 This is the first of a few lectures on elliptic functions. I'm not following any particular textbook, but Cambridge University Press has recently reprinted the classic book Course on Modern Analysis by Whitaker and Watson, and this contains a rather in-depth collection of chapters on elliptic functions. So first of all, what is an elliptic function? Well, a periodic function is a function like sine of x. So you know sine of x plus 2 pi is equal to sine of x, and here this 2 pi is the period. It's also got other periods, so obviously 4 pi, 6 pi, and so on are also periods. But all the periods are essentially multiples of 2 pi, and you can ask, can you have a function with two independent periods, say the periods are omega 1 and omega 2? Well, at first sight it seems you can't really, because if omega 1 over omega 2 is rational, then you can just take the greatest common denominator, greatest common divisor of these, and there's essentially only one period generating them. If it's an irrational real, this means the periods are dense, and this means the only function with all those periods are essentially constant if they're continuous. So there doesn't seem to be any way of having two independent periods. Well, that's if you're talking about functions of a real variable. So what we're going to do is to talk about functions of a complex variable. So we're going to be looking at elliptic functions f of z for z complex such that f of z plus omega 1 is equal to f of z, and f of z plus omega 2 is equal to f of z, where the ratio omega 2 over omega 1 is not real. As if it is real, nothing very interesting happens. Unfortunately, one of the main problems in the theory of elliptic functions is that notation for elliptic functions is a complete mess and not standardized at all. So in every author who writes about elliptic functions invents that own system of notation. So I'm using omega 1 and omega 2 for two of the periods, but other books will use all sorts of different notation for this. So the periods form a sort of lattice. So if you take 0 here, and omega 1 might be this complex number, and omega 2 might be a complex number here, and then all the other integral multiples of omega 1 and omega 2. So m omega 1 plus m omega 2 form a lattice, which I'm going to denote by l for lattice. So we get a sort of grid in the plane. And this means that the elliptic function is going to be the same if you shift by any one of these lattice points. So I have here a picture of an elliptic function. This is from the book by Janke and Emder. And here's an elliptic function that we might discuss later. It's one of Jacobi's elliptic functions. You can see it's sort of periodic in this direction. It's also periodic in that direction. So what I want to do this lecture is basically find all elliptic functions. So the first question is, can we find any holomorphic elliptic functions? And the answer is these all have to be constants, so they're not very interesting. And the reason for this is that f is bounded in fundamental domain. So a fundamental domain is going to be a region that looks a bit like this. And this fundamental domain is compact, so f being continuous is bounded in it. So it's bounded in a fundamental domain. And since it's periodic, it must be bounded everywhere, because every point is equivalent to a point in the fundamental domain. And if a holomorphic function is bounded on the whole plane, it must be constant. And for this, you use Lueville's theorem from complex analysis, which says any bounded holomorphic function has to be constant. So we should we should we should allow poles. So f is going to be meromorphic. It's just mean that it's allowed some poles. So the next question is, how can we find examples of elliptic functions? Well, there's a very general method. If you've got a group g acting on some vector space v, and you want to find something in a vector space fixed by g. So you want to find a vector of gv equals v for all g in v in g. There's a very simple way of doing this, you just take any element v of the vector space and just sum over all things in g acting on. And this is fixed by g if this series converges. So this is the catch. It doesn't always work because this group g might be infinite and then it's not clear if this converges. Now what we're going to do is to take g to be our lattice, which is a sum of two copies of z under addition. And we're going to take v to be meromorphic functions. And the action is just translation. If lambda is an element of the lattice, then and and we've got some function f, then we can change f of z to f of z plus lambda. And you can easily check this gives us a group action. So what we want to do is to take sum over all lambda in the lattice l. So that's lambda equals m lambda 1 plus n lambda 2 of f of z plus lambda. So let's call this big f of z. And there's a pretty good chance that this might be elliptic as long as this converges. Well, we've got this convergence condition. And in order for this to converge, the terms must at least tend to zero. So we must have f of z tends to zero as the absolute value of z tends to infinity, because otherwise this series isn't going to converge. So there are some obvious examples of functions like this. We can try f of z is one over z to the n for some integer n. So let's have a look at the first few of these and see if it converges. So let's try f of z was one over z and check for convergence. So we're taking sum over lambda in lambda of one over z minus lambda z plus lambda. I don't really care. So in order to test this for convergence, what we do is we take the complex plane and we draw a lot of circles of radius one, two, three and so on. And then count the number of terms of lambda inside each circle. And there are going to be about some constant times n terms in the nth circle or the nth ring. And the terms are all going to be about one over n in absolute value. So lambda is going to be approximately n and lambda is very large. So it's much larger than z. So the sum we get looks like sum over n greater than equal to one of n over n. And this obviously diverges rather badly. It doesn't work if you try this function. Well, let's try f of z was one over z squared. Well, here we get the same argument except the terms are about one over n squared. So we get sum of n over n squared. And this still diverges, but only just. It's right on the borderline of divergence. You know, this is this is the harmonic series, which, which does tend to infinity, but very slowly. And if you change this exponent n to anything bigger than two, this will converge. So we can try f of z equals one over z cubed. And here the series sum over one over z minus lambda cubed now converges and gives and gives an example of an elliptic function. So you can see it's got a pole at every point of lambda. And this is this is why elliptic functions sort of look like this. You can see there are all these poles here and the poles kind of live on some sort of lattice. This picture actually has poles of order one at every point rather than order three. And we'll talk about that later. So so we've got plenty of elliptic functions. It turns out that the most convenient function to use is not this one here, but this function here. Well, there's a bit of a problem with it because it diverges. So it doesn't actually give us an elliptic function, but it's so close to converging that we can make it converge by by just nudging it a little bit. So let's figure out why why it doesn't converge. Well, the problem is that if we take one over z minus lambda squared, this is going to be approximately one over lambda squared. And the sum of one over lambda squared diverges. However, it's if you expand further, it's sort of one over lambda squared plus two z over lambda cubed plus something over lambda to the four and so on. And for these bits, there's no problem. The terms for one over lambda cubed and one over lambda four and so on can converge if you sum them over all lambda. So the thing we have to worry about is this bad term here. We notice this is just a constant. So what we can do is we can just subtract off this constant from all the terms and we get sum over lambda in lambda of one over z minus lambda squared minus one over lambda squared. And what we've done is this is formally periodic but isn't periodic because it doesn't converge. So what we've done is we've sort of subtracted an infinite constant from it to make it converge. Well, there's still one more problem because this is infinite if lambda equals zero. So we omit this constant if lambda equals zero. So the sum we get is a sum over all these things for lambda in the lattice except when lambda is zero, we just omit this term. And this is the this is the famous Weierstrass p function which is defined by this sum here. Weierstrass used a capital letter p from some funny script alphabet and every letter of the script alphabet seems to be lost except for p which lives on as this as this strange sort of fossil in the mathematical literature. So well the problem is is p periodic because you see it's not at all obvious that it is periodic. I mean these terms are not invariant under changing z to z plus an element of lambda because there's this sort of funny infinite constant and we have to worry about this a bit but I mean we can see that it is in fact periodic and this follows from the first from these two facts. First of all if we take p of z and differentiate it it's equal to minus two times sum over one over z minus lambda cubed which is periodic. So it's derivative is periodic. This doesn't mean that the function itself is periodic because you can get functions with periodic derivative that aren't periodic for instance you would have a function that sort of keeps going up in bumps like that. However p has another property that p of z is even so p of z is equal to p of minus c and that's kind of obvious if you look at this because the terms if you change z to minus c and then change lambda to minus lambda you just get the same terms and if a function is periodic and even so if it's if the derivative is periodic and the function is even you can easily check a little exercise this that means the function is periodic because you know p of z plus lambda must be equal to c of lambda plus p of z for some c depending on lambda and if you just look at the term for minus lambda and do a little bit of algebra you can see that this the fact that this is even implies c of lambda must actually be zero. So we found a periodic function. In fact I can draw and I have an explicit picture of Weierstrich's function. It looks like this. You can see the pole is a little bit fatter than the poles in the previous picture because these were poles of order one and this is a pole of order two which is kind of a bit bigger and heftier. So we found one periodic function. We found several periodic functions because we can take various other inverse powers of z. So let's show that we've in fact found all periodic functions or at least we can do fairly simple manipulations to these to find all all periodic functions because we notice that if fc is periodic then so is f of z plus any constant c and so is p of f of z over q of f of z where p and q are polynomials. We need p and q to be polynomials so this is still meromorphic. So if we've got one periodic function we can get lots of others by these simple tricks and we now want to show that we can in fact get all periodic functions just by applying this method here to the ones we've already found. So first of all we can get rid of of poles not on the lattice l and here if you've got a periodic function we just multiply it by p of z minus p of c if f is a pole at z each c or maybe you need to raise it to some power to get rid of a pole. So by multiplying f by by polynomials in viastrisis function we can get rid of all poles that are not in the lattice l and we just left with an elliptic function whose only poles are an l. Next we we can make f even so f of z is equal to an even function even periodic function plus an odd periodic function the usual way you write a function is the sum of an even and odd function and the odd one is equal to p prime of z times an even function. So this is the derivative of viastrisis function which you can easily see is an odd function so if you divide an odd function by an odd function you get an even function. So it's enough to do an even periodic function so we we can assume that f is even the only poles are on l and now if we look at viastrisis function p of z it's equal to one over z squared plus some other terms I don't care about so if you raise it to the power of n it's equal to one over z to the two n plus something or other so taking f of z minus a linear combination of p e prime and so on we can kill the pole at zero so so I can assume f has no poles because we've got rid of them all and by what we said earlier any elliptic function with no poles at all is constant so what we've shown is that any elliptic function is some elliptic function whose only pole is on the lattice points divided by a polynomial in p and we've shown that any elliptic function whose only poles are at lattice points is a polynomial in p plus the derivative of p times a polynomial in p so we've got a complete description of all elliptic functions so let's write that this out explicitly elliptic functions with poles on l can be written are are just all polynomials in x and y where x is equal to the viastris function y is equal to the viastris function differentiated except we've got to question out by a relation and you notice that if we take the viastris function derivative and square this is even and it looks like z to the minus six plus something so it's equal to sorry four z to the minus six so it's equal to four times p cubed plus something times p squared plus something times p plus some constant because any even function whose only poles are at lattice points is a polynomial in p this term is in fact zero as you can check without too much difficulty and these are called minus g2 and minus g3 um for no particularly good reason so um we have to quotient out by x q 4x cubed minus g um 2x minus g3 minus y squared so here we have a explicit description of all elliptic functions the ones with poles only and l are just this polynomial ring quotient out by this ideal and the ones with poles anywhere are just um are given by the field of quotients of this ring here so so so we we know all elliptic functions in a in a reasonably explicit way um this actually gives us a sort of um isomorphism um if we take the complex numbers and modulo out by this lattice so we identify any two complex numbers if they differ by an element of this lattice then we can actually identify this with the curve consisting of um numbers such that y squared equals 4x cubed minus um g2x minus g3 so this is um some sort of a curve in the plane which might look something like this uh should say c over l minus l minus the zero because we're mapping the point z to the point p of z e prime of z is even the point x y so here z is a point of the of the complex numbers or the complex numbers modulo l minus zero um so in other words um the the the space of complex numbers modulo lattice simply can be identified with a cubic curve uh in the plane um well there's a question that um um might have occurred to you which is we managed to get an elliptic function of the pole of order two um at um lattice points why can't we get an elliptic function with a pole of order one at lattice points by taking this series sum over one over z minus lambda and um getting rid of the divergent terms just as we did with with the other one so um if we take one let's take one over z plus lambda so this is going to be approximately one over lambda minus c over lambda squared plus c squared over lambda cubed and so on and these terms here are small and these terms here cause a problem so let's just subtract them as we did before and take sum over lambda of one over z plus lambda and then subtract these difficult terms plus c over lambda squared and this now converges and so we get a function which is sometimes called the zeta function it's rather confusing it's it's the viastri zeta function which has nothing to do with the reamon zeta function and let's see if this is an elliptic function well first of all zeta prime of z is um minus um um the viastrice function so it's periodic um and this means zeta of z plus a period is equal to zeta of z plus um some constant which is traditionally um has a factor of two in there for obscure historical reasons and similarly this can be zeta of z plus another constant eight of one and eight of two and the question is are these a zero well in the case of viastrice's p function managed to prove that corresponding constant was zero because p was even however you can easily check that zeta of z is odd and this doesn't imply that this vanishes because it's perfectly easy to find odd functions which satisfy relations like this and in fact these constants can't both be zero um you can see this as follows um there's actually a relation between these um constants which is that um two pi i is equal to the integral over c of zeta of z z here um c is going to be the integral round fundamental domains and not omega one omega two and so on and we better avoid the um poles on the um fundamental domain so I'm actually going to integrate round something it looks a bit like this and this integral is equal to two pi i because this is two pi i times the um sum of the residues um and um we can also work out the integral explicitly because the integral along the bottom of the top almost cancel out by almost periodicity and the integral along the top bottom in the top turns out to be essentially um omega one times eight of two and similarly these two integrals here almost cancel out apart from the factor of omega two times eight of one and so what would this turns out to be if you keep careful tracking the signs is two eight one times omega two minus two eight two times omega one this gives le genre's relation between these um these four numbers here so you see eight of one and eight two can't possibly both be zero because this this factor here is non-zero um if you look this up in books by the way you will sometimes find this differs by factors of two from um formulas you sometimes find and that's because periods are sometimes taking to be two omega one rather than omega one as i've taken them here um i just finished with a little historical note about why elliptic functions are called elliptic functions they don't actually very much to do with ellipses and the name comes from elliptic integrals which are integrals of the form dt over square root of a cubic or quartic polynomial in t and you get integrals like this if you try and work out the arc length of an ellipse which is where the name elliptic comes from um if it's if there's a quadratic polynomial here you can easily do this in terms of arc signs and arc cosines and so on um but if this as degree greater than two you can't usually do that um however you can do it using elliptic functions so suppose i want to integrate um say dx over the square root of four x cubed minus g two x minus g three so here we've got a cubic in x what we can do is we can write x is p of z you remember that p and z satisfies a funny differential equation so p prime z squared is equal to four x cubed minus g two x minus g three where x is p so this just turns out to be the integral of dx um divided by p prime of of um z um so this is um d p prime of z so d p of z over p p prime of z um which is is is is just dz so um this mess here just turns out to be really just a differential dz and this means you can sort of integrate this and if you do this you find z is actually equal to the integral from p of um z to infinity of this integral here dx over square root of four x cubed minus g two x minus g three so you see that the elliptic function actually appears in a slightly funny way it's not the value of this integral from from something to something else it's what you have to put in to this interval to get z um and elliptic intervals were originally discovered in this rather roundabout way and this caused tremendous complications because defining them as the inverse of some function is really rather a mess and this function you're defining them as an inverse of is a rather messy function of branch points and so on and everywhere so um it's when you're doing elliptic functions it's probably best to forget about the the historical origins of them because that's that's that's just a rather unnecessarily complicated way of doing things um okay the next lecture will be more about properties of the via strice elliptic function