 words)

	The Laplace Transform
		There is a special type of substitution, called an integral transform that simplifies the task of solving differential equations.
		Since the Laplace transform of a function is defined as an improper integral, the integral may not converge.
		Another important fact about Laplace Transforms is that the Laplace transform is a linear transformation from "nice" functions to functions, where "nice" means functions that have a Laplace transform.
	www.ltcconline.net /greenl/courses/204/PowerLaplace/laplaceTransform.htm (269 words)

	Numerical Inversion of Laplace Transform with Multiple Precision Using the Complex Domain -- from Mathematica ...
		The Laplace transform should be provided as a function ready for multiple-precision evaluation in the complex plane.
		There are two main groups of methods for the numerical inversion of the Laplace transform.
		Laplace transform, Numerical inversion, Multiple-precision, Talbot method, Complex arithmetic
	library.wolfram.com /infocenter/MathSource/5026 (220 words)

	Laplace Transform Examples
		These are some of the Laplace transforms in the table in many texts.
		Check the syntax: the first part of the command is the function to be transformed; the second part is the variable of the function to be transformed (usually t); the third part is the variable of the transformed function (usually s).
		The second one is Laplace transform of 1/t evaluated at s+1, which is what the Laplace transform of Exp[-t]/t would be.
	www.ma.iup.edu /projects/CalcDEMma/laplace/laplace1.html (295 words)

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