 So far, we have introduced how we can construct new complex valued functions using the concept of infinite series. Now in this part, we will explore further about different properties of these functions. For example, we will see how to calculate the domain of these functions. Now let's recall how did we define complex valued functions using infinite series. So we consider this infinite series that we can see on the screen. This is an infinite series of the form cn z minus alpha raise to power n. Now for some values of z, this infinite series will converge and for some values of z, this infinite series will diverge. Now we want to calculate, we want to find those values, those complex numbers z for which this infinite series converge. And of course, it will be only those complex numbers for which the series converges. That will be the domain of this complex valued function. Because we don't want to include those complex numbers in the domain for which the output is undefined or infinity. So there is a simple notation related to this expression as well. So in this expression, instead of taking the first term to be c0 z minus alpha raise to power 0, we take the first term to be c0. Because it will avoid the situation of 0 raise to power 0, which is an undefined expression. Now to calculate the domain of these complex valued functions, we have our first result which says that if we have an infinite series n is equal to 0 to infinity cn z minus alpha raise to power n, then the point z for which the series converges is one of the following. So that we have three situations, either it can be a single point z is equal to alpha or it can be the disk. So basically it is the set of all points whose distance from alpha is less than rho. So basically rho is some positive number and we are just saying that this disk d rho alpha is set of all points whose distance from alpha is less than rho. Basically if we take this distance to be rho, then it is basically collection of all points inside this disk. This is the second scenario that can happen when we are trying to find the domain of this complex valued function f of z. And in this scenario the domain or the set of points for which the series converge is basically all points inside this disk. Now what about the points on the boundary of this disk? So we cannot say anything about those values of z which are at the boundary. So the infinite series it can either converge for some parts or diverge for all of the parts or it can converge for all of the parts or it can converge for none points on this boundary. So anything can happen about the points on the boundary. The third scenario is basically the entire complex plane. So we take the complex plane and take any complex number in the plane that the series converges for that z. So these are the three possible scenarios when we are trying to find the points the complex number z for which the series converge. Now let's prove that why these are the only three possibilities. So to prove these things we are going to need the root test that we have discussed in our earlier modules. So basically test says that if we have an infinite series of the following form then we calculate the lim sub of the nth root and in fact the mod of the nth root of its terms. And if this expression is equal to L some number L then the series absolutely converges if L is less than 1 it diverges if L is greater than 1 and for L is equal to 1 there is no conclusion. So we cannot say anything when the number L is equal to 1 the series may diverge the series may converge so anything can happen when the result of this lim sub is equal to 1. Now let's use this result on the infinite series because we have this infinite series and these terms are basically this zeta n. Now let's use this root test. So in other words we want to calculate this lim sub and this zeta in this case is basically Cn z minus alpha raised to power n okay now let's calculate it so the expression becomes Cn z minus alpha raised to power n and of course its mod value raised to power 1 over n now we want to calculate this limit now in this case we can see that this z minus alpha so it does not involve any n so in other words this is a kind of constant for this limit so we can write it down outside this limit. So this is our expression of interest if we want to find the conditions on z under which the series converges and the series diverges okay now there are three possibilities of course when we calculate this expression so it could be infinity it could be some number that lies between infinity and zero okay some number L that lies between zero and infinity or it could be zero so this gives us three cases to discuss so first case second case and third case now we will discuss these cases one by one starting from the first possibility if this lim sub value is equal to infinity what will happen so this implies that the series diverges for all z except for the value z is equal to alpha why is it is the case because if this is infinity then z minus alpha okay so lim so okay so Cn 1 over n so definitely this will be infinity and it will be of course greater than 1 but if z is not equal to alpha because if z is equal to alpha what will happen now let's see if z is equal to alpha then the series okay so now this expression becomes zero okay so this will be equal to zero so definitely this expression becomes zero and definitely the series will converge okay so there is another way to look at the situation if we consider this infinite series Cn z minus alpha n okay so n is equal to zero to infinity then for z is equal to alpha this series is nothing but you know a bunch of zeros so for example apart from the first expression C0 the rest of the things the rest of the terms are zero because alpha minus alpha is zero so that is why the series must converge when z is equal to alpha and for the rest of the values for z not equal to alpha the series diverges because this expression is definitely greater than one okay so we have proved the first case that the first possibility is that the series converges only at one point that is equal to alpha and this was the case that this lim so value is equal to infinity now let's consider the second scenario the second scenario is that the value of this lim soap is less than infinity it is going to be some finite number but not equal to zero the zero case will be discussed in our third possibility okay so what will happen in this case okay now if this is some finite number then we can easily find the condition using z minus alpha mod the condition on z for which this series converges of course the condition is that this expression should be less than one so what do we get so we get the following condition on z okay so this directly implies this expression because this is some positive number because we are calculating the lim soap of this positive non-negative numbers in fact so that is why it is going to be non-negative and it will not disturb the sign of this inequality it will remain less than so we have the modulus of z minus alpha is less than one over this expression once again this expression is some finite real number okay so we have a condition on the z so if we take a complex number such that the modulus z minus alpha is less than this number then the series is going to be convergent for that and if we take a number z for which this expression is not less than this then it is going to be divergent for this so we have a kind of condition on the z okay so this expression or this condition can also be written in the following form okay so because over here we are imposing the condition that whenever the distance between z and alpha is less than this then it will converge so we can we can name this distance to be for example row okay so let's call this expression row then the set of values the set of complex numbers for which the series converge is going to be a disc with center alpha and radius is going to be equal to this number row okay so this is going to be row this is the disc and this is the center okay so this is alpha and we have this disc which is so whenever we take z from this disc then the series is going to be convergent for that complex number now in this case we have proved the second part but we are left with one simple situation okay so let's go back to our disc the center is alpha the radius is row that we have just calculated now what about the points at the boundary okay so the points at the boundary are not discussed earlier because we don't know what will happen when the value of l is equal to 1 okay so due to that condition so the previous discussion does not say anything about the points at the boundary so it only says the for the points which are in such that the distance of those points from alpha is strictly less than row so that's why anything can happen for these points so the series may converge for all of the points it can converge for some of the points or it can converge for none points at the boundary of this disc okay so this proves the second case the second scenario that can happen so what can happen so the series converges for all of the points in this disc and it does not say anything about the points which are at the boundary of this disc now let's discuss the third possibility of this limit so if the value of this limit is equal to zero what will happen okay so this expression is always going to be less than zero because this expression this limits of value is zero so that's why this expression is equal to zero and zero is always less than one so whatever value of z whatever complex number z you take this expression is going to be less than one so we can say that this expression implies that for every complex number z the series is going to be convergent so we can say that the third scenario is here this series converges for each and every complex number in the entire complex plane so we have the following scenario you know so given any power series function okay so which is basically given over here so its domain has three possibilities so its domain consists of a single point or it consists of a disc and we don't know some parts or all of the all of the boundary of this disc may be included in the domain and the entire complex plane these are the three possibilities for the domain of this power series function now the number row such that the series converges for all z satisfying this thing and diverges for all set z satisfying this thing is called the radius of convergence because this row is basically is defining a disc so we can say that it is basically a disc of convergence and this row is basically the radius of this disc or we can say that it is the radius of convergence so the series converges whenever z lies inside this disc and diverges whenever z lies outside this disc so we can say that this row is basically defining or giving us the radius of convergence of that series okay now of course if we are getting a single point because there only three scenarios for the domain or for the points of convergence of this power series so in the first case when we have only one point then the radius of convergence is zero because if we are having only one point then there is no point outside this alpha so we can say that it is a circle of radius zero and the radius of convergence in this case is zero the second scenario that we have discussed was we have a disc now with some radius row and in this case of course the radius of convergence is row okay and once again let me remind you that sometimes some for some points at the boundary the series converges sometimes for all of the points the series converges and sometimes for none of the points on the bound at the boundary of this disc the series converges so the points at the boundary can have different behavior but overall the radius of this disc is row so we say that the radius of convergence of this series is row and of course the third possibility is the entire complex plane and we can say that in this case the radius of convergence is infinity because it converges for each and every complex number in the complex plane so obviously the whole complex plane has we can say that has radius infinity so the radius of convergence in this case is infinity now in this part we have discussed how to find the domain of a complex valued function which is defined using infinite series and we have called these functions power series functions now in our further discussions we will calculate how to find the domain of some particular power series function