Weierstrass Products and More on Analytic Continuation

Transcription

1 Weierstrass Products and More on Analytic Continuation Thursday, December 05, :58 PM Hard deadline for Homework 4: Wednesday, December 11 at 5 PM. rapid grading on or after Friday, December 6 Final exam on Thursday, December 12 at 6:30 PM. Weierstrass products These are attempts to represent entire functions in factored form, based on their zeros. This generalizes factoring polynomials to factoring entire functions, just as the Mittag-Leffler expansion generalized partial fractions for rational functions to meromorphic functions. Weierstrass products will often involve infinitely many factors, so one needs to discuss converging of infinite products. There are theorems similar to the Mittag-Leffler theorems that tell you that by doing it carefully enough, you can write entire functions in factored forms (Weierstrass product). We'll construct an example of a Weierstrass product to illustrate the content of the theorem. Let's try to get a Weierstrass product representation for This can be built systematically from noticing that has simple zeros (meaning whenever n is an integer, and at no other location. But we'll do a shortcut that still illustrates the key issues. Realize that: New Section 2 Page 1

2 Integrate both sides: But this was too quick; the sum doesn't converge. We'll have to go back and do this right, but the idea is that if you can get a MLE-type expansion for log f(z) then by exponentiating that expression, you get a Weierstrass product representation for f(z). A convergent way to write the sum is: Exponentiate both sides: One can check that this expression for the sum is convergent. New Section 2 Page 2

3 This product as written is absolutely convergent, so we can reorder the product by multiplying out the terms in the order: n=1,-1,2,-2,.. New Section 2 Page 3

4 This representation is also absolutely and uniformly convergent on any compact subset of the complex plane. We have here a typical situation which is that the Weierstrass product theorem tells us we can write the entire function in factored form with suitable convergence factors, and then this canonical form can be reorganized sometimes to produce a simpler but not canonical expression for the infinite product representation of the function. Analytic Continuation: A Practical Look The basic idea of the theorems of analytic continuation is that if we have two functions f(z) and g(z) that are analytic on different domains, and if they agree with each other on a common intersecting domain that contains a limit point, then one can build an analytic function of the union of the domains of f and g by simply defining it piecewise. The practical content of this concept is that an analytic function is completely determined globally by its behavior over any set that has a limit point. In particular, the real axis is a set with a limit point. So, the principle of analytic continuation says there is at most one way to extend a function on the real axis to an analytic function in any connected region containing the New Section 2 Page 4

5 real axis in the complex plane. So there's only one way to extend exponentials, and trig functions into the complex plane. Also, it means that if you have analytic relationships between analytic functions that are valid on the real axis, then those relationships are also valid wherever those functions can be analytically continued. How is analytic continuation used in a specific practical sense? The basic idea that we described in a previous lecture of just re-expanding the given function as Taylor series near the boundary of the domain of analyticity works in principle but is very unwieldy in practice. The actual ways in which analytic continuation is done is ad hoc, and to understand these procedures, it helps to see some simple examples. New Section 2 Page 5

6 Notice for example that: Then by the principle of analytic continuation, we would say that is the analytic continuation of the analytic function represented by the power series from the domain D to the entire complex plane (other than the isolated singularity). In practice, what this is usually interpreted as is that there is in the background the analytic function and when we write down a power series for it, it is the representation of that analytic function that fails outside the domain D, not the analytic function itself. So when one has a breakdown of a representation of an analytic function outside some restricted domain, one way of invoking analytic continuation is to rewrite the power series in such a way that the new representation converges over a broader domain. The intuition behind this first example is what is behind what's known as Pade approximation: Often when you solve a problem, say by perturbation theory, you have your answer represented as a power series of some parameter: Solve for the constants in terms of the by expanding the rational function as a power series and matching the first few terms. New Section 2 Page 6

7 (The resulting equations can be solved easily in an interactive fashion.) The resulting rational function (Pade approximant) often gives better approximations than the power series representation because nearby singularities (possibly in the complex plane, not in the actual parameter space of interest) can spoil convergence of the direct power series. But the denominator of the rational function can account for singularities. Integral representations: So we can think of as the analytic continuation of the integral representation from the domain D to the whole complex plane. Again, just as with power series, a particular integral representation may have some problems that cause it to fail to represent the underlying analytic function everywhere. In practice, one often has integral representations (sometimes from solutions to differential equations, etc.) that are only valid over some restricted domain. One tries to manipulate them in a way to get a representation that agrees with them but is valid more generally. This can be seen with the function. A standard definition for the function is: New Section 2 Page 7

8 One reason it's useful is that it generalizes the factorial to non-integers. The integral representation is well defined for the domain: is an analytic function on D because of an extension of the theorems about integrals with complex parameters. Can it be analytically continued into the LHP and if so, how? One way to do this analytic continuation is to realize that on D, This is a nice analytic relationship, it allows to refer to the value of the Gamma function at shifted values. Why not use this to define over a broader domain? New Section 2 Page 8

9 This can be done arbitrarily many times but it doesn't lead to an expression that is valid over whole complex plane. To get a global representation, there are at least two possibilities: One is based on MLE: New Section 2 Page 9

10 Or one can write down a Weierstrass product representation that also is valid everywhere except at the isolated singularities: Landau damping In plasma physics, one is interested in the behavior of populations of electrons and ions interacting with electromagnetic fields. One framework for modeling such systems is with the Vlasov equations, which are a kinetic theory for plasmas. The fundamental object of study is the distribution function of the electrons: New Section 2 Page 10

11 These equations are analyzed by a linearized perturbation expansion about equilibrium. Then one obtains some linearized integro-pdes. Then you do a Fourier transform in space and Laplace transform in time. What results form all this is that the Fourier-Laplace transform of the electric field takes the form: One virtue of using Laplace transforms is that the singularities of the Laplace transform tell you about instabilities and other interaction modes. So in this equation, we want to know where D(k,s) vanishes. Vlasov simply took the above New Section 2 Page 11

12 expression for D(k,s) and looked for where it vanished -- no interesting solutions. But then several years later, Landau realized a mistake in the analysis. The function D(k,s) depicted above is not analytic over the whole complex plane, with just a few isolated singularities. Rather, it breaks down everywhere along the imaginary axis. Landau showed that the correct analytic continuation of the expression for D(k,s) into the left hand plane is given by: And this expression does vanish for some value in the left half plane, predicting a damping mode for energy exchange between the electric field and the plasma. New Section 2 Page 12