 Hello students, welcome to this course on Modeling Stochastic Phenomena. An important concept that emerges out of the study of Fourier transforms is the concept of a function called delta function. This function is not a regular function. In fact, it has to be understood in a slightly different way and not as a normal function. The idea of such a function was introduced by Dirac in his study of quantum systems. And this concept has found great applications in many engineering problems, subsequent to its use. It has been accepted now as a mathematical function, but in the sense of what is called as a distribution not as a function. That is a quantity which is meaning only when it is inside an integral and not as a standalone function. We will examine some of its properties, some of its uses and study its characteristics. Let us first examine in which context such a function is likely to be useful. We have earlier seen a quantity called Kronecker delta. For example, Kronecker delta is defined as delta let us say n, m Kronecker delta. We used this concept in Fourier series which basically means this quantity exists when n equal to m or else it is 0. There is no problem with this concept, it is regular and you can always use it when n and m are discrete. But however, if my variables are not discrete and I want to describe the situation when the quantities say n and m are very close to each other and their continuous variables then this concept of Kronecker delta is no longer valid and that is where delta function becomes a matter of use. There are many applications today which involve delta functions. For example, if you want to describe a point charge in space and you want to find out its distribution the effect of the point charge on the potential distribution in space then delta function is a very useful concept. Similarly, if I want to solve the problem of heat conduction due to a point source the concept of delta function can be used to describe that point source and then you can operate it with like any other extended source once you have understood the meaning of this delta function. So, we will now explore some of its properties. One way one simple way to understand delta function is as a limit of a rectangular distribution. Consider a distribution whose width is b and height is h such that breadth into height equal to 1 that is it is a shape of unit area. Now I half it in the base so it becomes b by 2, but I want to conserve the areas as 1 so I will have to double the height. So, it will be approximately it will look like this. So, it will have the height of 2 h and the base area of 2 b by 2. So, increasing I extend this process at infinitum then I will arrive at a situation when my function could look for example, very very narrow, but very very tall it is a w will be very very narrow and it is a height will be very very tall, but still it is area equal to w into h remains 1. So, in the limit when when w tends to 0 this function would look one cannot really describe it, but it would look like a very steepish rising function and in the limit it would be actually a line vertical line of height infinity and width 0, but we are supposed to assume its area is 1. So, such a function which conserves the area, but whose width is 0 and that is called as a delta function. So, delta x is a limit of the rectangle function as its base b tends to 0. This is one simple physical way to understand this quantity called the delta function. Another way to understand the meaning of delta function is the so called limit of the Gaussian distribution. For example, let us let us consider a Gaussian distribution described by a function f x equal to 1 by sigma root 2 pi e to the power minus x square by 2 sigma square, where sigma is a measure of width half width of this distribution. Now, supposing I take the limit sigma tending to 0, we would expect that the function its height at x equal to 0 will increase, because height at x equal to 0 f 0 equal to always 1 by sigma root 2 pi. So, as sigma tends to 0, f 0 will increase reciprocally. Since Gaussian always conserves its area, because integral f x dx minus infinity to infinity is 1. Hence, as we continue to decrease the sigma, the curve will go on becoming sharper and sharper more and more peaked, still conserving its area every time, thereby tending to the delta function that we saw in the case of a rectangle. Hence, another representation of a delta function is it is actually the limit sigma tends to 0 of a Gaussian distribution that is 1 by sigma root 2 pi e to the power minus x square by 2 sigma square. The third representation is from the Fourier transforms, which is of course, little difficult to visualize, but it is a very well known result that comes from the study of Fourier transforms. It cannot be actually proved very easily, it deals with convergence of highly oscillatory functions at large values of the variable and we merely assume the result, which says that the Fourier transform of 1 by 2 pi will be a delta function with respect to the parameter k. This basically is a Fourier transform of a variable whose value is 1 by 2 pi. We have used the definition Fourier transform as simply the integral of the function with respect to e to the power i k x extending from minus infinity to infinity. We can exactly not prove it, but we can see why it should diverge at k equal to 0 because that is what the property of delta function is. It is infinity at k equal to 0 and it is supposed to be 0 elsewhere. If you put k equal to 0, this becomes an integral of simply dx and integral dx at minus infinity to infinity is going to be infinity because it is the area under this rectangle, area between these 2 lines. So, we can see why it would be infinity. However, as for any other value of k, it is an integral over psi n and cosine functions, highly oscillatory functions and it is assumed that the limit value of these functions will tend to 0. In fact, it is taken as 0. So, this is a very useful representation at times to resolve the delta function in terms of its so called Fourier components. In stochastic phenomena, the importance of Fourier representation is manifold. It basically means if I want to see the various frequencies that is contained in a sharply peaked function, then it will have almost equally valid or equally strong frequencies of all values. That is why sometimes it is connected to what is called as a white noise problem, white meaning all frequencies occur with equal intensity and delta function therefore, is a conjugal representation of such a spectrum. We now discuss some of the properties of the delta function which is useful to apply it to different contexts. One important property upon which as I said the very definition is based is an integral of delta function from minus infinity to infinity dx that is equal to 1. Basically, the area under the this function is 1 and you cannot plot this function. We again like to emphasize that you cannot plot a delta function, because it is then infinitely high curve with 0 width. We can slightly generalize this property supposing I have defined a delta function at the point x equal to a, I just represent it by a line although it cannot be represented by a spike. There is a significant conceptual difference between a spike and a delta function. The area under a spike is 0 whereas, area under a delta function is 1. So, such a function which it is only represented for want of better way is denoted by delta x minus a. In fact, since the function is 0 outside the point a you can actually integrate from any value outside of the point a say some point a minus epsilon to a plus epsilon of the delta function and still you would get the same value 1. It is not necessary it is not wrong, but it is not necessary to have it as minus infinity to plus infinity. A small deviation from the point where the delta function is defined that is sufficient for us to carry out the integral. So, that is an important property. So, this we can say as property 2 if we denote the previous property as property 1. Now, the third important property of the delta function is that it is an even function delta x equal to delta of minus x. This comes from the symmetry of the limiting functions that we have chosen for representing the delta function with respect to the point where the delta function is defined. So, it is like therefore, like any other even function which is very important utility. Most important advantage of delta function is its ability to select values. So, if I have a function fx and if I multiply this function by delta x and then I integrate over the it is an entire domain then this will give me f of 0. So, what I did is like this I had a function fx. Some function fx which was going like this I had a delta function at some let us say value x equal to a then if I multiply by this delta function with this function and integrate it selects the value at a that is I generalize equation 4 that is minus infinity to infinity delta of x minus a fx dx this selects the value of the function at f at x equal to a. It is a very very important property there is selection property. Next useful property of the delta function is to solve some simple algebraic equation. We have always been told that one should not divide 0 by 0 supposing you have to solve an equation x into phi x equal to 0 then we would often say either x is 0 or the function phi should be 0. More elegant way of writing the same function is to write the solution phi x equal to some constant c into delta x this almost carries the same implication. So, this is a possible solution to the problem. So, it is one of the advantages of delta function because it helps you to generalize and write through a common framework. The next important property is delta of some constant a into x is 1 by mod a of delta x how do you prove this supposing this is we will just check that this is really true how do you do that supposing I have to integrate delta ax dx minus infinity to infinity this will be 1 by mod a of minus infinity to infinity delta x dx as per the above relation. This will be 1 by mod a because this integral is 1 whereas, if you do it by transforming ax equal to u we can note that same thing will come to minus infinity to infinity delta of u then dx will be du by a which again will be 1 by a and since delta was an even function sign of a did not matter. So, it will be replaced with 1 by mod a. So, we see this are equal and therefore, this representation is valid. Another important property of the delta function is supposing the argument of the delta function is not just x it is a function of x supposing there is a function g x which looks like this. So, it cuts the x axis let us say at 3 points and I have to define this function delta of g x. So, what it basically means is how will this function contribute to an integral because delta function is defined only within an integral. So, as a standalone representation it is written as if there are let us say n roots and none of which has its derivative 0 then it is g prime x n. So, g prime should exist otherwise that function cannot be the argument of the delta function it has no meaning physically it is very easy to understand this. So, it uses the property that delta function it is integral need not be from minus infinity to infinity it will give you a value of 1 when you are just to the left of the point to the right of the point supposing my function that I have spoken become 0 here and I have multiplied the delta function. So, when I integrate this function from supposing this is my x naught is the 0 point then my delta function integrated from x naught minus epsilon to x naught plus epsilon dx will also yield 1. Let us first consider the case of a g x function which has 1 0 at point x equal to x naught that is g of x naught equal to 0. We can then write the function g x as g x equal to g x naught we are actually Taylor expanding x minus x naught g prime x naught plus higher order terms x minus x naught whole square by 2 factorial which is 2 g double prime x naught plus higher order terms. Since g x naught is 0 this will be rewritten as x minus x naught into g prime x naught plus x minus x naught by 2 into g double prime x naught. Let us assume that g prime x naught is not 0 then we know that when x approaches x naught the second derivative term will not contribute especially within a delta function. Hence for our purpose we can stop the expansion at the first term itself and write this as g prime x naught with this we have delta of g x will become delta of x minus x naught g prime x naught. We have seen earlier that delta function of a product can be reduced to delta of x minus x naught divided by mod g prime x naught. So, this basically proves the result for a case where we have 1 0. So, if you have a series of 0's finite number of 0's then this Taylor expansion will be true for every one of the around every one of those 0's and one get the result that we wrote down in the just before this. Let us take an example of a function g x equal to x minus a into x plus b. So, it is basically a quadratic function it has 2 0's if a and b are 0 it has 1 0 at minus b and another 0 is a. So, it could be a function like this. Now we want to express delta of g x. So, from the formula that we derived it will be delta of x plus b divided by the derivative of the g function at that point b of course modulus similarly plus delta of x minus a and the derivative of the function at the point a. So, g prime x equal to you can show that it is a 2 x plus b minus a. Hence g prime b modulus is going to be a plus b modulus and g prime a modulus is also going to be a plus b modulus and the solution that we are looking at the delta representation for the function delta of x minus a into x plus b can be written as delta of x minus a plus delta of x plus b divided by modulus of a plus b. So, you have now reduced it to delta functions with respect to the normal coordinates. Thank you.