You see we have been looking at a point at infinity and what we have seen is situation
when infinity is an isolated singularity okay and I have already told you what happens when
infinity is a removable singularity or infinity is a pole and even more generally we have
also seen the so-called residue theorem for the extended complex plane okay, so the point
that one has to remember is that the residue theorem for the extended complex plane which
we discussed in the last lecture in that the contribution of the residue at infinity will
always be there okay irrespective of whether the point at infinity is a removable singularity
or a pole or an essential singularity.
So the issue is that you know even if infinity is a removable singularity okay for example
for the function f of w equal to 1 by w okay infinity is w equal to infinity is a removable
singularity okay because the function tends to 0 as w tends to infinity but still the
residue at infinity is not 0 which does not happen for a point in the usual plane, in
the complex plane at a point at which the function has a removable singularity is a
point where the function can be redefined so that it becomes analytic and therefore
if you calculate the residue at that point you will get 0 okay whereas this does not
happen for the point at infinity that is 1 point that you should always keep in mind.
Now we are going to talk about the situation when infinity is an essential singularity
okay, so so what does it mean to say that infinity is an essential singularity, so basically
you are looking at a function for which the point at infinity is a singular point and
isolated singular point that means it is defined in a neighbourhood of infinity in a deleted
neighbourhood of infinity and it is analytic in a deleted neighbourhood of infinity which
means that basically you are looking at an analytic function which is analytic outside
a circle of sufficiently large radius okay and because that is what neighbourhood of
infinity is okay you know that the stereographic projection.
Now what does it mean to say that the function has infinity as an essential singularity,
it is the same as sayingůthere are 2 ways of saying it of course in fact 3 ways of saying
it and all the 3 ways of saying it are correct, so there is one way which is using the Laurent
expansion at infinity, so so one way of saying that infinity is an essential singularity
is by saying that if you take Laurent expansion at infinity then the singular part has
infinitely many terms okay and there what you must remember is that the singular part
is actually a polynomial part I mean it is theůif you expand it in powers of the variable
okay if w is the variable then you are going to expand the function in positive and negative
powers of w okay and you should look at an expansion which is valid outside a circle
of a sufficiently large radius mind you that is the Laurent expansion in a neighbourhood
of infinity okay and because you are looking at a point at infinity the positive powers,
the terms involving positive powers of w that is a singular part okay that is a singular
part and the term involving the constant and the negative powers of w is the analytic part
because negative powers of w behave well at infinity okay.
So the condition for the function to have infinity as an essential singular point is
that if you write out its Laurent expansion in a which is valid outside at all points
outside a sufficiently circle of sufficiently large radius centred at the origin okay then
you must get you must get the positive powers of the variable that occur, they must be there
must be infinitely many terms which is the same as saying that the singular part at infinity
has infinitely many terms, so what should not happen is that the singular part has only
finitely many terms at infinity which means the singular part is actually a polynomial
okay. So in other words what we are saying is that you know how do you recognize that
action has infinity as an essential singular point, you write out the Laurent expansion
at infinity valid in a neighbourhood of infinity then what happens is that the singular part
is not just a polynomial okay.
In particular what this means is that if you take a polynomial functions they are certainly
not going to have infinity as an essential singularity and you know that you have seen
it last time a polynomial of degree polynomial is in fact have infinity as a pole and the
order of the pole is actually the degree of the polynomial okay, so this is one way of
defining infinity as an essential similar point of course the other way is that the
limit of the function as you approach infinity does not exist okay that is also correct okay.
So usually there are 3 characterisations one of any singular point there are 3 characterisations
one is you look at the Laurent series and look at the condition of the Laurent series
whether it has no terms at all in the singular part or whether it has finitely many terms
in the singular part or whether it has infinitely many terms in the singular part and the 2nd
condition is on the limit of the function as you approach that point okay. Of course
the 3rd condition is about the bounded mass of the function okay or the unboundedness
of the function but of course the boundedness is equivalent to the point being a removable
singularity okay and the unboundedness means that it is either it could either be a pole
or an essential singularity okay.
So so let me write these these points down infinity as an essential singularity of f
of W, so you see so f of w is assumed to be analytic in a deleted neighbourhood of infinity
okay that is a standard assumption blanket assumption and well so, so what are the conditions
well you write a Laurent expansion expansion of f at infinity that is f of w equal to Sigma
and equal is to minus infinity to infinity a n w power n valid in a deleted neighbourhood
of infinity has in finitely many terms in its singular part and what is the singular
part?
The singular part is Sigma, the singular part is the positive powers of so it is Sigma n
equal to 1 to infinity a n w power n and that is the same as saying that a n not equal to
0 for many in finitely many n for infinitely many n, so and you see this definition is
just the same as usual definition that in order to say that a certain point for example
in the complex plane and isolated singularity is an essential singularity all you have to
do is write out the Laurent expansion at that point then you find that the Laurent expansion
has in finite singular point okay and there it will be in finitely many negative powers
okay.
So so of course I should tell you that this is equivalent to g of Z is equal to f of 1
by Z has an essential singularity at 0 okay, so g of Z f of 1 by Z has an essential singularity
singularity at 0 at Z equal to 0 okay and this is of course you know this is of course
based on this philosophy that the behaviour of f of w at infinity is the same as the behaviour
of f of 1 by Z at 0 okay you put w equal to 1 by Z we already know that this is a homeomorphism
okay and it is a holomorphic isomorphism of the punctured plane, the plane minus the origin
onto itself okay and so this is one thing. Then the other thing is so this is one
condition the other condition is limit w tends to infinity f of w does not exist this is
this is the other condition and of course this is the same as saying that limit Z tends
to infinity g of Z does not exist.
So this is also something that you know and so well of course the usually we will
have another conditions which will be on the behaviour of the function in the neighbourhood
of that point and well certainly you cannot expect the function to be bounded in a neighbourhood
of that point okay because that is equivalent to the function being actually analytic at
that point and that is Riemannĺs removable singularity is theorem right. Anyway so so
let me go ahead and say some other things, so maybe I will put some I will change color
and put a few boxes here, so so this is equivalent to this and this part is the same as this
okay.
Alright so now what I want to say next is that you know let us analyse this condition
that infinity is an essential singularity okay, now theůso for example what are the
functions? What are the entire functions which have infinity as an essential singularity,
you can ask this question? Okay so we have already answered a similar question for poles
okay and for removable singularities. See you take an entire function mind you an entire
function means a function which is analytic on the whole plane okay and the moment it
is analytic on the whole plane the whole plane is also mind you a deleted neighbourhood of
the point at infinity you must not forget that there for a function is analytic on the
whole plane is also having infinity as an isolated singular point automatically okay.
If you think of the Riemannĺs stereographic projection you see that the infinity corresponds
to the North pole and the whole plane corresponds to the remaining part of the Riemann sphere
which is the Riemann sphere minus the North pole okay by this geographic projection, so
the plane itself is a neighbourhood of the point at infinity and therefore an entire
function will always have the point at infinity as an isolated singularity. Now what happens
if that isolated singularity is a removable singularity? Well then you are saying that
the entire function as infinity as a removable singularity okay which is the same as saying
that at infinity it is bounded or it has a limit okay and then by Liouvilleĺs theorem
it will reduce to a constant okay.
So the moral of the story is that a nonconstant entire function cannot have infinity as a
removable singularity we have already seen this okay and then you can ask the question
is when will an entire function have infinity as a pole okay and have seen that that will
happen if and only if the entire function is well is a polynomial okay and the degree
of the polynomial will be the order of the pole okay. So now we ask the question, when
will an entire function have infinity as an essential singularity? And the answer to that
will be that it should not be a polynomial okay basically if you write out the if you
write out its Taylor expansion at any point, the Taylor expansion of course you will get
you have to ride only a Taylor expansion because it is an entire function okay.
There is no question of Laurent expansion okay, so at any point you choose any point
in the plane it is analytic everywhere, the plane so you choose any point and write the
Taylor expansion at that point. That Taylor series will have in finite radius of convergence
because this function is entire okay and the point is that the Taylor series should be
a power series, it should not be a polynomial okay, so the moral of the story is an entire
function as infinity as an essential singularity if and only if its Taylor series is not a
polynomial okay.
So and the Taylor series of an entire function being a polynomial is the same as you entire
function itself being polynomial okay therefore all you are saying is that you know an entire
function if you wanted to have an essential singularity at infinity okay then it should
not be a polynomial alright and for this reason we call such entire function as transcendental
okay so usually a polynomial function and the Meromorphic functions which are given
by quotients of polynomials they are called algebraic and everything that is not algebraic
is called transcendental, so such entire functions which have infinity as an essential singular
point are called transcendental entire function okay.
So so let me write that down an entire function
that has infinity as an essential singularity is one that is not a polynomial okay, so and
you know what will happen if it is a polynomial, if it is a polynomial infinity is a pole okay,
if it is a polynomial of positive degree of course if youůwe also consider conscience
as polynomials, polynomial is of degree 0 and that is the case of a constant function,
so if you are looking at a non-constant entire function okay then it isůthe only way infinity
is an essential singularity is that it is not a polynomial of positive degree okay.
So or a polynomial, if constant has infinity has a removable singularity
and if nonconstant has infinity as a pole of order equal to its degree okay, so a polynomial
is certainly not an entire function that has an essential singularity at infinity and conversely
if an entire function as an essential singularity at infinity because you take any point and
you write out the retailers expansion at that point you will see that if infinity is
a is a pole then the Taylor expansion should terminate and it has to be a polynomial, so
if infinity is not a pole then your Taylor series will have infinitely many terms okay
which is the same as saying that there are in finitely many terms in the Laurent expansion
at infinity okay, mind you if you take a polynomial the polymer normal itself is the Taylor expansion
of the function that it represents at the origin okay and sense that expansion is valid
on the whole plane it is also a Lauren expansion at infinity.
The expression for the polynomial itself is a Lauren expansion at infinity and it is except
for the constant part the positive part terms involving positive part of the variable, that
is automatically the singular part at infinity and that it has only finitely many terms tells
you that infinity is a pole okay. So let me write that down, conversely if f of w has
infinity as an essential singularity then its Taylor expansion
at any point of C has to have, at any point of C has to have
in finitely many terms, so does not a polynomial okay, so say things in short a transcendental
entire function is something that is different from a polynomial.
Now I want to make the following statement that take any entire function okay exponentiated
a, so you take f of w to be an entire function okay, f of w may have infinity as you know
either a pole or may be an essential but if you take E power f of w okay if you take E
power f of W, I claimed that it will always be transcendental okay, so that means I am
saying that or the power f of W, w equal to infinity will always be an essential singularity
okay that is also very easy to see and how do you see it let me tell you the argument
and words you see so I know that f of w is entire and am looking at E power f of w okay.
Of course E power f of w is also entire because it is a composition of entire functions, exponential
function is of course entire okay and E power f of w is f of w allowed by the exponential
function it is composition of entire function, so it is entire okay and what other possibilities
of E power f of w at infinity? Infinity can either be removable singularity or it can
be a pole or it can be an essential singularity. If infinity is a removable singularity you
know then the power f of w must be a constant because of the Liouvilleĺs theorem and if
E power f of w is a constant then f has to be a constant because f is if E power
f of w is a constant that constant is a complex number which are not be 0 because exponential
function never takes the value 0 and f has to be log of that okay.
F can be one of the logarithms of that constant, nonzero constant okay and so so in that case
you power f so what am trying to say is that if you are looking at an entire function a
nonconstant entire function okay the power have has to be transcendental okay job the
only case you will have to worry about is the constant function. When of course E power
a constant is also a constant, so if you have a non-constant entire function okay the power
f also will be nonconstant right, so infinity or the power f if infinity is a removable
singularity then f is a constant, so if you are looking at a nonconstant f then he power
f will not have infinity as a removable singularity.
So it can be a pole now if E power f is a pole has a pole at infinity, infinity is a
pole for E power f then E power f has to be a polynomial because we have seen that the
only entire functions which have pole at infinity are the polynomials so E power f has to be
a polynomial but that is not possible because you see if you take a polynomial of constant
a nonconstant polynomial of positive degree the fundamental theorem of algebra says that
it will have zeros, so there areůit will assume the value 0 but that is equal to E
power f and E power anything can never be 0 okay therefore E power f cannot be a polynomial
alright at means that infinity cannot be a pole for E power f so therefore the only thing
that is left is E power f should have infinity as an essential singularity, so the moral
of the story is that you take any nonconstant entire function and you take E power that,
the resulting function will certainly have infinity as an essential singularity in other
words I am saying that the resulting function it will always be transcendental okay.
So so let me write that down okay so let me use a different color an entire function
that has infinity as an essential singularity is called okay, an entire function that has
infinity as an essential singularity is called transcendental and so all entire functions
which are not polynomials are transcendental okay. Thus the entire functions, the transcendental
entire functions are precisely the non-polynomials, precisely those that are not polynomials okay
if f of w is entire and nonconstant then he power f of w is always transcendental.
For if E power f w has w equal to infinity as a removable singularity
then E power f w reduces to a constant by Liouville and contradicts f being nonconstant
also E power f also infinity cannot be a pole as then he power f would be a polynomial which
must have zeros okay.
So that also cannot happen so as a result you take any entire function and you which
is not constant and the exponentiated what you get is a transcendental entire function
okay. So in this context let me also say that you know we entire function the functions
which are either polynomials or quotient of polynomials okay they are called Meromorphic
functions they are all called algebraic okay and the non-algebraic functions are the transcendental
ones okay.
So so let me write that in general, the polynomials and more generally the Meromorphic
functions in the extended plane are called algebraic okay at least in
the sense of algebraic geometry okay in the sense of complex algebraic the non-algebraic
ones are called transcendental, okay. Examples, examples of transcendental functions, this
is something that one should look ahead, so so you take for example E power Z this is
transcendental because Z equal to infinity is an essential singular point and why is
that so because basically if you write the expansion for E power Z Taylor expansion or
MacLaurin expansion which is a Taylor expansion in the origin you write the usual expansion
that we all know and you know it has infinitely many terms okay.
So E power Z and then similarly you can take the trigonometric functions you can take sin
Z you can take cos Z and so on okay, so these are all transcendental functions and the way
of course you know you can also see that infinity is an essential singular point because if
you change if you invert the variable you will get origin as the essential singular
points, so if you take E power 1 by Z origin will be an essential singular point, if you
take sin 1 by Z origin will be an essential singular point, if you take cos 1 by Z origin
will be an essential singular point okay, so these are examples of transcendental functions
and the other thing that I want to tell you is that I want to also recall the big Picard
theorem in this connection okay take now you see you know take an entire function, take
an entire function which is not constant okay and of course if infinity is a infinity cannot
be a removable okay because if infinity is a removable singularity then it will reduce
to a constant. Since I have taken a nonconstant entire function, infinity is not a removable
singularity.
The next possibility is infinity is a pole in which case the entire function is a polynomial
and you know the image of a polynomial map is the whole plane okay because a polynomial
can assume we will assume all values okay you take any value you can equate to the polynomial
and you can solve for it and you will get solutions that is because of the fundamental
theorem of algebra, so if you take a polynomial mapping it will be the image of the whole
plane under a polynomial mapping will be the whole plane then you look at the 3rd case
namely when infinity is an essential singular point okay. Now if infinity is an essential
singular point okay you see what does the big Picard theorem say?
The big Picard theorem says that you take any neighbourhood of an essential singular
point no matter how small, the image will be the whole plane and or it may be a punctured
plane it might miss at one point okay. Now you see this is the so what you are saying
is that if I take a transcendental entire function okay For example E power Z or sin
Z or cos Z okay then the big Picard theorem tell you that even if you take the image of
not the whole plane but the exterior of a circle no matter of how large radius you will
still get the whole plane or the punctured plane and you can compare it to the little
Picard theorem which tells you that the image of whole plane is either the whole plane or
the plane minus a point, so I want you to understand the significance.
See if I take for example take E power Z okay the little Picard theorem will tell you that
the image of E power Z is either the whole plane or it is the whole plane minus a point
because you know it is E power Z we know it is the plane minus the origin okay but the
little Picard theorem never tells you what is the image of anything other than the whole
plane okay but if you apply the big Picard theorem to E power Z okay you are applying
mind you when I apply the big Picard theorem to E power Z I am trying to apply I have to
apply it only to an essential singularity and where does E power Z have an essential
singularity at infinity.
So if I apply the big Picard theorem to E power Z at infinity okay then you see that
you take any neighbourhood of infinity okay which is you take the outside the reason exterior
to a circle of any radius, no matter how large, the image of that itself will be the whole
plane or a punctured plane okay and in fact every value is taken in finitely many times,
so you see you see in that sense how the big Picard theorem is far far stronger than the
little Picard theorem in the case of entire functions you can see that of course for polynomials
there is nothing special because your fundamental theorem of algebra which will tell you things
very clearly but if you are looking at non-polynomials.
If you are looking at transcendental entire functions you see you can really see the amount
of difference in the conclusion and you can see the strength of the theorem, you see the
great Picard theorem is you much more than the little Picard theorem okay, so if you
take E power Z and you take the image of the exterior of a circle matter how large radius
it will still be the punctured plane okay that is what the big Picard theorem applied
to E power Z will tell you okay whereas if you try to apply the little Picard theorem
you will not get anything it will give you only the image of the whole plane but it will
not tell you anything about the image of the exterior of a circle okay, so in that sense
you see how powerful the big Picard theorem is okay.
Now what I want to do next is tell you about tell you about Meromorphic functions okay,
so you see so I want to concentrate on Meromorphic functions and you know we need to study Meromorphic
functions in order to get to our main aim which is the proof of the Picard theorems
alright, so the 1st fact about Meromorphic functions is that you see if you take domain
either the domain may be in the plane or it may be a domain in the extended plane which
means it could include the point at infinity as an interior point. On a domain if you look
at the set of Meromorphic functions on the domain that is automatically a field okay,
so you get an algebraic structure called a field and it will be an extension field of
the field of complex numbers okay. So and this field extension its algebraic properties
are deeply connected with a geometric properties of the domain you are studying okay.
So so let me make that statement so let me go to the next thing the field of the Meromorphic
functions, so let me recall what is a Meromorphic function? So it is basically a function which
is analytic and has only isolated singularities which can be at most okay and of course they
could be removable singularities but if they are removable singularity is you really do
not consider them because they are actually points where the function is analytic okay
otherwise the only singularities that it has are poles okay, so so the moment am talking
about a Meromorphic function I have to remember that 1st of all what is allowed is ů what
kind of singularities are allowed us are only poles which means there are only isolated
singularities there are no non-isolated singularities okay.
There are only isolated singularities and they these isolated singularities are actually
only poles okay and the point is that you see if you look at a Meromorphic function
in the extended plane okay then you see that since an isolated set in the extended plane
has to have only finitely many points okay because the extended plane is compact okay
therefore there will be only finitely many poles okay, so the moral of the story is that
if you are looking at Meromorphic functions on the extended plane it will have only finitely
many poles okay. I think probably I will stop here.