Laplace transform of the unit step function

Online Calculus Solver ».

If you're seeing this message, it means we're having trouble loading external resources on our website. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. Search for courses, skills, and videos. Properties of the Laplace transform. About About this video Transcript. Introduction to the unit step function and its Laplace Transform.

Laplace transform of the unit step function

To productively use the Laplace Transform, we need to be able to transform functions from the time domain to the Laplace domain. We can do this by applying the definition of the Laplace Transform. Our goal is to avoid having to evaluate the integral by finding the Laplace Transform of many useful functions and compiling them in a table. Thereafter the Laplace Transform of functions can almost always be looked by using the tables without any need to integrate. A table of Laplace Transform of functions is available here. In this case we say that the "region of convergence" of the Laplace Transform is the right half of the s-plane since s is a complex number, the right half of the plane corresponds to the real part of s being positive. As long as the functions we are working with have at least part of their region of convergence in common which will be true in the types of problems we consider , the region of convergence holds no particular interest for us. Since the region of convergence will not play a part in any of the problems we will solve, it is not considered further. The unit impulse is discussed elsewhere , but to review. The area of the impulse function is one. The impulse function is drawn as an arrow whose height is equal to its area. Now we apply the sifting property of the impulse. So the Laplace Transform of the unit impulse is just one. Therefore the impulse function, which is difficult to handle in the time domain, becomes easy to handle in the Laplace domain. It will turn out that the unit impulse will be important to much of what we do.

So let me draw some arbitrary f of t.

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Online Calculus Solver ». IntMath f orum ». In engineering applications, we frequently encounter functions whose values change abruptly at specified values of time t. One common example is when a voltage is switched on or off in an electrical circuit at a specified value of time t. The switching process can be described mathematically by the function called the Unit Step Function otherwise known as the Heaviside function after Oliver Heaviside.

Laplace transform of the unit step function

If you're seeing this message, it means we're having trouble loading external resources on our website. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. Search for courses, skills, and videos. Properties of the Laplace transform. About About this video Transcript. Introduction to the unit step function and its Laplace Transform. Created by Sal Khan. Want to join the conversation? Log in. Sort by: Top Voted.

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But recall that we can't use generally use multiplication of functions we have no multiplication property , only addition by use of the linearity property. I don't know, we're not using an x anywhere. So let me draw some arbitrary f of t. So what if I-- my new function, I call it the unit step function up until c of t times f of t minus c? Want Better Math Grades? This is an exponential function see Graphs of Exponential Functions. What does this mean? Posted 9 years ago. That's my x-axis right there. I just paused the video because it was having trouble recording at some point on my little board.

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Disclaimer: IntMath. Let me pick a nice variable to work with. So let's say that just my regular f of t-- let me, this is x. And obviously, nothing can move it immediately like this, but you might have some system, it could be an electrical system or mechanical system, where maybe the behavior looks something like this, where maybe it moves it like that or something. So let's say the Laplace transform, this is what I was doing right before the actual pen tablet started malfunctioning. We could take the integral-- let me write it here. So it's included, so I'll put a dot there, because it's greater than or equal to c. What's going to happen? Let's say that instead of it going like this-- let me kind of erase that by overdrawing the x-axis again-- we want the function to jump up again. I could write this as the integral from 0 to infinity of e to the minus sy times f of y dy. Flag Button navigates to signup page.