The strength of the Laplace transform technique, is to solve the differential equations where you have a discontinuous right-hand side. So, either the form of the function changes at some time T, or you have an impulse force on the right-hand side. So, in the case when the form of the function changes, it's useful to introduce a new function that can model that. This function is called the Heaviside step function. So, what is the Heaviside step function? It's very easy to see what it is if we draw a graph. So, here is our graph. Let's call the X axis T and the Y axis X. So, the Heaviside step function, is a function that is zero until you hit some value, C and then it becomes one. Okay. So, this is the value of C. This is the value of one. We call this the Heaviside step function. It has the form, this would be X equals U sub C of T, that's the notation we'll use. So, the Heaviside step function is a function of T, but it has a parameter, which is called C such that it's zero. So, we can write down the mathematical formula. So, U sub C of T is equal to zero, when T is less than C and one when T is greater than or equal to C. It actually doesn't matter where you define it. At C, how you define it whether to be zero or one but here we'll just use the convention that we define it to be one at C. Okay. We'll see in a moment why this is useful for modelling discontinuous functions. What I want to do first, is to show you how to take the Laplace transform of the Heaviside step function. So, the Laplace transform of UC of T, from the definition of the Laplace transform, is the integral from zero to infinity, E to the minus ST times U sub C of T, DT. How do you do an integral like this? Well, you know U sub C is zero between zero and C, and then it becomes one. So, you can change the limit of integration and get rid of the Heaviside step function. So, this becomes, since the integral is zero between zero and C, it becomes the integral from C to infinity of E to the minus ST, DT, which is just the integral of an exponential function, that will be minus one over S into the minus ST. Putting in the upper limit will get zero, and then putting in the lower limit will get one over S times E to the minus CS. So this will be an E to the minus CS, divided by S. Okay. That will be in our table and I'll show you where that is in a moment. So, how do we use this Heaviside step function? One use, is the following. If we have some function, let me draw it here. So, we have some function maybe looks like this. Instead of a function, let's call that F of T. We want to shift that function a distance C to the right. So, we want it to look like this one instead, and make sure that it's zero up until the shift. So, this shift occurs at C. How do we do that in terms of the Heaviside step function? This would then be UC of T to make sure it's zero for T less than C, right, this axis here is T. Then, the shifted function will be F of T minus C. So, when T is equal to C, we get the value, and the shifted function of what F would be at zero. Okay. So, this is one of the uses of the Heaviside step function. We're going to have to know how to take the Laplace transform of this bit. So, we need the Laplace transform of U sub C of T times F of T minus C. You can do that integral. So, you substitute this into to the definition of the Laplace transform. I won't do that for you here. You can try that at home, where you end up with E to the minus CS times F of S. That will also be in the table and useful for the table. Okay. Useful for our future calculations, but you'll just read it off of the table. Okay. One more point that I need to make about the Heaviside step function. I said that you can use the Heaviside step function to model a discontinuity in the inhomogeneous term of the equation. So, let's say we have F of T here, which is composed of two functions. Let's say it's F1 of T. When T is less than C, and as F2 of T a different function. When T is greater than or equal to C. So, how would you model that using the Heaviside step function? The idea being, we want to take the Laplace transform of F of T but if we write it in this expression, we don't know how to do that, but if we write it in terms of one expression that contains the Heaviside step function, then will be able to do that using the table. So, the way that to write this, is to write this as F1 of T, plus F2 of T, minus F1 of T, times the Heaviside step function U sub C of T. Why does that work? When T is smaller than C, the second term is zero and you have F1 of T. Then when T is larger than C, UC of T will be one and you have F1 of T plus F2 of T minus F1 of T and you get F2 of T. So, you can model a discontinuous function piecewise, it's called a piecewise continuous function. So, there are two functions that are glued together making use of the Heaviside step function. Okay. Let me review here then. I've introduced a new function that will need when we model a discontinuous inhomogeneous term in the differential equation, that function is called the Heaviside step function, is written as U sub C of T, which is zero for T less than C and one for T greater than C. You can take the Laplace transform of the Heaviside step function. You can use the Heaviside step function to shift the function to the right a distance C, and you can use the Heaviside step function to glue together two functions at the value of C. The other point I should make that I don't need to go into the detail here is that, you can also use the Heaviside step function to model a step down function. This one we call a step up function. You can model a step down function or you can model a step up and a step-down function. I'm Jeff Chasnov. Thanks for watching and I'll see you in the next video.