 Good morning. Today, we start our module on complex analysis. We start with analytic functions. We define the function of a complex variable z as a rule, which will associate a unique complex number like this for every z in a domain in the z plane. And as for any calculus that is for example, calculus of real function, we start with the definition of limit continuity and derivative. The first limit, if the function f of z is defined in a neighborhood of a point z 0, except possibly at z 0 it tells. What it means is that for the definition of the limit at point z 0, it is not necessary for the function to be defined at that point. You can define a limit without the function being defined at that point. So, if the function is defined in a neighborhood everywhere in a neighborhood of z 0, except possibly at z 0 itself. And if there is a complex number i such that for every epsilon every positive epsilon whatever small, you can find a delta such that taking the point z in a delta neighborhood will keep the difference of the function value from the value i within a an epsilon neighborhood. Then we say that limit exists that is the around z 0, you can find a small neighborhood such that you can approach the function value as i with any tolerance required. Whatever tolerance you want accordingly you would ask for the delta neighborhood to be closer and closer. If you can achieve this then we say that the function has a limit at z 0 and that value i is the corresponding limit. Now, this limit can be defined even if the function f is not defined at z 0 that is at z 0 even if the function does not have a value associated with it. And even if the function has a value associated with it at z 0, but the function value is not i, but along every path if this i value is achieved is approached rather than achieved even then we say that i is the limit. That means the function value at that point and the limit at that point may be different as in the function of real variable as well. Now, here you will note that there is a very important difference with real functions. In the case of real functions of a single variable x the point x 0 could be approached only along one direction from the left or from the right of course. So, x 0 point x 0 could be reached either from x 0 minus delta or x 0 plus delta left side and right side, but in the case of complex variable the description of the variable is not along a line, but on a plane. And therefore, there are infinite directions along which the point can be approached and this is the crucial difference which makes the definition of the limit in the case of a complex function much more restrictive. That is it is not only not enough for the limit to be same along two directions usually opposite, but along all paths in the complex plane. So, all paths leading to the point z 0 should give us the same limit and in that case we will say same limit i in that case we will say that limit exists and that limit is i. Now, this makes it extremely restrictive definition and apart from that if the value of the limit and the value of the function also at that point is same then we say that the function is continuous at that point. That is the function should be defined at that point and limit should exist in this sense and the function value and the limit must be same in that case we say that at z equal to z 0 the function is continuous. Continuity in a domain means continuity at every point of the domain. Now, after defining limit and continuity at the next step we define differentiability and the derivative. So, just like real calculus we make the first step in the definition of a of the derivative of a complex function and that is from z 0 if we take a small difference and reach the point z then this is the difference of the values of the function at the two point and this is the difference of the two values of z. Now, this limit as z tends to z 0 will give us the derivative if this limit exists. So, as z tends to z 0 if we show z as z 0 plus delta z then it will look like this. Now, when this limit exists that means along all path in the z plane as z 0 is approach is made closer and closer to 0 that means as the point z approaches z 0 along all path then if all the limits turn out to be same then we say that this limit exists and correspondingly we say that f z is differentiable and that value of that limit is called the derivative at z equal to z 0. This again is extremely restrictive definition that means that for a complex function of a complex variable to be differentiable it must be extremely nice that is it has to satisfy such a restrictive requirement and that within itself brings in a lot of desirable properties to the function. So, in that case we find that the function that we are talking about just by being differentiable it brings in a lot of additional properties all of them together is the concept of analyticity. So, we call a function analytic in a domain D if it is defined and differentiable at all points in D if there is a domain in which at every point a function is defined and is differentiable in this sense then we say that the function is analytic and it can be shown that the function being analytic at that point means that it can be expanded in an infinite series of powers of z that can be shown and now there are a few points which can be said in continuation of this set of nice properties that are brought in and these are points which will be established or settled later as we study the integration also. So, one is that if we can establish that a function is analytic at a point that means it is differentiable at a point then that will also imply that the derivative itself also will be analytic that means it will possess a derivative of its own and then by continuing on this argument we can show that an analytic function will possess derivatives of all order and that means that in the case of the real valued functions differentiability did not mean too much a function could be differentiable once, but that would not mean immediately that the derivative itself will be differentiable. Now, in the case of complex functions the existence of the first derivative itself requires so much that once the first derivative is established then it implies a lot of other things which will finally mean that once that function is differentiable then it will be differentiable as many times as you need and that is the entire implication of analytic. So, this is the great qualitative difference between functions of a real variable and those of a complex variable when we were studying series solution at that stage when we were talking about the coefficient functions being analytic at a point the sense was this analyticity that is it is not only differentiable, but it has derivatives of all order and that means that it can be expanded in a power series around that point and that will be convergent. Now, there are a pair of conditions called Cauchy-Riemann conditions which are satisfied by a function if it at a point if at that point it is analytic and to appreciate that we consider this situation suppose a function f u plus i v is analytic then we have f prime which is limit of delta f by delta z as delta z tends to 0. Now, since z is x plus i y so delta z will be delta x plus i delta y. So, delta z tending to 0 we mean both delta x and delta y tending to 0 and this is delta f from here and this is the corresponding delta z. Now, if the function is analytic then it will mean that the limit of this along all paths should be same that means for example, suppose this is a point z 0 then 1, 2, 3, 4, 5 along all paths the point is approach the limit is same in particular let us consider these two paths horizontal and vertical. In the horizontal case delta z is equal to delta x because y does not change and in the case of the vertical path delta z will mean i delta y because delta because x does not change. Now, if all these limits are same then in particular these two limits also will be same. Now, as we consider delta z along this path then delta z is delta x. So, this limit immediately will be del u by del x plus i del v by del x that is this. If we consider this path then delta z will be i delta y so keep the i delta y here. So, this will be 1 by i del u by del y that is this 1 by i which is minus i because minus i square is 1 plus i by i will go out will get cancelled the rest is del v by del y that is here. So, along this path we will have this derivative expression along this path we will have this derivative expression. Now, from analyticity we know that along all these paths the limit is same in particular along these two paths the limit is same. So, we get two expressions for the derivative and as we equate these two expressions and separate out real and imaginary part then we get del u by del x is equal to del v by del y and del v by del x is equal to minus del u by del y. These two conditions are called Cauchy-Riemann conditions or T R conditions or T R equations. So, Cauchy-Riemann conditions are simply this and in this derivation we first assumed that the function is analytic at z 0 and then we found that these two should hold at z 0. That means that these are Cauchy-Riemann conditions are necessary for analyticity of the function at that point. Immediately the second question that will arise that are they sufficient also that is do the Cauchy-Riemann conditions imply analyticity? Answer turns out to be yes and to establish that we consider two functions u and v which have first order continuous partial differential coefficients and these conditions among those partial derivative hold that is Cauchy-Riemann conditions hold. Then we want to show that the function is analytic for that we construct delta u and delta v. So, what is delta u? Delta u will be u at the change point minus u at the current point and up to first order we will get this as delta x into del u by del x at the point x 1 y 1 plus delta y into del u by del y at the point x 1 y 1. What is x 1 y 1 here? x 1 y 1 is a point in the line segment joining the original point to the change point. This is by mean value theorem. So, that means that by mean value theorem and that is why we could use this equality without any plus dot dot dot because we have we are not including the second order terms. So, that is why we have to use the mean value theorem and the remainder form of the Taylor series that is remainder form of the Taylor's theorem. So, we are keeping only the first order change. So, we use the mean value theorem up to the first order now which is Lagrange's theorem for that matter. So, this x 1 y 1 is a point which is in the line segment joining x y to x plus delta x y plus delta y. That means for a psi in 0 to 1 x 1 is this and y 1 is this. Now, this gives us the expression for delta u. Similarly, we get the expression for delta v that is v at the change point minus v at the original point. So, that gives us this expression for x 2 y 2 being a point joining a point on the line segment joining this point to that point. Now, x 1 y 1 and x 2 y 2 can be two different points that is x y is here x plus delta x y plus delta y is here. As we join these two we get this line segment and x 1 y 1 could be somewhere on that this line segment x 2 y 2 could be somewhere on this line segment need not be at the same point may be at different point. So, now this delta u and this delta v we have got in hand. So, what will be the corresponding delta f that will be delta u plus i into delta v that is this. We take this and add to that i times this as we do that we get we club together appropriate terms that we club together appropriate terms this plus i into this you will find here plus this into i into this you will find here. Now, an i has been kept outside and that is why in this case we sorry in this case in this case this plus 1 we replace with minus i square. So, that is why this term has come here i into this i is outside this is sitting here and this is minus i square into this minus i square is 1. So, out of minus i square 1 i is outside the rest of it minus i and this whole thing is here. Now, we try to simplify this for simplification consider this if Cauchy-Riemann conditions hold which is part of the hypothesis here if Cauchy-Riemann conditions hold then del v by del y can be replaced with del u by del x here already del u by del x is there we will find del u by del x here also. So, then this first bracket term you will find here. So, the del v by del y there has been represented with del u by del x. Similarly, here in this case we find del u by del y here which can be replaced with minus del v by del x. So, as we replace this del u by del y here with minus del v by del x this minus will become plus and we will have del v by del x here that is this whole thing that is this bracket term. Now, we concentrate on this this is del u by del x this is also del u by del x. Now, we note that this is at x 1 y 1 and this is at x 2 y 2 if this is also if this where also at x 1 y 1 then this whole thing we could have taken common and what would come inside delta x plus i delta y that is delta z. So, what we can do is that for the time being here in place of x 2 y 2 we take x 1 y 1 and that will mean that this term will remain and from there remain outside and from there we will subtract this same thing with x 1 y 1 here. That means i delta y del u by del x at x 1 y 1 we add and subtract add to this and subtract from here. As we do this we get the next expression in the process of simplification. Now from here if we add i delta y del u by del x x 1 y 1 to this term then this will come common and along with that we will get delta x plus i delta y and that you get here. Now then whatever we added here i delta y del u by del x at x 1 y 1 that same thing we subtract from here and for that we will get common i delta y and inside the bracket we will get del u by del x at x 2 y 2 minus del u by del x at x 1 y 1 that is the term here. A similar exercise we do on this here you see i del v by del x x 1 y 1 and then here there is a little mistake let me make a small correction delta y delta y is missing here delta y is missing here. Now here we have i delta y del v by del x x 1 y 1 del i del v by i delta y del v by del x and outside there is another i. So, i i delta y del v by del x at x 1 1 y 1 and along with that we would like to have del x delta x into del v by del x at x 1 y 1. So, that we add to this and we will subtract from here as we add to this we get the i common outside as it as it is already there del v by del x at x 1 y 1 we take common and then we get in bracket delta x plus i delta y that is here and whatever we added to this that we subtract from here. So, we added delta x del v by del x at x 1 y 1. So, that same thing we subtract from here and then the result is here. So, we find that delta f expression has come to this stage and now we want to divide with delta z note that this is delta z this is delta z. So, as we divide by delta z we get delta f by delta z as this plus i into this that is these two things plus a lot of things from here. So, here what we will get i delta x by delta z into this whole thing plus i delta y by delta z into plus into this whole thing. Now, we want to take the limit of this as delta z tends to 0 and if that limit exists then we will say that the function is analytic. Now, we ask this question whether that limit exist these derivatives are all existing that we already know. Now, note this that as delta z tends to z that means z tends to z plus delta z tends to z and that means that the two points between which we consider the line segment in the in which we found x 1 y 1 and x 2 y 2 as two points and that means that as delta z tends to 0 this line segment shrinks and we do not get too much space to get two points x 1 y 1 x 2 y 2 that means all these points shrinks to x y x 1 x y itself that is z. So, that means that as z tends to as delta z tends to 0 these points have to get shrunk over a length of 0 that means these points have to collapse together and that means these will have limit 0, but what about these two phases. Now, since delta z is delta x plus i delta y that means if this is z and this complex number is delta z then this length is delta z this length is delta x this length is delta y. So, delta x by delta z is cosine of this angle and delta y by delta z in the size sense in the absolute sense that is the sign of this angle. So, both cosine and sign are less than 1 in magnitude. So, these two are less than 1 in magnitude. So, as delta z tends to 0 these bracketed terms will tend to 0 and these terms are bounded that is they do not turn to infinity if at the same time these fellows could turn to infinity then this indeterminate form would remain. So, as delta z tends to 0 these are anyway bounded bounded by 1 the magnitude of these two and these turn to this approach 0 and therefore, in the limit these terms will vanish and these will remain. So, therefore, the limit exists and that limit happens to be this at x y itself. So, then whether you write it like this or you write it like this using the Kuschelmann conditions it is the same and the limit exists. So, we find that Kuschelmann conditions are not only necessary, but also sufficient for analyticity. That means the moment some function is known to be analytic you can immediately use Kuschelmann conditions. On the other hand the moment you can establish Kuschelmann conditions for a function you can immediately conclude that the function is analytic and all properties of analyticity you can assume immediately. Now if we take the Kuschelmann conditions and if the function possesses second derivatives also it will possess because if the first order this Kuschelmann condition is satisfied then analyticity is established and we have already discussed that analytic function is again it derivative is also differentiable that means derivatives of all order will exist. So, if we differentiate this then you find del 2 u by del x square is equal to del 2 v by del x del y. Similarly, del 2 u by del y square is minus del 2 v by del y del x then from here you can also find del 2 u by del x del y or del y del x as equal to del 2 v by del y square and similarly from here you get this. So, the second order derivatives will satisfy these requirements and then you can note that if we add these two then we get del 2 u by del x square plus del 2 u by del y square and that is this plus this is d u. Similarly, if you add these two if you add this and this if you add these two then you will find or subtract rather in this case you have to subtract from here you have to subtract this del 2 v by del y square plus del 2 v by del x square then as you subtract this minus this you will get 0 that means u and v the real and imaginary components of f in that case both will satisfy the Laplace equation that is both will be harmonic functions. So, this is a great property of analytic functions that both the real and imaginary components of analytic functions satisfy the Laplace equation that is their harmonic functions and in that case the two harmonic functions are also called the conjugate harmonic of each other that is conjugate harmonic of u is v. Now, we already know that families of curves u equal to c and another family of curves v equal to k are two mutually orthogonal families of curves in the x y plane except possibly at points where the derivative turns out to be 0. This you can see because if you take the function u of x y equal to c and then from there you try to find out the slope of a curve from this family then you will consider delta u as del u by del x delta x plus del u by del y delta y and the slope of this will be given as minus this derivative by this derivative. Similarly, if you take the family if you take the take a curve from the other family v equal to k then this will give you its slope as m k which will be. Now, if you multiply these two you will note that del u by del x will exactly cancel with del v by del y and del v by del x will cancel with minus del u by del y leaving minus 1 in the product. In all those cases where the four derivatives only two of which are unequal because Cauchy-Mann conditions are satisfied in the case that both of them are non-zero this is obvious even if say one of them is 0 that is suppose del u by del x is 0 in that case this will be infinite. That means the curves of the u equal to c family will be vertical that is the tangents will be vertical, but in that case if del u by del x is 0 then del v by del y is also 0. That means the slope here is 0 that means the curves of the other family are horizontal at that point curve of the other family at that point is horizontal. So, this vertical and this horizontal is again orthogonal are again orthogonal with respect to each other. Similarly, if del u by del y is 0 then the curve of the first family is horizontal and correspondingly del v by del x is 0 in that case the curve of the other family through that point is vertical again orthogonal. The only problem will arise if both the derivatives are 0 that is del u by del x as well as del u by del y is 0 in that case this slope is undefined and this slope is also undefined. So, it may happen that you may not be able to figure out that the product is minus 1 or what and therefore, we say that that kind of a situation can arise only at a point where del u by del x and del v by del x both are 0 in that case del f by del z is actually 0 and that is what we say here that families of curves u equal to c and v equal to k are mutually orthogonal at all points except possibly at those points where this derivative turns out to be 0. Before proceeding further please note this correction the slope of the curve u of x y equal to c was written on the blue board as m c is equal to minus del u by del y by del del u by del x this is not right it should be minus del u by del x by del u by del y. Similarly, the slope of the curve v of x y equal to k was written in the board and as m k is equal to minus del v by del y by del v by del x this will be corrected to minus del v by del x by del v by del y thank you. Now, you can continue further in the rest of the lecture now a good question a very important question is that if u of x y is given then how to develop the complete analytic function this is actually and exercise the basic work regarding which we did much earlier when we were solving the first order differential equations when we were studying first order differential equations in that context we have actually studied this particular problem and what we do for that is that we construct del u by del x and del u by del y from the given u and using Cauchy Riemann conditions we get del v by del y and del v by del x and using del v by del x del v by del y we construct v of x y. So, that way after constructing v of x y we get the complete analytic function that means if one of the components real or imaginary of the complex analytic function is given then the other one can be derived using Cauchy Riemann conditions and integration. Now, another important concept in the case of analytic functions is conformal mapping conformal the word means of similar shape conformal conformal mapping means shape similar mapping. So, a conformal mapping is defined by an analytic functions except at those points where the derivative is 0 f prime z is 0. So, now function will give you the mapping of elements in domain to their images in the domain is the z plane and the corresponding core domain is the w plane. So, from points in the z plane as you map the points to the w plane you get the mapping. Now, here depiction of the complete in the case of real variables you plotted the independent variable x in the horizontal axis and dependent variable that is the function y along the vertical axis you cannot do this here because the depiction of a variable itself over its domain will require a full plane. So, here how you show the mapping you take two planes z plane and w plane. So, depiction of a complex variable will require a plane. So, depiction of a mapping will require two planes together. So, in this manner say this is a z plane in which we take the domain and this is the w plane and between z plane and w plane we consider this function w equal to e to the power z. Now, every point here will give you a corresponding point here. Let us consider four points here a b c d a rectangle. So, the point a from here which is origin that will give you e to the power 0. So, there you will get 1. So, 1 plus i 0 the point b that will give you that is here 1 that means 1 plus i 0. So, e to the power 1 will give you e this is 2.718 and so on that is b prime c is 1 plus i into pi by 2 say 1.57 pi by 2 c is 1 plus i pi by 2. So, as you write e to the power 1 plus i pi by 2 e to the power 1 is i sorry e to the power 1 is e into e to the power i theta is cos theta plus i sin theta. So, cos pi by 2 is 0 and i sin pi by 2 is 1. So, you get e into i. So, that is why you get e magnitude i that is in the vertical direction c prime comes here. Similarly, d prime is simply i pi by 2. So, that will be e to the power 0 into i pi by 2 sin i sin pi by 2 that will bring you here. If you try to draw the diagonal you will find that diagonal a c will come like this. Now, this line segment a b comes like this line segment b c will come like this c d will come like this and d a will come like this. The shape of this rectangle has changed, but you will note one important issue a b and b c were orthogonal mutually perpendicular at b. Here also a prime b prime and b prime c prime the curves are perpendicular to each other here. Similarly, b c c d are perpendicular here also b prime c prime and c prime d prime meeting at e are perpendicular. So, all the edges have gone to the w plane in such a manner that these between the tangents you are all getting you are getting all the right angles. This diagonal a c has been mapped to this curve a prime c prime, but note emerging from a whatever angles you are getting here a b a c a d. Similar same angles you get here a prime b prime a prime c prime a prime d prime that is along the tangents. You will get the same sector here and that will happen everywhere that is because this happens to be a conformal mapping that is it is same shape mapping similar shape mapping that and that similarity of shape is in terms of the local shape only make that point very clear. We can very easily establish this fact the demonstration of which we just saw through these figures. The conformal mapping is a mapping that preserves the angle between any two directions in magnitude as well as sense. Now, we verified this fact for this particular mapping w equal to e to the power z. So, this analytic function defines a conformal mapping. So, we find that through relative orientations of curves at a point at points of intersection the local shape of the figure is preserved at every point whatever rays we draw here and the corresponding rays we map to the target plane the core domain the range. We find that the relative angles here and the relative angles there are preserved. So, why should this happen? We take the curve we take a curve apart from rays we were taking rays earlier. Now, we take a curve the z of t in the z plane passing through this point z 0 at t equal to 0 corresponding image is w of t which is f of z of t because w is f of z and passing through w 0 which is f of z 0 at t equal to 0. Now, if the function f is analytic then we can have its derivative then w dot from here through chain rule will be f prime z evaluated at that point into z dot that is this. So, w dot evaluated at t equal to 0 will be f prime at z 0 into z dot evaluated at the corresponding t equal to 0 and this will imply that this side and this side these two are equal in magnitude as well as direction. So, magnitude equality is here and direction equality will be here that is argument of this is equal to argument of this the argument of this. Now, as we draw several points through the same point z 0 then their directions will be different here five curves through z 0 will have five different angles here, but for all of them this is same because this does not depend on the those curves these are property of the function itself f itself and therefore, whatever are the differences of angles among the curves here as we map them as we map those curves to the w plane the differences here will be same that is every curve from this plane to that plane turns through this angle and this angle is same for all the curves because all of them are passing through z 0. So, for several curves through z 0 image curves pass through w 0 and all of them turn by the same angle and that turning is this. So, through z 0 if in the z plane we draw four curves call them 1 2 3 4. So, if curve 1 turns through the mapping through an angle 30 degree that means this argument is 30 degree. So, curves 2 3 4 also have to turn through the same 30 degree which is this and this depend only on the function and not on the curves that we are drawing through z 0 and this shows that the local shape gets preserved if one of them turns by 30 degree then all of them turn by 30 degree through the mapping and the magnitude changes like this. So, magnitude changes direction also changes, but all the directions these curves from here say these are four curves drawn from a particular point in z plane. Now, as they as these raise go to the w plane their lengths may all change by this factor and they all may turn by this angle. So, as all of them turn they look like now this. So, that means all of them turn together. So, their shape does not change, but the important points to note in this regard is that this will happen only at those points where this magnitude is non 0 because if this magnitude is 0 then this will collapse. So, this analyticity is must apart from that for conformality of the mapping the value of the derivative should be non 0. Now, if f prime is 0 at that point then the argument is undefined and conformality will be lost or may be lost. Now, one point to another point to notice that the derivative varies from point to point and therefore, we say that the shape does not change locally. So, local shape is preserved. So, as around this point all of them turn by 30 degree around another point where f prime may be something else the raise may be turning by 35 degree another point the raise may be turning by 45 degree and so on. And therefore, the scaling and turning effects at different points are not the same at different points of the z plane are not the same. And therefore, though the local shape at every point is preserved through the conformal mapping the global shape is not preserved the global shape may change in general that does change. And that is what we saw here even though locally the collection of every raise through a collection of all raise through every point preserved their mutual angles, but the overall shape of the region defined by a b c d is not preserved here it was a rectangle here it turns out to be a part of an sector of an analysis. So, global shape may change because f prime z 0 at different points z 0 will be different in general. So, from the foregoing discussion we can conclude that an analytic function defines a conformal mapping at all points except at its critical point where its derivative is 0. Now, except at critical points we find that analytic function is invertible also. So, that means that for any conformal mapping we can establish an inverse and this fact is of enormous practical importance. So, we are coming to that practical point later first let us see a few examples if you quick examples of conformal mapping linear functions like this will define conformal mapping for all non 0 a linear fractional transformation like this will define conformal mapping except for the case when a d minus b c is 0 why so because if you try to differentiate this you will find that in the case a d minus b c you will have 0 derivative. Now, other elementary functions like z to the power n e to the power z etcetera though they have completely different meanings in the case of complex functions as we put e to the power x plus i y we will find that turns out to be e to the power x into cos y plus i sin y. So, that turns out to be a complex function in which the real part is e to the power x cos y and imaginary part is e to the power x sin y. So, it is quite different from the real function e to the power x which is all through exponential. So, even then these elementary functions with similar expressions similar meanings that we define in the case of real calculus. Now, as we put those same formulas here we get quite I mean similar formulas will yield different meanings here yet all of these will define in the case of conformal mappings except for those situations where the derivative vanishes. Now, these are analytic functions and you can show that in whichever case the expression of f of z you can put in terms of z only after collapsing after coalescing all the x y terms such that x and y do not appear alone separately such functions you can always show that they define the satisfy Cauchy-Riemann conditions as long as the derivative expression does not become undefined. So, these will establish conformal mappings and special significance practical significance of conformal mappings is that a harmonic function phi of u v in the w plane that is in the w plane a function which satisfies the Laplace equation is also a harmonic function in the form phi of x y in the z plane as long as the two planes z plane and the w plane are related through a functional relationship which itself defines a conformal mapping. So, this fact gives us an advantage in solving a lot of important problems underlying the solution in such cases is the famous Riemann mapping theorem and that tells us that if d is a simply connected domain in z plane in z plane you take a domain simply connected domain which is bounded by a closed curve. Now, whatever closed curve it is whatever is its shape as long as the region that it encloses is a simply connected domain then there will exist a conformal mapping that will give you a 1 to 1 correspondence between this curve phi and a unit circle. That means also the in between the interior of this curve interior of this region with the unit disc that is interior of the unit circle. So, such a conformal mapping will give us this 1 to 1 correspondence there will be a conformal mapping which will give us a 1 to 1 correspondence between this domain d and the unit disc which is there as well as between the boundaries. Now, this important fact gives us a very handy tool to solve boundary value problem. For example, suppose we have got a boundary value problem in which the domain is of a very complicated shape, but as long as it is simply connected what we can do we first establish a conformal mapping between the given domain and the domain of simple geometry. For example, the unit disc next solve the boundary value problem in this simple domain and then in the case of the conformality the mapping will also have an inverse. So, after the solution is available in the simple domain we use the inverse of the conformal mapping and thereby we construct the solution for the original domain. Now, one particular advantage one particular application of this is through the Poisson's integral formula which is this. Now, let us first see what this integral formula tells us R e to the power i theta is a point z in the z plane expressed in the polar coordinate. You see x plus i y in polar coordinate will mean R cos theta plus i R sin theta. So, if you take R common then you get cos theta plus i sin theta which is this e to the power i theta. So, this R e to the power i theta is nothing but x plus i y in polar coordinate and the formula tells you that this value of the function at z can be found through this integral 1 by 2 pi into integral from 0 to pi over the full circle this integral if we evaluate then we get the value of f of z. Now, what is this integral and what does this involves it involves capital R that is radius value smaller the radial coordinate of z and it involves theta that is the polar this theta coordinate of z and it involves phi. Now, phi this circle at a particular point or at this circle is r phi and a point in the interior is r theta. So, this Poisson's integral formula tells us that a the function can be evaluated at an interior point here through the cyclic integral 0 to 2 pi of this integral over the full circle. This circle over the circle and for that the function value is required only at the circle right. So, that means if we know the boundary values all the boundary values then by using those boundary values here for different phi running from 0 to 2 pi for constant r we can evaluate this integrand at every point for any interior point r. So, interior point r where we want the function value gives us the values of r and theta smaller and smaller and theta. So, smaller and theta and every point here has radial value r capital R and the value of phi changes from 0 to 2 pi. That means if we know all the boundary values then we need these function values. So, by using the boundary values through this integral we can find the value of the function at any point in the interior. Practically that means that we can solve the Dirichlet problem for the function f that is boundary point value we know and in the interior we want to find out the function. So, this formula itself we will be able to establish after we study a little integral calculus of complex functions also. Now, apart from that what else is the application of conformal mappings we have already seen that the relationship between one family of curves u of x y equal to c and another family v of x y equal to k is established through Kuschelmann condition. So, in the case of the analysis of two dimensional potential flows if we have the velocity potential phi of x y that gives us the velocity components in this manner and we know that a streamline is a curve in the flow field the tangent to which at any point is along the local velocity vector. So, stream function is a function which remains constant along a streamline. So, psi of x y that is a stream function turns out to be a conjugate harmonic function of the velocity potential function and this complex potential function consisting of phi and psi together defines the flow completely. So, in the fluid flow problems if we encounter a solid boundary of a complicated shape conformal mapping allows us to transform the boundary conformally to a simple boundary a boundary of a simple shape and this transformation helps us facilitates us facilitates the study of the flow pattern through analysis of the simple boundary. This is what we do in the case of complicated stream line shapes and also in the case of the airfoil studies. So, these are the points which we studied in this particular lesson Kuschelmann conditions conformality and applications of the complex analytic functions in the case of boundary real problems and flow description. In the next lecture we will take up the question of integral in the complex plane integral of complex functions. Thank you.