Hello Students, today I am going to discuss Taylor Series and this is part of complex analysis and Taylor series and Maclaurin series are used to find the function of any series we study this but in Complex also there are 2 types of series one is Taylor and the second one is Laurent Series So before this, I have already made some videos on Complex Analysis in which first was made of what is cauchy demand conditions and how we identify function analysis are identified or not and then in my second lecture I have taught you that if any function is analytic and if its real part is given then how we will find the imaginary part if its imaginary part is given then how we will find the real part and after that in my third video I have taught you that if any function is analytic and if real part is given then how we will construct the function by Milne Thomson method and after that in my last lecture I taught by a linear transformation, so students If you want to see all these videos so here is the I tab you can click on this and watch my all videos if you watch them then these concepts will also be clear because it is the it is the continuous part of this topic so now we are going to take the topic Taylor series so what is Taylor series is if a function is analytic inside a circle C with a center at z=a then it can be expanded in the series But we have to keep in mind that the point should lie inside the analytics function and should not lie in the outside function and if it lies in the outside function so we are not to express it in this series, so now there is a question and it is given as I will tell you this question by two methods one we have is the series method and the other method is expanding the binomial form but look here for the first method so what method 1st says then here we have to find f' A so we will differentiate it once Now we will differentiate it once again and in this same way we can also do differentiation for the 3rd time and in this way we will find their values Now we will place all these values and we have to find the value along with z=1 point so first we write the formula Now we will place the value the value of a here is 1 now the value of the function is kept so the value of Now If we want then we can take out 1/2 common I am writing it slightly here So in this way we will solve this Now we will solve this question by the second method In my opinion the second method is easier and if this question is asked in the competition exam then you should always use this method only so here you have to do it for z=1, so in the second method what will do so students look here in which point it is given bring here and assume it as t, clear so now we will write it in the form of t the value for z is t+1 so what we will keep here here I want to explain you a bit about binomial formula which we study in 11th class the binomial expansion we will use here so what binomial expansion says, look here so if we simplify it sorry I have written the wrong formula by mistake let me correct the formula so this is the correction so I am writing again I told you wrong so students with the help of formula I have taught you the expansion and this I have taught you with the help of binomial and this is the second method and in both the methods the answer is approximately the same so I will take 2 more questions on this topic and then we will move forward So here we will discuss the second question and that is Find the first four terms of the Taylor series expansion of a complex variable function about z=2 and find the region of convergence you also have to find the region of convergence from what to what point it is convergence it means from where to where the function the is analytic right. if the center of a circle let me tell you here about z=2 it is given then the center is 2 right if the center of a circle is z=2 then the distance of the singularities z=3 and z=4 from the center are 1and 2 you can see here the singular point the singular points are those points on which the function is not analytic and here these points are 3 and 4 so we have to show that these points 3 and 4 must be outside the circle right because if it is inside then the function will not be analytic on these points and we will not be able to find the Taylor series whatever the points are there must be analytic and must not be singular points right and we have to show that 3 and 4 should lie on the circle or it is outside clear so the singular point here is 3 and 4 so we have to take that region for 3 and 4 must look outside and the center here is given 2 the equation for the circle is if we take radius= 1 right then these 2 points will be outside so their radius here will be this and this will be its region of convergence and in this region of convergence this series will be defined So now I will tell you how we will take it out so 1st of all we have to do the partial fraction so how we do the partial fraction look here we will take out the series Now we will do the partial fraction I will tell how to do this is it this is the shortcut method to find the partial fraction we will take out constant common because when we do the binomial expansion we know what happens in binomial expansion so in this way, we solve this question So students one last question more I will take on this topic and that is the same to the same question and let me tell you that to solve the questions of Taylor series we have 2 methods the one method is in the form of series by doing differentiate but it is very long and complicated method and it also has an alternate method and I am telling you this alternate method and you can solve this question by the alternate method in the exam and no questions will be deducted so look here find the Taylor series expansion of a function of the complex variable and also find the region of convergence right so in what region this series will be defined so what will do is so we have to take that region of convergence whose center is less than 3 and 1 or equal so that these two points don't lie in this circle and if they lie in this circle then these points will not be analytic and then this series will not be defined because the Taylor series theorem say s the function should be analytic we can see here that the function is not analytic on 1 and 3 if it is coming in the region right so what will do is we will take that region of convergence so that this region gets defined we will take the radius 1so that these two points can lie on or outside the circle so the region of convergence will be Now we will expand this like I have done in the previous question and you can also do it by differentiating both are same but differentiation is a long process So look so in this way, we get the series and we simplify it and we get the answer so, friends, I hope this concept is understood by you and after this, I will take the next topic that is the Laurent series and in that, we will learn how to expand the series in the given region and I have uploaded many videos which will be seen by in I tab in the last part of the series in videos that what all videos I have uploaded for BSC and Engineering students you can have a look there. Thank you so much for watching my videos thank you, and keep commenting, sharing and liking my videos So that I can be here with the same energy level. Thank You so much