# Why laplace transform. Laplace transform 2019-01-22

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## When to use Laplace & Fourier Series/Transforms

We will also see that, for some of the more complicated nonhomogeneous differential equations from the last chapter, Laplace transforms are actually easier on those problems as well. Thanks for contributing an answer to Electrical Engineering Stack Exchange! Let me delete this here, this equal sign. And don't worry about that right now. I'll show you in a few videos, there are whole tables of Laplace Transforms, and eventually we'll prove all of them. } The unknown constants P and R are the located at the corresponding of the transfer function. Ideally you want a linear system that does not depend on time like a spring or a circuit or something like that , and then you apply some kind of forcing function.

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## The Laplace Transform and the Important Role it Plays

These responses have cleared the clouds for me, and like what you said rbj, I knew the theory and its magic, but am not sure when to apply which to which situations. This section covers the Laplace transform - one of the most important concepts in system analysis and theory. Fourier series are like the squares of many branches of math, much easier to understand at first than a rhombus. Likewise, Laplace and Z transforms turn nasty differential equations into algebraic equations that you have a chance of solving. So, if you can a function as a combination of them, you can solve the heat or wave equation with that function as the initial condition. He then went on to apply the Laplace transform in the same way and started to derive some of its properties, beginning to appreciate its potential power. You could use them on both sides too, the result will work out to be the same with some mathematical variation.

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## The Laplace Transform

Because I could actually use some of that real estate. Another, often unspoken, 'big deal' is that the transform is unique in some sense eg, if the transforms of two continuous functions agree, then the functions agree in the original domain. However, if you want to understand what a system does when you flip a light switch, you typically need Laplace transforms. I see Laplace and Fourier and Jacobian, etc. The original differential equation can then be solved by applying the inverse Laplace transform. } into the simpler operations of {adding and subtracting}. This makes the problem much easier to solve.

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## The Laplace Transform

However for some physical systems, you only have the data of what happened until then. The Laplace Transform allows us to do: Differential Equations Algebraic Equations It allows us to turn certain differential equations into equivalent algebraic equations, we can then solve the corresponding algebraic equation and use the Inverse Laplace Transform to get the corresponding solution to the original differential equations. It means in the signal there is always a change. So if you can solve the problem in the s-domain, then you have solved it, in some sense, in the original domain. Now we took the anti-derivative. This difference may not matter to study the properties of the system at this level, but that got me thinking about this question.

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## 2. Definition of the Laplace Transform

And especially if you're going to go into engineering, you'll find that the Laplace Transform, besides helping you solve differential equations, also helps you transform functions or waveforms from the time domain to this frequency domain, and study and understand a whole set of phenomena. But with the Laplace transform, you have another dial that allows you to express things in terms of damped sine and cosine functions, rather than just sine and cosine. Keeping track of the zeros and poles then is a way to easily keep track of information about the original function. In this method we don't have to put the values of constants by our self. The Laplace transform is invertible on a large class of functions. Now you can analyze the signal at the transform domain. The Laplace Transform is widely used in engineering applications mechanical and electronic , especially where the driving force is discontinuous.

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## 2. Definition of the Laplace Transform

Once in the s-domain, we may begin discussing the components in terms of impedance. I've never quite decided if it was a good one or not. So what is the Laplace Transform? I think the reason for caring about Fourier series is that they are sort of canonical solutions to the heat equation and the wave equation they are eigenvalues of the relevant differential operator, and they have a certain physical interpretation in each case. For differential equations, first use the Laplace transform to make the equation malleable, then move things around to your hearts content, then undo the Laplace transform. Well, the Laplace Transform, the notation is the L like Laverne from Laverne and Shirley. The expression for the Fourier transform, i.

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## Differential Equations

There is a limitation here: Laplace will only generate an exact answer if initial conditions are provided â€¦. We're transforming our information into another form that showcases what we need in a way that makes the problem easy. What you asked can be plotted in one graph and understood easily. I know I haven't actually done improper integrals just yet, but I'll explain them in a few seconds. Fourier series were originally invented to solve for heat flow in bricks and other partial differential equations.

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## Differential Equations

It's like f t x , but using more sophisticated means to transform it. This makes sense in electrical engineering applications for example, where you consider sinusoidal signals and you have an idea of what is going to come. We only work a couple to illustrate how the process works with Laplace transforms. As Wiki points out, the Fourier transform measures the periodic components of a function, whereas the Laplace transform measures the exponential components of a function. The Laplace transform is very similar to the.

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