 Okay, so welcome again to lectures on advanced complex analysis, so what we are going to do today is ask a very basic question okay, so let me switch to the writing board. So suppose f from D to C is an analytic function, of course analytic means the same as holomorphic okay, so analytic is the same as holomorphic and always as usual we assume D is a domain in the complex plane okay, so D domain in the complex plane, so that means that D is subset of the complex plane, D is open, D is connected and of course you know D is certainly non-empty because by definition you know the empty set is also open okay, so of course we are not interested in looking at the empty set, so D is connected and of course you know in this context that for an open set connectedness is equivalent to path connectedness, so D is also path connected okay, so you have a function f, f is analytic, f is a function which is a complex valued function, it is a complex valued function of one complex variable and that one complex variable you might call it as z, so you can think of the function as f of z and z varies over D okay and what is the fundamental question that we are asking, the fundamental question that we are asking is what is the image of f okay, so that is the question, so here is the, so let me change colour to something else, so here is the question, what is the image of f, so this is the question, so in other words so that is, so you look at f of D okay, what is f of D, it is a set of values of f okay, so this is equal to the set of all f of z where z varies in D okay, this is the set of values of f, this is all the values that f takes on D okay and obviously it takes complex values, so f of D is a subset of the complex variable and the question is what kind of a subset is this okay, so yeah, so what is the image of f, that is if f of D is equal to this then what is the nature of f of D, so when I say what is the nature of f of D, you know you ask what do you mean by nature, nature of course one can ask a couple of things, one is of course topological nature okay, so f of D is a subset of the complex plane and you know complex plane is a topological space, so you can ask whether f of D is open, whether f of D has any one of these properties that subsets of a topological space satisfy okay, the properties that you know are open sets, sets being open, sets being closed, sets being connected, sets being path connected, sets being compact and so on okay, so you can ask what is the nature of this set f of D in the topological sense okay, then you can ask another question is how big is f of D okay, what is this, how much of it, how big it is when compared to the whole complex plane okay, so when I say nature I can ask topological and the other thing is how big, how big is it okay, so and it so happens that complex analysis gives you several nice theorems which answer go along way in answering these questions okay, so let me you know let me go ahead and you know look at just a minute let me resize the screen so that I get, so let me look at the following, let us ask some simple questions, so let us ask some simple questions, so there are first a few obvious things that you can say, see f is an analytic function, so you know analytic functions are in fact infinitely differentiable that is what you learn in the first course in complex analysis, once differentiability implies infinite differentiability on an open set and that is one of the characteristic properties of an analytic function and therefore in particular they are certainly continuous and you know continuity preserves certain properties for example the continuous image of a connected set is connected, continuous image of a path connected set is path connected, the continuous image of compact set is compact, so you can say immediately that f of D is certainly a connected set okay, so that is very very basic topology, it just uses the fact that the continuous image of a connected set is connected okay, so let me say the following thing of course f is continuous, so you know I will use some abbreviations CTS stands for continuous, so f is continuous, so f of D is connected, this is of course very very basic but then we want to know more, so you know let us try to see, let us ask a few questions, so for example we ask a question like can f take values on a line or can f take all values only on a curve okay, so let us ask this question, so let me again change curve, can f take values only on a curve, of course by a curve I mean any simple curve that you can think of like a parabola or a circle or something like that and in particular also it could very well be a straight line which is also a curve okay, so can f take values on a curve, so say for example you know suppose I take, so example, so let me look at a few examples, can f take values on the real line, I mean take only values on the real line, this means you are saying that the image of f is a subset of the real line okay and you know if see if f takes values only on the real line this is equivalent to saying that the imaginary part of f is 0 okay, because f is a complex value function normally we write f is equal to u plus i v where u is the real part of f and v is the imaginary part of f and you know very well from a first course in complex analysis that you and we have to you know be harmonic and in fact they will satisfy the Cauchy-Riemann equations okay because f is analytic but the point is that if you say that f takes only values on the real line it means you are saying that v is always 0 that means the imaginary part of f is 0, so this is equivalent to saying that you know imaginary part of f is 0 okay and you know I can also ask can f take only values on the imaginary axis okay that is y axis considered as the imaginary axis on the complex plane okay and the that is equivalent to saying that the real part of f is 0 okay, so this is another case of f taking values only on the line then of course I can ask can f take values on a circle okay, so let me ask that also can f take only values on a circle so you know if you think that the circle is centered at the point w0 and has radius r0 this is equivalent to saying that modulus of f of modulus of f minus w0 is equal to r0 this is what it means this is this is the condition that f takes values on a circle okay. Now surprisingly not surprisingly in fact this is something that you should have seen if you just recall that these are all the conditions that will ensure that the derivative of f vanishes and therefore f has to be a constant okay, so if the imaginary part of f is 0 that is the same as saying that the imaginary part is a constant okay it is a special case of the fact that the imaginary part is a constant and if the constant is 0 this is equivalent to saying that amounts to saying that the imaginary part of f is 0 which means it takes only real values and if the real part of f is 0 that is a special case of the real part being constant okay and the f taking values on a circle is the same as saying that the function f minus w0 which is f added to minus w0 which is a constant function okay that has constant modulus that modulus is r0 okay. Now you would have done this in a first course in complex analysis probably by using the Cauchy Riemann equations that you know if the function has imaginary part constant or the real part constant or modulus constant then the function has to be constant okay, so all these things can happen only if f is constant okay, so all these can happen only when f is a constant, f is a constant function okay takes the same value, so of course therefore the question is certainly we are not interested in studying constant functions because there is nothing special about these constant functions the constant function maps the whole plane onto a single point which is the value of the function at constant we are not interested in, so we are not interested in such constant functions we are worried about non-constant functions okay, so what this will tell you immediately is that if you have a non-constant analytic function okay it cannot take values at least on a line or something like a circle okay but then so what is it, so what this tells you is that the either you have the case that you are looking at a constant function in which case the image is single point okay it is that single constant value that it takes or the image cannot be a subset of just the real line or it cannot be a subset of only the imaginary axis it cannot be a subset of the circle that such things cannot happen that is what this says okay but what is it that you have more generally, so more generally we have you know we have very nice theorem, so here is a theorem and it is called the open mapping theorem and it is a very very fundamental theorem what it says is that a non-constant analytic map is always an open map okay, so if f from D to C is a non-constant analytic then f is an open map it is an open map, so let us try to understand what this means it means that see what is an open map an open map is a map which for which the image of any open set is again an open set okay, so in particular what this will tell you is that f of D is an open set because D is already a domain the D is a domain, so D is already an open connected set, so f of D will become open okay and it is already connected, so it is the same as it being path connected, so f of D is again a domain okay, so what this so what is it, so let us try to understand what this means f of D is open, so it is a domain so that is something that comes immediately okay and mind you f is not a constant function so it takes more than one value, so f of D is non-empty of course okay and then the other important thing is the following thing what is this condition of open mapping, if you take U as subset of D which is an open subset then f of U will also continue to be open okay, so if U is subset of D is open then f of U is open, this is what an open map means it maps open sets to open sets, the image of an open set under an open map is again an open set that is the definition of what an open map is okay and so let us go a little bit more into this and you know try to see what it really means, what is the meaning of an open set, an open set is a set for which every point is an interior point okay that means you take any point in the open set then there is a small disc open disc surrounding that point which is also in that set that is what an interior point means okay, so what does f being open mean, suppose f takes a certain value W, let us say it takes a value W not okay then there is a that means you are saying W not belongs to the image of f okay, if f takes the value W not okay that means W not is in the image of f because image of f just consists of the values of f okay and then but since the image of f is open W not is a point of an open set therefore there is a small open disc centred at W not which is also in the image so it means that if f takes a certain value it will take all values in a small disc surrounding that value okay, so this should immediately tell you that f cannot simply take values on a curve because the moment f takes values at a point it will take all values in a small disc surrounding that point and you know no curve can accommodate a small disc however small okay therefore you immediately get this idea that you know the image of an analytic mapping non constant analytic mapping cannot go into a curve, we saw special cases we saw that it cannot go on into the real axis it cannot go into the imaginary axis it cannot be a circle okay cannot go into the circle but it is more general the reason is that the image is open okay and of course curves are closed sets okay, so let me write that down, if W not is equal to f of Z not for Z not in D that is the same as saying that W not is in image of f which is f of D then f of D being open implies that there exists small open disc in f of D containing W not and that implies that f takes all values in a small disc centered at W not, so this is what is very very important if f takes a certain value it will take all values in a disc about that value okay, this is a very very important property and this is true for of course for a non constant analytic function okay fine, so this is about at the moment this is about the topological property of f of D okay this theorem tells you that f of D the image of f is certainly a domain it is an open connected set it is very very important that it is an open set and in fact going into a higher geometric point of view okay what actually happens is this, so let me tell it to you in words what actually happens is that the mapping f from D to f of D becomes what is called here ramified cover of Riemann surfaces okay, so it means that it there are set of points these are the points where the derivative of f vanishes okay these are called the points of ramification and outside those points the complement of those points this is actually a covering map okay it is a covering map in the topological sense and also in the analytic or holomorphic sense okay, so this open mapping theorem is so important that it tells you that essentially every analytic non constant analytic mapping is a ramified cover okay fine, so now what I am going to do is I am going to go to ask a more specific question, so we are trying to look at the image of a domain under analytic function, so let us look at the cases where first at the case where you know the function is analytic on the whole plane, so these are the entire functions, so what is an entire function? An entire function is a complex value function which is analytic on the whole plane okay and then the question is what is the image of such a function, so there is a very deep theorem namely it is the so called little Picard theorem which says that the image is either the whole complex plane or it is a punctured plane namely it is a complex plane minus a simple point that means an entire function okay will take all values except for omitting at most one value okay and this is called the little Picard theorem okay, so let me state that, so here is a little Picard theorem, sometimes people also use the adjective small Picard theorem, so what is this, if f from c to c is analytic that is so let me write it here, f is entire then either f of c is equal to c okay or f of c is equal to c minus w not for some w not in c, so this is a little Picard theorem, so you know it is a very tremendous theorem it says that you take an entire function, you take the image, the image is huge I mean the image is literally everything, at the worst if the image omits it can omit only one value okay and the case where the image omits a single value is of course the simplest example is that of the exponential function, you know if you take the function z going to e power z then that is an entire function okay and the image will not, it will be the whole punctured it will be the punctured plane, it will be the complex plane minus the origin because exponential function will never take the value 0 because 0 does not have a logarithm okay, so if you take any non-zero complex number you can find a logarithm and the exponential of that logarithm will give you back that complex number of course you will get many logarithms okay but you can find at least one for a non-zero complex number, so it means that the exponential function will take all values except 0 okay and that is the so in that case it is an example that illustrates Picard's theorem if you take f of z equal to e power z then the image of f is actually c minus 0 which is a punctured plane normally if you take the whole complex plane and remove a single point that is called a punctured plane okay with puncture at that point because that point is being removed okay and of course there is also the case when a function an entire function can take all values the simplest case is that of a polynomial, so if you take a polynomial if you take f of z equal to p of z where p of z is a polynomial then it will take all values because I can always solve for p of z equal to w not for any w not and that is because of the fundamental theorem of algebra namely that the complex numbers are algebraically closed, so I can always solve a polynomial equation in one variable okay. So a polynomial is also an entire function and it is it gives the case the first case namely the image of the whole complex plane is the whole complex plane okay fine so this is the little Picard's theorem. Now somehow what I want to do is I want to really try to prove this okay it is a deep theorem normally this theorem is only stated in a first course in complex analysis but since this is an advanced course in complex analysis I think it is fitting to look at a proof of this. Now well you know interestingly it is very interesting that the proof of this that I am going to present is actually gotten by deriving this as a corollary to a much more deeper theorem which is called the Big Picard's theorem and the funny thing is that the Big Picard's theorem is a theorem which deals which again has the same kind of question it answers the same it answers the same kind of question namely what is the image of a domain under analytic map okay but the point is that if the domain you are looking at is a disk around an essential singularity of an analytic function okay. So you know so let me state that so here is so let me use something else this will be deduced from the from the Big or Great Picard's theorem and that is let so here is a statement of the theorem let z not be an isolated essential singularity an analytic function f okay then f of so I am so let me write this 0 less than mod z minus z not less than epsilon is equal to c or c minus a single value for every epsilon greater than 0 in of analyticity okay. So I have just stated a part of the theorem there is still more to it so I want you to look at this it is see what I want you to appreciate is I want you to appreciate the following thing to deduce the little Picard's theorem which is a theorem about a function is analytic on the whole plane if a function is mind you if a function is analytic on the whole plane it has no singularities okay it has no singular points okay so the little Picard's theorem is a theorem about a function which has no singular points okay and it says that the image of the whole plane under such a function is either the whole plane or a punctured plane okay but we are deducing it from a theorem about the image of a function with a singularity so that is the funny thing okay so it is like you know even to answer question about an entire function you are forced to study singularities this is the point I want you to understand okay see normally we would not like to dirty our hands with singularities why study singularities when there are functions without singularities but the point is you know sometimes mathematics and theory teaches us that even to study good things you have to study bad things okay so if you want to prove the little Picard's theorem is a theorem about good things I mean a function is analytic entire you have to still study functions which are having singularities okay and so here is a great the big Picard's theorem and obviously you know the adjective great or big should tell you that this big Picard's theorem has to be a big brother of the little Picard's theorem and therefore you know the little Picard's theorem can be deduced from the support of the big brother okay and what is this big Picard's theorem what does it say see you are looking at an analytic function okay and you are looking at a singularity of an analytic function okay now so I will come later to what a singularity is okay because that is motivation for me to recall these things okay so you look at a function f which at a point has isolated singularity okay isolated means there is a whole disc surrounding that point where there are no other singularities okay and a deleted disc surrounding that point is given in this form as I have written here in the on my board 0 strictly less than mod Z minus Z naught strictly less than epsilon is actually the disc centered at Z naught radius of epsilon it is an open disc but I have thrown out Z naught that is the reason for putting 0 strictly less than I am not allowing Z equal to Z naught that means I this is a punctured disc okay it is a punctured disc centered at Z naught and the puncture is exactly at Z naught I have thrown out Z naught okay and on this disc the I am assuming that this disc is full of points where function is analytic okay and that will be true at least for small values of epsilon greater than 0 because the point Z naught is an isolated singularity okay and look at what the theorem says it says you take the image of this when I write f of something okay it means f of this set which is the punctured disc that is the whole complex plane or it is a complex plane minus a single point and this is true for epsilon sufficiently small and therefore it will be true for even larger epsilon so long as this deleted disc is in the domain of analysis T of f because larger discs larger deleted discs will contain smaller deleted discs and therefore their images will contain images of smaller deleted discs okay so you see this is again the result of the conclusions of the theorem both the big Picard theorem and the little Picard theorem they are the same I mean the conclusion always says that the image under analytic function of a certain domain okay is either the whole complex plane or it is a complex plane minus a point okay and in the case of the little Picard theorem you are looking at the domain is the whole complex plane but in the case of the great Picard theorem the domain is a very small neighbourhood deleted neighbourhood of an essential singularity of an analytic function and what is really amazing is in fact there is more to this Picard theorem what it says is you see so I what I wanted to observe is the following thing it is a very very deep result it says take a very small neighbourhood of the essential singularity okay deleted neighbourhood that means of course you do not take the neighbourhood that you take should be a domain where the function is analytic so it cannot include the singularity so when I say take a neighbourhood of an essential singularity of course I mean delete that essential singularity so you are taking a deleted neighbourhood of the essential singularity and mind you take a neighbourhood as small as I want you see this epsilon can be extremely small okay and the theorem is amazing it says you take no matter how small a neighbourhood you take the image of that neighbourhood is still the whole plane no matter how small your neighbourhood is the image of that very small neighbourhood no matter how small is still the whole plane it still takes all those values so what this tells you is you know it tells you that it tells you how the values of the analytic function change in a neighbourhood of an essential singularity in a neighbourhood of an essential singularity this analytic function is taking all values at the worst it can omit one value okay and of course you know the example for this is just as in the case of the little Picard theorem where you for the example of an entire function omitting a value is exponential e power z okay which omits the value 0 here you can take e power 1 by z okay you can take the function e power 1 by z and this e power 1 by z and e power 1 by z at z equal to 0 has an essential singularity and if you take any small deleted neighbourhood of 0 however small and you take the image and e power 1 by z you will get the whole plane except the origin because exponential function will never take the value 0 so you know it is an amazing result it is an amazing result and in fact there is a stronger version of the Picard theorem which says that not only does the image of any small neighbourhood however small of an essential singularity under an analytic function is a whole plane or plane minus point it says it takes the every complex value that it takes it takes infinitely many times so there is in fact so let me write that down just to tell you how powerful the theorem is so let me write that for every epsilon greater than 0 such that 0 less than mod z minus z naught less than epsilon is in the domain of analyticity of f f assumes each complex value with at most one exception w naught infinitely many times so in fact this is infinitely many times part of it which tells you the more I mean it tells you with lot of force what is happening so the first part of the great Picard theorem says that you take an essential singularity and take a very small deleted neighbourhood about that take a very small disc surrounding the essential singularity and take its image under the analytic function of course you do you do not take the analytic function is not defined at a singularity okay so you do not take the value at the singularity there is no such thing so you are actually taking the image of a deleted neighbourhood but the point is no matter how small the deleted neighbourhood is your image will be the whole complex plane or it may be complex plane minus a single point that is the first part of the theorem in fact what this part of the theorem says is that you know you take any value any of the values in the complex plane except possibly for that one value w naught which it will not take okay take that take any other of the values that it takes that value itself if you take the pre-image of that value in that neighbourhood the pre-image is an infinite set okay that means there are infinitely many points even in that small neighbourhood there are infinitely many points at which the function takes that prescribed that value that you that you are pointing at and this happens for every value that it takes so what it does it is very funny it looks as if you take a very small neighbourhood around the essential singularity the function not only maps that very small neighbourhood onto the whole plane or whole plane minus a point but it maps it infinitely many times okay it is like it maps it thousands and thousands of times okay and that is an amazing thing okay it is not that for every complex value there is one value here which goes to that the fact is you take any complex value other than the exceptional value w naught then there are infinitely many points in this very small disc however small where that value is taken by f okay so that small neighbourhood it is really amazing to think of it think of a very small infinitesimally small neighbourhood which is being again and again you know it mapped thousands of times I mean probably uncountably many number of times onto the whole plane or the whole plane minus a point that is how the function behaves in a neighbourhood of an essential singularity and this is the key to this theorem on singularities is the key to proving or deducing as a corollary the little Picard theorem okay. So we will try to in the forthcoming lectures we will try to give a proof of this theorem okay and so I will tell you roughly I will give you an idea of where we are going to go so you know first of all I want to recall something about singularities some you would have studied singularities but I would like to recall them and some basic theorems about singularities especially the Riemann's theorem on removable singularities and then I want to deduce from that what is called the weak version of the big Picard theorem which is called the Casorati-Weierstrass theorem and the Casorati-Weierstrass theorem is slightly weaker what it says is that while the big Picard theorem says that a function assumes an analytic function assumes all values except with possibly one exception in every neighbourhood of an essential singularity what the Casorati-Weierstrass theorem says is that it says it will come arbitrarily close to every value okay. So the Casorati-Weierstrass theorem is a slightly weaker version of the great Picard theorem and that can be more or less deduced using the Riemann's theorem on removable singularities which I will prove okay I will state and prove okay so I will have to recall something about singularities but then as we move towards the proof of the big Picard theorem what you will have to do is that we will have to study not one function but you will have to study a space of functions and we have to study functions with singularities and the kind of functions we are going to study are functions with singularities as poles and these are called Meromorphic functions so what I am going to do is I am going to study topology of a space of Meromorphic functions and prove some fundamental theorems like Montel's theorem okay and these are the keys to unlocking the proof of the big Picard theorem okay so what I am going to do in the next few lectures is first recall singularities then tell you something about Riemann's removable singularity theorem prove the Casorati-Weierstrass theorem and then go on to Meromorphic functions studying Meromorphic functions and then trying to study families of Meromorphic functions topologically whether that space is compact and things like that okay so that is the direction in which we will be proceeding so at least the first part of these lectures our aim is to prove the great Picard theorem and you will see on the way we will prove several other important theorems okay.