welcome to my new series on the llas transform in this video in the videos going forward we're going to study some of the reasons why the llast transform is an incredible powerful piece of mathematics in this first video we're going to define the llast transform we're going to see a couple different examples where we're going to compute the llast transform but why might we be interested in something called a llast transform in the first point one reason why I really like the llast transform is it's useful for solving differential equations like this one here I have an ordinary differential equation together with two different initial conditions in general differential equations can be very challenging although this specific one can be solved as it's a constant coefficient homogeneous one relatively straightforward however the point of a llast transform is to convert differential equations like this ones that have derivas in them into somehow an algebraic equation this is some new equation it's got a new Y in terms of a new variable s but there's no deres involved it's just a purely algebraic equation and indeed in this particular case it's very simple to solve for why I can just divide through and so one of the powers of the class transform is going to be able to convert differential equations into algebraic equations that were're able to solve and then we're going to learn how we can convert backwards and take the solution to the algebraic equation and figure out a solution to the original differential equation so that's our plan for the future but right now in this first video we're simply going to Define find the llast transform and we're going to compute what the llast transform is in three different examples the notation for the llast transform is I do a sort of squiggly L and what this squiggly L does is it has an input which is some function of T and then you're going to take that F of T and via the llas transform you're going to transform it into some other function now a capital f and this other function depends on some other variable now s so theast transform is is kind of like a function which would take a point to a point but aast transform takes a function to a different type of function here's how it's defined the llast transform this capital F ofx is defined to be the improper interal from Zer up to Infinity of e to the minus St notice that's where the S comes in of f of T DT because we're doing an intergral with respect to T the t is going to go away so that what you get for your F of s is only depends on the value of the S this thing that comes in by this negative exponential because this is an improper interval we're going to have questions like when does this converge and when does it not converge nevertheless this is the definition so let's see some examples the first one I want to begin with is just going to be the llast transform of an exponential function so I'm going to have e to the a t where a is just some constant all right so what's its laass transform we can just simply write it down so this is the proper integral from 0 up to Infinity e to the minus St I just copy that and then for the Ft I plug it in I plug in the e to the a t this is two different exponentials so I can combine the exponentials the powers of them and make it just A- s * T this is a straightforward enough integrant so when you integrate that you're going to get e to the A- s * T and then you have the Divide out by A- s and we're evaluating this between 0o and infinity by the way because we're doing improper integrals the technical way to define these is to replace the infinity sign with say a b and then take the limit as B goes to Infinity nevertheless we have this short hand of to sort of evaluating it zero and infinity okay so uh putting this together and getting rid of the mess this is my claim how do I evaluate that well it depends on the A and the S and the question of whether this is going to converge when you take this limit as some value goes to Infinity is that going to be a finite number or or not well it depends on the A and the S so for example if s is greater than a we're going to get one case and if f is less than or equal to a you'll get another when s is greater than a in the exponent you have e to well a negative number time t as T goes to Infinity S he's just really windy and Howling outside but you know what we're in the middle of covid-19 I got to record the video now nevertheless so if you can hear some random wind noises just ignore that anyways and then if you take T to Infinity of a negative exponential it just goes away to zero and so indeed in this case it just converges as zero and then when you plug in the zero you get an e to the 0 over the Aus s and and the end just 1 / s minus a in the other case where s is less than or equal to a well if it's equal then just in the denominator you have a division by zero it diverges and if the S is strictly less than a it's going to be well it's going to be an exponential that has a positive times a t going to Infinity likewise it's going to diverge so in this example we figured out how to compute the little transform of this exponential function and we've noted that the answer that we get does depend on what the value of s is relative to this constant a and you sometimes get it diverging and sometimes get it converging before we get to our second example I want to introduce a new function to us I'm going to call this a step function or sometimes called the heavy side function what this function does is it's zero when T is less than zero and it's one when T is bigger than zero and then the example I want to compute out is actually the pl transform not just of the step function but of the step function U of T minus a what this U of T minus a looks like is well something like this the idea is that when your T is going to be less than a then it's just going down to zero and then when your T is greater than a it's going to be of height one this is a very useful function because if you have any function that you think should have a discontinuity at some particular spot when you multiply by this U of T minus a where a is the spot where you want to have a discontinuity it introduces a discontinuity and so this is a wonderful way to be able to Model A lot of functions that have step discontinuities you just sort of multiply by this heavy side function or this step function nevertheless our goal is to compute the llast transform of this function well let's write it down so again it's our integral but now our integral doesn't start at 0 to Infinity because the function is just the function zero all the way up to the value of a so the portion of the integral that's zero up to a is just going to be equal to zero here by the way I'm assuming that a is a positive constant so the integrand has that negative exponential that is part of the definition of the lass transform and then it just multiplies by one but what really is happening with the step function is it's restricting though don't main so it's now it's a to Infinity not 0 to Infinity nevertheless it's an easy integran we can compute that and when you plug in the infinity because it's a negative exponential it goes away to zero our assumption here is that a is positive and as a result it's just going to be e to the minus sa a / by a it's what happens when you plug in the lower limit that t is equal to a all right so now let's do our third example but before we do that I want to introduce a new piece of terminology I want to introduce something called called the gamma function the gamma function looks a little bit related to the llast transform indeed it will be in just one moment but for now it's just an improper intergral from 0 to Infinity very similar the specific exponential e to the minus t and then multiplied by T to x -1 DT this is some function and it turns out to be important enough we give it a name and so we call it the gamma function okay the gamma function has actually many very pleasing properties the first sort of simple one is is just that the gamma function of One turns out just to be one if you plug in xal to 1 then you get T just to the zero which just becomes a negative exponential you can integrate that and get one but perhaps more interestingly is I want to see what happens to the gamma function when I evaluate it at an integer n + 1 so if I do that and I plug this in well t to the N +1 minus one just becomes a t the N so now this is some sort of expression here if you thought about how you might have to integrate this well you could do integration by parts and different times to reduce that t to the N down to a one but I'm actually only going to do it once and try to get some sort of recursive behavior that I can use so that's what I'm going to do I'm going to set u equal T the N I'm going to set DV equal to e to the minus tdt and I'm going to do an integration of my parts what this gives me is the following long expression so I get a u * a v evaluated at the end point0 to the infinity and then then I subtract off a v du and that gives me this expression now for the first part that's evaluated at infinity and zero well e to the negative Infinity is going to dominate the polinomial and it's going to give a zero and when you plug in zero you get well just zero to the end you're likewise going to get zero so the the first portion of this is just all zero what about the second portion well the second portion kind of looks like another gamma function there is an N mixed in there so we have to take that n and bring it out but other than that it's just n * the gamma fun function of n so what I've done here is I've related the gamma function of n plus1 to the gamma function of N and it brought up as coefficient N I could do the exact same argument again and I could say well look I could take the gamma n and relate it to gamma of n minus one and that will bring out an N minus one and then I can just keep going in this way time and time again going all the way down gamma n gamma n min-1 gamma nus 2 and so forth all the way down to one we'd seen that Gamma 1 was one and so really what we just have just n factorial n * n-1 * n- 2 and so forth so it's quite pleasing that for n factorial which is never been a function that we've thought of as sort of this thing you could go say take the of or integrate now can be expressed as this particular improper interval one of the nice things that we have is a bit of a generalization of the factorial for example you could ask what gamma of say 1/2 is you don't have an answer for what half factorial should be defined as but now you can express it in terms of this particular inter and compute up that intergral and it turns out to have some very nice properties all right so with the gamma function defined now let me turn to this problem of computing theast transform of just a polinomial t to the N it turns out that we're going to be using the gamma function in our computation indeed if we write out what our expression is going to be well F of s the laass transform is just the improper integral eus s t * T the N DT I'm going to try to clean this up a little bit to make it look more like the gamma function by doing the substitution U equ Al to S * T if I do that well the exponential is just to become eus U but then further I can say that the T to the N is going to become a u to the N but it divides out by an S to the N that goes on the bottom and then and then likewise when the DT turns into a du you also have to divide by one more copy of s so what you get is 1 / s n + 1 stuck out the front and then some integral entirely in terms of U now we can recognize this interval as just being the gamma function so I have that same coefficient up the front but now I just have the gamma function evaluated at n +1 and we had computed that out that was just n factorial so our final answer for the llast transform of T the N is n factorial over s to the N +1 so that is that for so those are my three examples of computing out the class transform and in the next video we're going to see three different properties that class transform has if you have questions about this video leave them down in the comments if you want to see more differential equations or any of my other courses links to those playlists will be down in the description give the video a like for that YouTube algorithm and we'll do some more math in the next video