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By an odd coincidence, I stumbled upon a second question in as many weeks about power series, and once again the only way I know how to prove the result is by complex methods; once again, I am leaving it here as a challenge to any interested readers, and I would be particularly interested in knowing of a proof that was not based on complex analysis (or thinly disguised versions thereof), or for a reference to previous literature where something like this identity has occured. (I suspect for instance that something like this may have shown up before in free probability, based on the answer to part (ii) of the problem.)
Here is a purely algebraic form of the problem:
where we use to denote the -fold derivative of with respect to the variable .
- (i) Show that can be formally recovered from by the formula
- (ii) There is a remarkable further formal identity relating with that does not explicitly involve any infinite summation. What is this identity?
To rigorously formulate part (i) of this problem, one could work in the commutative differential ring of formal infinite series generated by polynomial combinations of and its derivatives (with no constant term). Part (ii) is a bit trickier to formulate in this abstract ring; the identity in question is easier to state if are formal power series, or (even better) convergent power series, as it involves operations such as composition or inversion that can be more easily defined in those latter settings.
To illustrate Problem 1(i), let us compute up to third order in , using to denote any quantity involving four or more factors of and its derivatives, and similarly for other exponents than . Then we have
multiplying, we have
and hence after a lot of canceling
Thus Problem 1(i) holds up to errors of at least. In principle one can continue verifying Problem 1(i) to increasingly high order in , but the computations rapidly become quite lengthy, and I do not know of a direct way to ensure that one always obtains the required cancellation at the end of the computation.
Problem 1(i) can also be posed in formal power series: if
is a formal power series with no constant term with complex coefficients with , then one can verify that the series
makes sense as a formal power series with no constant term, thus
For instance it is not difficult to show that . If one further has , then it turns out that
as formal power series. Currently the only way I know how to show this is by first proving the claim for power series with a positive radius of convergence using the Cauchy integral formula, but even this is a bit tricky unless one has managed to guess the identity in (ii) first. (In fact, the way I discovered this problem was by first trying to solve (a variant of) the identity in (ii) by Taylor expansion in the course of attacking another problem, and obtaining the transform in Problem 1 as a consequence.)
The transform that takes to resembles both the exponential function
and Taylor’s formula
but does not seem to be directly connected to either (this is more apparent once one knows the identity in (ii)).
In the previous set of notes we introduced the notion of a complex diffeomorphism between two open subsets of the complex plane (or more generally, two Riemann surfaces): an invertible holomorphic map whose inverse was also holomorphic. (Actually, the last part is automatic, thanks to Exercise 40 of Notes 4.) Such maps are also known as biholomorphic maps or conformal maps (although in some literature the notion of “conformal map” is expanded to permit maps such as the complex conjugation map that are angle-preserving but not orientation-preserving, as well as maps such as the exponential map from to that are only locally injective rather than globally injective). Such complex diffeomorphisms can be used in complex analysis (or in the analysis of harmonic functions) to change the underlying domain to a domain that may be more convenient for calculations, thanks to the following basic lemma:
- (i) If is a function to another Riemann surface , then is holomorphic if and only if is holomorphic.
- (ii) If are open subsets of and is a function, then is harmonic if and only if is harmonic.
Proof: Part (i) is immediate since the composition of two holomorphic functions is holomorphic. For part (ii), observe that if is harmonic then on any ball in , is the real part of some holomorphic function thanks to Exercise 62 of Notes 3. By part (i), is also holomorphic. Taking real parts we see that is harmonic on each ball in , and hence harmonic on all of , giving one direction of (ii); the other direction is proven similarly.
Exercise 2 Establish Lemma 1(ii) by direct calculation, avoiding the use of holomorphic functions. (Hint: the calculations are cleanest if one uses Wirtinger derivatives, as per Exercise 27 of Notes 1.)
Exercise 3 Let be a complex diffeomorphism between two open subsets of , let be a point in , let be a natural number, and let be holomorphic. Show that has a zero (resp. a pole) of order at if and only if has a zero (resp. a pole) of order at .
From Lemma 1(ii) we can now define the notion of a harmonic function on a Riemann surface ; such a function is harmonic if, for every coordinate chart in some atlas, the map is harmonic. Lemma 1(ii) ensures that this definition of harmonicity does not depend on the choice of atlas. Similarly, using Exercise 3 one can define what it means for a holomorphic map on a Riemann surface to have a pole or zero of a given order at a point , with the definition being independent of the choice of atlas.
In view of Lemma 1, it is thus natural to ask which Riemann surfaces are complex diffeomorphic to each other, and more generally to understand the space of holomorphic maps from one given Riemann surface to another. We will initially focus attention on three important model Riemann surfaces:
- (i) (Elliptic model) The Riemann sphere ;
- (ii) (Parabolic model) The complex plane ; and
- (iii) (Hyperbolic model) The unit disk .
The designation of these model Riemann surfaces as elliptic, parabolic, and hyperbolic comes from Riemannian geometry, where it is natural to endow each of these surfaces with a constant curvature Riemannian metric which is positive, zero, or negative in the elliptic, parabolic, and hyperbolic cases respectively. However, we will not discuss Riemannian geometry further here.
All three model Riemann surfaces are simply connected, but none of them are complex diffeomorphic to any other; indeed, there are no non-constant holomorphic maps from the Riemann sphere to the plane or the disk, nor are there any non-constant holomorphic maps from the plane to the disk (although there are plenty of holomorphic maps going in the opposite directions). The complex automorphisms (that is, the complex diffeomorphisms from a surface to itself) of each of the three surfaces can be classified explicitly. The automorphisms of the Riemann sphere turn out to be the Möbius transformations with , also known as fractional linear transformations. The automorphisms of the complex plane are the linear transformations with , and the automorphisms of the disk are the fractional linear transformations of the form for and . Holomorphic maps from the disk to itself that fix the origin obey a basic but incredibly important estimate known as the Schwarz lemma: they are “dominated” by the identity function in the sense that for all . Among other things, this lemma gives guidance to determine when a given Riemann surface is complex diffeomorphic to a disk; we shall discuss this point further below.
It is a beautiful and fundamental fact in complex analysis that these three model Riemann surfaces are in fact an exhaustive list of the simply connected Riemann surfaces, up to complex diffeomorphism. More precisely, we have the Riemann mapping theorem and the uniformisation theorem:
As we shall see, every connected Riemann surface can be viewed as the quotient of its simply connected universal cover by a discrete group of automorphisms known as deck transformations. This in principle gives a complete classification of Riemann surfaces up to complex diffeomorphism, although the situation is still somewhat complicated in the hyperbolic case because of the wide variety of discrete groups of automorphisms available in that case.
We will prove the Riemann mapping theorem in these notes, using the elegant argument of Koebe that is based on the Schwarz lemma and Montel’s theorem (Exercise 57 of Notes 4). The uniformisation theorem is however more difficult to establish; we discuss some components of a proof (based on the Perron method of subharmonic functions) here, but stop short of providing a complete proof.
The above theorems show that it is in principle possible to conformally map various domains into model domains such as the unit disk, but the proofs of these theorems do not readily produce explicit conformal maps for this purpose. For some domains we can just write down a suitable such map. For instance:
Exercise 6 (Cayley transform) Let be the upper half-plane. Show that the Cayley transform , defined by
is a complex diffeomorphism from the upper half-plane to the disk , with inverse map given by
Exercise 8 Show that for any real numbers , the strip is complex diffeomorphic to the disk . (Hint: use a branch of either the complex logarithm, or of a complex power .)
We will discuss some other explicit conformal maps in this set of notes, such as the Schwarz-Christoffel maps that transform the upper half-plane to polygonal regions. Further examples of conformal mapping can be found in the text of Stein-Shakarchi.
My colleague Tom Liggett recently posed to me the following problem about power series in one real variable . Observe that the power series
has very rapidly decaying coefficients (of order ), leading to an infinite radius of convergence; also, as the series converges to , the series decays very rapidly as approaches . The problem is whether this is essentially the only example of this type. More precisely:
Problem 1 Let be a bounded sequence of real numbers, and suppose that the power series
(which has an infinite radius of convergence) decays like as , in the sense that the function remains bounded as . Must the sequence be of the form for some constant ?
As it turns out, the problem has a very nice solution using complex analysis methods, which by coincidence I happen to be teaching right now. I am therefore posing as a challenge to my complex analysis students and to other readers of this blog to answer the above problem by complex methods; feel free to post solutions in the comments below (and in particular, if you don’t want to be spoiled, you should probably refrain from reading the comments). In fact, the only way I know how to solve this problem currently is by complex methods; I would be interested in seeing a purely real-variable solution that is not simply a thinly disguised version of a complex-variable argument.
(To be fair to my students, the complex variable argument does require one additional tool that is not directly covered in my notes. That tool can be found here.)
In the previous set of notes we saw that functions that were holomorphic on an open set enjoyed a large number of useful properties, particularly if the domain was simply connected. In many situations, though, we need to consider functions that are only holomorphic (or even well-defined) on most of a domain , thus they are actually functions outside of some small singular set inside . (In this set of notes we only consider interior singularities; one can also discuss singular behaviour at the boundary of , but this is a whole separate topic and will not be pursued here.) Since we have only defined the notion of holomorphicity on open sets, we will require the singular sets to be closed, so that the domain on which remains holomorphic is still open. A typical class of examples are the functions of the form that were already encountered in the Cauchy integral formula; if is holomorphic and , such a function would be holomorphic save for a singularity at . Another basic class of examples are the rational functions , which are holomorphic outside of the zeroes of the denominator .
Singularities come in varying levels of “badness” in complex analysis. The least harmful type of singularity is the removable singularity – a point which is an isolated singularity (i.e., an isolated point of the singular set ) where the function is undefined, but for which one can extend the function across the singularity in such a fashion that the function becomes holomorphic in a neighbourhood of the singularity. A typical example is that of the complex sinc function , which has a removable singularity at the origin , which can be removed by declaring the sinc function to equal at . The detection of isolated removable singularities can be accomplished by Riemann’s theorem on removable singularities (Exercise 35 from Notes 3): if a holomorphic function is bounded near an isolated singularity , then the singularity at may be removed.
After removable singularities, the mildest form of singularity one can encounter is that of a pole – an isolated singularity such that can be factored as for some (known as the order of the pole), where has a removable singularity at (and is non-zero at once the singularity is removed). Such functions have already made a frequent appearance in previous notes, particularly the case of simple poles when . The behaviour near of function with a pole of order is well understood: for instance, goes to infinity as approaches (at a rate comparable to ). These singularities are not, strictly speaking, removable; but if one compactifies the range of the holomorphic function to a slightly larger space known as the Riemann sphere, then the singularity can be removed. In particular, functions which only have isolated singularities that are either poles or removable can be extended to holomorphic functions to the Riemann sphere. Such functions are known as meromorphic functions, and are nearly as well-behaved as holomorphic functions in many ways. In fact, in one key respect, the family of meromorphic functions is better: the meromorphic functions on turn out to form a field, in particular the quotient of two meromorphic functions is again meromorphic (if the denominator is not identically zero).
Unfortunately, there are isolated singularities that are neither removable or poles, and are known as essential singularities. A typical example is the function , which turns out to have an essential singularity at . The behaviour of such essential singularities is quite wild; we will show here the Casorati-Weierstrass theorem, which shows that the image of near the essential singularity is dense in the complex plane, as well as the more difficult great Picard theorem which asserts that in fact the image can omit at most one point in the complex plane. Nevertheless, around any isolated singularity (even the essential ones) , it is possible to expand as a variant of a Taylor series known as a Laurent series . The coefficient of this series is particularly important for contour integration purposes, and is known as the residue of at the isolated singularity . These residues play a central role in a common generalisation of Cauchy’s theorem and the Cauchy integral formula known as the residue theorem, which is a particularly useful tool for computing (or at least transforming) contour integrals of meromorphic functions, and has proven to be a particularly popular technique to use in analytic number theory. Within complex analysis, one important consequence of the residue theorem is the argument principle, which gives a topological (and analytical) way to control the zeroes and poles of a meromorphic function.
Finally, there are the non-isolated singularities. Little can be said about these singularities in general (for instance, the residue theorem does not directly apply in the presence of such singularities), but certain types of non-isolated singularities are still relatively easy to understand. One particularly common example of such non-isolated singularity arises when trying to invert a non-injective function, such as the complex exponential or a power function , leading to branches of multivalued functions such as the complex logarithm or the root function respectively. Such branches will typically have a non-isolated singularity along a branch cut; this branch cut can be moved around the complex domain by switching from one branch to another, but usually cannot be eliminated entirely, unless one is willing to lift up the domain to a more general type of domain known as a Riemann surface. As such, one can view branch cuts as being an “artificial” form of singularity, being an artefact of a choice of local coordinates of a Riemann surface, rather than reflecting any intrinsic singularity of the function itself. The further study of Riemann surfaces is an important topic in complex analysis (as well as the related fields of complex geometry and algebraic geometry), but unfortunately this topic will probably be postponed to the next course in this sequence (which I will not be teaching).
We now come to perhaps the most central theorem in complex analysis (save possibly for the fundamental theorem of calculus), namely Cauchy’s theorem, which allows one to compute (or at least transform) a large number of contour integrals even without knowing any explicit antiderivative of . There are many forms and variants of Cauchy’s theorem. To give one such version, we need the basic topological notion of a homotopy:
Definition 1 (Homotopy) Let be an open subset of , and let , be two curves in .
- (i) If have the same initial point and final point , we say that and are homotopic with fixed endpoints in if there exists a continuous map such that and for all , and such that and for all .
- (ii) If are closed (but possibly with different initial points), we say that and are homotopic as closed curves in if there exists a continuous map such that and for all , and such that for all .
- (iii) If and are curves with the same initial point and same final point, we say that and are homotopic with fixed endpoints up to reparameterisation in if there is a reparameterisation of which is homotopic with fixed endpoints in to a reparameterisation of .
- (iv) If and are closed curves, we say that and are homotopic as closed curves up to reparameterisation in if there is a reparameterisation of which is homotopic as closed curves in to a reparameterisation of .
In the first two cases, the map will be referred to as a homotopy from to , and we will also say that can be continously deformed to (either with fixed endpoints, or as closed curves).
For a similar reason, in a convex open set , any two closed curves will be homotopic to each other as closed curves.
- (i) Prove that the property of being homotopic with fixed endpoints in is an equivalence relation.
- (ii) Prove that the property of being homotopic as closed curves in is an equivalence relation.
- (iii) If are closed curves with the same initial point, show that is homotopic to as closed curves if and only if is homotopic to with fixed endpoints for some closed curve with the same initial point as or .
- (iv) Define a point in to be a curve of the form for some and all . Let be a closed curve in . Show that is homotopic with fixed endpoints to a point in if and only if is homotopic as a closed curve to a point in . (In either case, we will call homotopic to a point, null-homotopic, or contractible to a point in .)
- (v) If are curves with the same initial point and the same terminal point, show that is homotopic to with fixed endpoints in if and only if is homotopic to a point in .
- (vi) If is connected, and are any two curves in , show that there exists a continuous map such that and for all . Thus the notion of homotopy becomes rather trivial if one does not fix the endpoints or require the curve to be closed.
- (vii) Show that if is a reparameterisation of , then and are homotopic with fixed endpoints in U.
- (viii) Prove that the property of being homotopic with fixed endpoints in up to reparameterisation is an equivalence relation.
- (ix) Prove that the property of being homotopic as closed curves in up to reparameterisation is an equivalence relation.
We can then phrase Cauchy’s theorem as an assertion that contour integration on holomorphic functions is a homotopy invariant. More precisely:
- (i) If and are rectifiable curves that are homotopic in with fixed endpoints up to reparameterisation, then
- (ii) If and are closed rectifiable curves that are homotopic in as closed curves up to reparameterisation, then
This version of Cauchy’s theorem is particularly useful for applications, as it explicitly brings into play the powerful technique of contour shifting, which allows one to compute a contour integral by replacing the contour with a homotopic contour on which the integral is easier to either compute or integrate. This formulation of Cauchy’s theorem also highlights the close relationship between contour integrals and the algebraic topology of the complex plane (and open subsets thereof). Setting to be a point, we obtain an important special case of Cauchy’s theorem (which is in fact equivalent to the full theorem):
An important feature to note about Cauchy’s theorem is the global nature of its hypothesis on . The conclusion of Cauchy’s theorem only involves the values of a function on the images of the two curves . However, in order for the hypotheses of Cauchy’s theorem to apply, the function must be holomorphic not only on the images on , but on an open set that is large enough (and sufficiently free of “holes”) to support a homotopy between the two curves. This point can be emphasised through the following fundamental near-counterexample to Cauchy’s theorem:
As a consequence of this and Cauchy’s theorem, we conclude that the contour is not contractible to a point in ; note that this does not contradict Example 2 because is not convex. Thus we see that the lack of holomorphicity (or singularity) of at the origin can be “blamed” for the non-vanishing of the integral of on the closed contour , even though this contour does not come anywhere near the origin. Thus we see that the global behaviour of , not just the behaviour in the local neighbourhood of , has an impact on the contour integral.
One can of course rewrite this example to involve non-closed contours instead of closed ones. For instance, if we let denote the half-circle contours and , then are both contours in from to , but one has
In order for this to be consistent with Cauchy’s theorem, we conclude that and are not homotopic in (even after reparameterisation).
In the specific case of functions of the form , or more generally for some point and some that is holomorphic in some neighbourhood of , we can quantify the precise failure of Cauchy’s theorem through the Cauchy integral formula, and through the concept of a winding number. These turn out to be extremely powerful tools for understanding both the nature of holomorphic functions and the topology of open subsets of the complex plane, as we shall see in this and later notes.