How do we tell the temperature of some piece of material? I've written about temperature and thermometry a couple of times before (here, here, here). For ordinary, every-day thermometry, we measure some physical property of a material or system where we have previously mapped out its response as a function of temperature. For example, near room temperature liquid mercury expands slightly with increasing \(T\). Confined in a thin glass tube, the length of a mercury column varies approximately linearly with changes in temperature, \(\delta \ell \sim \delta T\). To do *primary* thermometry, we don't want to have some empirical calibration - rather, we want to measure some physical property for which we think we have a complete understanding of the underlying physics, so that \(T\) can be inferred directly from the measured quantity and our theoretical expressions, with no adjustable parameters. This is particularly important at very low temperatures, thousandths of a Kelvin above absolute zero, where the number of things that we can measure is comparatively limited, and tiny flows of power (from our measurements, say) can actually produce large percentage temperature changes.

This recent paper shows a nice example of applying three *different* primary thermometry techniques to a single system, a puddle of electrons confined in 2d at a semiconductor interface, at about 6 mK. This is all the more impressive because of how easy it is to inadvertently heat up electrons in such 2d layers. All three techniques rely on our understanding of how electrons behave at low temperatures. According to our theory of electrons in metals (which these 2d electrons are, as far as physicists are concerned), as a function of energy, electrons are spread out in a characteristic way, the Fermi-Dirac distribution. From the theory side, we know this functional form exactly (figure from that wikipedia link). At low temperatures, all of the electronic states below a highest-filled-state are full, and all above are empty. As \(T\) is increased, the electrons smear out into higher energy states, as shown. The three effects measured in the experiment all depend on \(T\) through this electronic distribution:

Fig. 2 from the paper, showing excellent, consistent agreement between experiment and theory, showing electron temperatures of ~ 6 mK. |

**Current noise**in a quantum point contact, the fluctuations in the average current. For this particular device, where conduction takes place through a small, controllable number of quantum channels, we think we understand the situation completely. There is a closed-form expression for what the noise should do as a function of average current, with temperature as the only adjustable parameter (once the conduction has been measured).**"Coulomb blockade" in a quantum dot**. Conduction through a puddle of electrons connected to input and output electrodes by tunneling barriers ("pinched off" versions of the point contacts) shows a very particular form of current-voltage characteristic that is tunable by a nearby gate electrode. The physics here is that, because of the mutual repulsion of electrons, it takes energy (supplied by either a voltage source or temperature) to get charge to flow through the puddle. Again, once the conduction has been measured, there is a closed-form expression for what the conductance should do as a function of that gate voltage.**"Environmental"**Coulomb blockade in a quantum dot. This is like the situation above, but with one of the tunnel barriers replaced by a controlled resistor. Again, there is an expression for the particular shape of the \(I-V\) curve where the adjustable parameter is \(T\).

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 […]

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).

** — 1. Laurent series — **

Suppose we are given a holomorphic function and a point in . For a sufficiently small radius , the circle and its interior both lie in , and the Cauchy integral formula tells us that

in the interior of this circle. In Corollary 18 of Notes 3, this was used to form a convergent Taylor series expansion

in the interior of this circle, where the coefficients could be reconstructed from the values of on the circle by the formula

Now suppose that is only known to be holomorphic outside of . Then the Cauchy integral formula no longer directly applies, because the interiors of contours such as are no longer contained in the region where is holomorphic. To deal with this issue, we use the following convenient decomposition.

Lemma 1 (Cauchy integral formula decomposition in annular regions)Let be a holomorphic function. Let , be simple closed anticlockwise contours in such that is contained in the interior of (or equivalently, by Exercise 49 of Notes 3, that is contained in the exterior of ). Suppose also that the “annular region” is contained in . Then there exists a decompositionon , where is holomorphic on the union of and the interior of , and is holomorphic on the union of and the exterior of , with as . Furthermore, if is connected, then this decomposition is unique.

In addition, we have the Cauchy integral type formulae

for in the interior of , and

for in the exterior of . In particular, we have

*Proof:* We begin with uniqueness. Suppose we have two decompositions

on , where and holomorphic, and both going to zero at infinity. Then the holomorphic functions and agree on the common domain , and are hence restrictions of a single entire function . But goes to zero at infinity and is hence bounded; applying Liouville’s theorem (Theorem 28 of Notes 3) we see that vanishes entirely. This gives and on the non-empty open set , and then we have and on and by analytic continuation (Corollary 23 of Notes 3).

Now for existence. Suppose that we can establish the identity (1) for in . Then we can define on by

and on by

noting from (1) that this consistently defines on

From Exercise 36 of Notes 3 we see that is holomorphic. Similarly if we define on by

and on by

One can then verify that obey all the required properties.

Thus it remains to establish (1). This follows from the homology form of the Cauchy integral formula (Exercise 63(v) of Notes 3), but we can also avoid explicit use of homology by the following “keyhole contour” argument. For , we have

and

and so to prove (1), it suffices to show that

By the factor theorem (Corollary 22 of Notes 3) it thus suffices to show that

By perturbing using Cauchy’s theorem we may assume that these curves are simple closed polygonal paths (if one wishes, one can also restrict the edges to be horizontal and vertical, although this is not strictly necessary for the argument). By connecting a point in to a point in by a polygonal path in the interior of , and removing loops, self-intersections, or excursions into the interior (or image) of , we can find a simple polygonal path from a point in to a point in that lies entirely in except at the endpoints. By rearranging and we may assume that is the initial and terminal point of , and is the initial and terminal point of . Then the closed polygonal path has vanishing winding number in the interior of or exterior of , thus contains all the points where the winding number is non-zero. This path is not simple, but we can approximate it to arbitrary accuracy by a simple closed polygonal path by shifting the simple polygonal paths and slightly; for small enough, the interior of will then lie in . Applying Cauchy’s theorem (Theorem 52 of Notes 3) we conclude that

taking limits as we obtain (2) as claimed.

Exercise 2Let be a simple closed anticlockwise contour, and let be simple closed anticlockwise contours in the interior of whose images are disjoint, and such that the interiors are also disjoint. Let be an open set containing and the regionShow that for any , one has

(

Hint:induct on using Lemma 1.)

Exercise 3 (Painlevé’s theorem on removable singularities)Let be an open subset of . Let be a compact subset of which has zero length in the following sense: for any , one can cover by a countable number of disks such that . Let be a bounded holomorphic function. Show that the singularities in are removable in the sense that there is an extension of to which remains holomorphic. (Hint:one can work locally in some disk in that contains a portion of . Cover this portion by a finite number of small disks, group them into connected components, use the previous exercise, and take an appropriate limit.) Note that this result generalises Riemann’s theorem on removable singularities, see Exercise 35 from Notes 3. The situation when has positive length is considerably more subtle, and leads to the theory of analytic capacity, which we will not discuss further here.

Now suppose that is holomorphic for some open set that contains an annulus of the form

for some and . From Lemma 1, we can split , where is holomorphic in , and is holomorphic in the exterior region , with going to zero as . From Corollary 18 of Notes 3, one has a Taylor expansion

for some coefficients that is absolutely convergent in the disk . One cannot directly apply this Taylor expansion to . However, observe that the function is holomorphic in the punctured disk , and goes to zero as one approaches zero. By Riemann’s theorem (Exercise 35 from Notes 3), this function may be extended to to a holomorphic function that vanishes at the origin. Applying Corollary 18 of Notes 3 again, we conclude that there is a Taylor expansion

for some coefficients that is absolutely convergent in the punctured disk . Changing variables, we conclude that

for all in (3), with the doubly infinite series on the right-hand side being absolutely convergent. This series is known as the Laurent series in the annulus (3). The coefficients may be explicitly computed in terms of :

Exercise 4 (Fourier inversion formula)Let be holomorphic on some open set that contains an annulus of the form (3), and let be the coefficients of the Laurent expansion (4) in this annulus. Show that the coefficients are uniquely determined by and , and are given by the formulafor all integers , whenever is a simple closed curve in the annulus with . Also establish the bounds

The following modification of the above exercise may help explain the terminology “Fourier inversion formula”.

Exercise 5 (Fourier inversion formula, again)Let .

- (i) Show that if is holomorphic on the annulus , then we have the Fourier expansion
for all , where the Fourier coefficients are given by the formula

Furthermore, show that the Fourier series in (7) is absolutely convergent, and the coefficients obey the asymptotic bounds (5), (6).

- (ii) Conversely, if are complex numbers obeying the asymptotic bounds (5), (6), show that there exists a function holomorphic on the annulus obeying the Fourier expansion (7) and the inversion formula (8).

The Laurent series for a given function can vary as one varies the annulus. Consider for instance the function . In the annulus , the Laurent expansion coincides with the Taylor expansion:

On the other hand, in the exterior region , the Taylor expansion is no longer convergent. Instead, if one writes and uses the geometric series formula, one instead has the Laurent expansion

in this region.

Exercise 6Find the Laurent expansions for the function in the regions , . (Hint:use partial fractions.)

We can use Laurent series to analyse an isolated singularity. Suppose that is holomorphic on a punctured disk . By the above discussion, we have a Laurent series expansion (4) in this punctured disk. If the singularity is removable, then the Laurent series must coincide with the Taylor series (by the uniqueness component of Exercise 4), so in partcular for all negative ; conversely, if vanishes for all negative , then the Laurent series matches up with a convergent Taylor series and so the singularity is removable. We then adopt the following classification:

- (i) has a
*removable singularity*at if one has for all negative . If furthermore there is an such that and for , we say that has a*zero of order*at (after removing the singularity). Zeroes of order are known as*simple zeroes*, zeroes of order are known as*double zeroes*, and so forth. - (ii) has a pole of order at for some if one has , and for all . Poles of order are known as
*simple poles*, poles of order are*double poles*, and so forth. - (iii) has an essential singularity if for infinitely many negative .

It is clear that any holomorphic function will be of exactly one of the above three categories. Also, from the uniqueness of Laurent series, shrinking does not affect which of the three categories will lie in (or what order of pole will have, in the second category). Thus, we can classify any isolated singularity of a holomorphic function with singularities as being either removable, a pole of some finite order, or an essential singularity by restricting to a small punctured disk and inspecting the Laurent coefficients for negative .

Example 7The function has a Laurent expansionand thus has an essential singularity at .

It is clear from the definition (and the holomorphicity of Taylor series) that (as discussed in the introduction), a holomorphic function has a pole of order at an isolated singularity if and only if it is of the form for some holomorphic with . Similarly, a holomorphic function would have a zero of order at if and only if for some with .

We can now define a class of functions that only have “nice” singularities:

Definition 8 (Meromorphic functions)Let be an open subset of . A function defined on outside of a singular set is said to be meromorphic on if

- (i) is closed and discrete (i.e., all points in are isolated); and
- (ii) Every is either a removable singularity or a pole of finite order.

Two meromorphic functions , are said to be *equivalent* if they agree on their common domain of definition . It is easy to see that this is an equivalence relation. It is common to identify meromorphic functions up to equivalence, similarly to how in measure theory it is common to identify functions which agree almost everywhere.

Exercise 9 (Meromorphic functions form a field)Let denote the space of meromorphic functions on an open set , up to equivalence. Show that is a field (with the obvious field operations).

Exercise 10 (Order is a valuation)If is a meromorphic function, and , define theorderof at as follows:

- (a) If has a removable singularity at , and has a zero of order at once the singularity is removed, then .
- (b) If is holomorphic at , and has a zero of order at , then .
- (c) If has a pole of order at , then .
- (d) If is identically zero, then .
Establish the following facts:

- (i) If and are equivalent meromorphic functions, then for all . In particular, one can meaningfully define the order of an element of at any point in , where is as in the preceding exercise.
- (ii) If and , show that . If is not zero, show that .
- (iii) If and , show that . Furthermore, show if , then the above inequality is in fact an equality.
In the language of abstract algebra, the above facts are asserting that is a valuation on the field .

The behaviour of a holomorphic function near an isolated singularity depends on the type of singularity.

- (i) If is a removable singularity of , then converges to a finite limit as .
- (ii) If is a pole of , then as .
- (iii) (Casorati-Weierstrass theorem) If is an essential singularity of , then every point of is a limit point of as , that is to say there exists a sequence converging to such that converges to (where we adopt the convention that converges to if converges to ).

*Proof:* Part (i) is obvious. Part (ii) is immediate from the factorisation and noting that converges to the non-zero value as . The case of (iii) follows from Riemann’s theorem on removable singularities (Exercise 35 from Notes 3). Now suppose is finite. If (iii) failed, then there exist such that avoids the disk on the domain . In particular, the function is bounded and holomorphic on , and thus extends holomorphically to by Riemann’s theorem. This function cannot vanish identically, so we must have on for some and some holomorphic that does not vanish at . Rearranging this as , we see that has a pole or removable singularity at , a contradiction.

In Theorem 56 below we will establish a significant strengthening of the Casorati-Weierstrass theorem known as the Great Picard Theorem.

Exercise 12Let be holomorphic in outside of a discrete set of singularities. Let . Show that the radius of convergence of the Taylor series of around is equal to the distance from to the nearest non-removable singularity in , or if no such non-removable singularity exists. (This fact provides a neat way to understand the rate of growth of a sequence : form its generating function , locate the singularities of that function, and find out how close they get to the origin. This is a simple example of the methods of analytic combinatorics in action.)

A curious feature of the singularities in complex analysis is that the order of singularity is “quantised”: one can have a pole of order , , or (for instance), but not a pole of order or . This quantisation can be exploited: if for instance one somehow knows that the order of the pole is less than for some integer and real number , then the singularity must be removable or a pole of order at most . The following exercise formalises this assertion:

for all . Show that the singularity of at is either removable, or a pole of order at most (the latter option is only possible when is positive). (

Hint:use Lemma 4 and a limiting argument to evaluate the Laurent coefficients for .) In particular, if one hasfor all , then the singularity is removable.

As mentioned in the introduction, the theory of meromorphic functions becomes cleaner if one replaces the complex plane with the Riemann sphere. This sphere is a model example of a Riemann surface, and we will now digress to briefly introduce this more general concept (though we will not develop the general theory of Riemann surfaces in any depth here). To motivate the definition, let us first recall from differential geometry the notion of a smooth -dimensional manifold (over the reals).

Definition 14 (Smooth manifold)Let , and let be a topological space. An (-dimensional real) atlas for is an open cover of together with a family of homeomorphisms (known ascoordinate charts) from each to an open subset of . Furthermore, the atlas is said to besmoothif for any , thetransition map, which maps one open subset of to another, is required to be smooth (i.e., infinitely differentiable). A map from one topological space (equipped with a smooth atlas of coordinate charts for ) to another (equipped with a smooth atlas of coordinate charts for some ) is said to besmoothif, for any and , the maps are smooth; if is invertible and and are both smooth, we say that is a diffeomorphism, and that and arediffeomorphic. Two smooth atlases on are said to beequivalentif the identity map from (equipped with one of the two atlases) to (equipped with the other atlas) is a diffeomorphism; this is easily seen to be an equivalence relation, and an equivalence class of such atlases is called asmooth structureon . Asmooth -dimensional real manifoldis a Hausdorff topological space equipped with a smooth structure. (In some texts the mild additional condition of second countability on is also imposed.) A map between two smooth manifolds is said to besmooth, if the map from (equipped with one of the atlases in the smooth structure on ) to (equipped with one of the atlases in the smooth structure on ) is smooth; it is easy to see that this definition is independent of the choices of atlas. We may similarly define the notion of a diffeomorphism between two smooth manifolds.

This definition may seem excessively complicated, but it captures the modern geometric philosophy that one should strive as much as possible to work with objects that are *coordinate-independent* in that they do not depend on which atlas of coordinate charts one picks within the equivalence class of the given smooth structure in order to perform computations or to define foundational concepts. One can also define smooth manifolds more abstractly, without explicit reference to atlases, by working instead with the structure sheaf of the rings of smooth real-valued functions on open subsets of the manifold , but we will not need to do so here.

Example 15A simple example of a smooth -dimensional manifold is the unit circle ; there are many equivalent atlases one could place on this circle to define the smooth structure, but one example would be the atlas consisting of the two charts , , defined by setting , , , , for , and for . Another smooth manifold, which turns out to be diffeomorphic to the unit circle , is the one-point compactification of the real numbers, with the two charts , defined by setting , , , to be the identity map, and defined by setting for and .

Exercise 16Verify that the unit circle is indeed diffeomorphic to the one-point compactification .

A Riemann surface is defined similarly to a smooth manifold, except that the dimension is restricted to be one, the reals are replaced with the complex numbers, and the requirement of smoothness is replaced with holomorphicity (thus Riemann surfaces are to the complex numbers as smooth curves are to the real numbers). More precisely:

By considering dimensions greater than one, one can arrive at the more general notion of a complex manifold, the study of which is the focus of complex geometry (and also plays a central role in the closely related fields of several complex variables and complex algebraic geometry). However, we will not need to deal with higher-dimensional complex manifolds in this course. The notion of a Riemann surface should not be confused with that of a Riemannian manifold, which is the topic of study of Riemannian geometry rather than complex geometry.

Clearly any open subset of the complex numbers is a Riemann surface, in which one can use the atlas that only consists of one “tautological” chart, the identity map . More generally, any open subset of a Riemann surface is again a Riemann surface. If are open subsets of the complex numbers, and is a map, then by unpacking all the definitions we see that is holomorphic in the sense of Definition 17 if and only if it is holomorphic in the usual sense.

Now we come to the Riemann sphere , which is to the complex numbers as is to the real numbers. As a set, this is the complex numbers with one additional point (the *point at infinity*) attached. Topologically, this is the one-point compactification of the complex numbers : the open sets of are either subsets of that were already open, or complements of compact subsets of . As a Riemann surface, the complex structure can be described by the atlas of coordinate charts , , where , , , is the identity map, and equals for with . It is not difficult to verify that this is indeed a Riemann surface (basically because the map is holomorphic on ). One can identify the Riemann sphere with a geometric sphere, and specifically the sphere , through the device of stereographic projection through the north pole , identifying a point in with the point on collinear with that point, and the point at infinity identified with the north pole . This geometric perspective is especially helpful when thinking about Möbius transformations, as is for instance exemplified by this excellent video. (We may cover Möbius transformations in a subsequent set of notes.)

By unpacking the definitions, we can now work out what it means for a function to be holomorphic to or from the Riemann sphere. For instance, if is a map from an open subset of to the Riemann sphere , then is holomorphic if and only if

- (i) is continuous;
- (ii) is holomorphic on the set (which is open thanks to (i)); and
- (iii) is holomorphic on the set (which is open thanks to (i)), where we adopt the convention .

Similarly, if a function is a map from an open subset of the Riemann sphere to the Riemann sphere, then is holomorphic if and only if

- (i) is holomorphic on ; and
- (ii) is holomorphic on , where we again adopt the convention .

We can then identify meromorphic functions with holomorphic functions on the Riemann sphere:

Exercise 18Let be open, let be a discrete subset of , and let be a function. Show that the following are equivalent:

- (i) is meromorphic on .
- (ii) is the restriction of a holomorphic function to the Riemann sphere.
Furthermore, if (ii) holds, show that is uniquely determined by , and is unaffected if one replaces with an equivalent meromorphic function.

Among other things, this exercise implies that the composition of two meromorphic functions is again meromorphic (outside of where the composition is undefined, of course).

Exercise 19Let be a holomorphic map from the Riemann sphere to itself. Show that is a rational function in the sense that there exist polynomials of one complex variable, with not identically zero, such that for all with . (Hint:show that has finitely many poles, and eliminate them by multiplying by appropriate linear factors. Then use Exercise 29 from Notes 3.)

Exercise 20 (Partial fractions)Let be a polynomial of one complex variable, which by the fundamental theorem of algebra we may write asfor some distinct roots , some non-zero , and some positive integers . Let be another polynomial of one complex polynomial. Show that there exist unique polynomials , with each having degree less than for , such that one has the partial fraction decomposition

for all . Furthermore, show that vanishes if the degree of is less than the degree of , and has degree otherwise.

** — 2. The residue theorem — **

Now we can prove a significant generalisation of the Cauchy theorem and Cauchy integral formula, known as the residue theorem.

Suppose one has a function holomorphic on an open set outside of a singular set . If is an isolated singularity of , then we have a Laurent expansion

which is convergent in some punctured disk . The coefficient plays a privileged role and is known as the residue of at ; we denote it by . Clearly this is quantity is local in the sense that it only depends on the behaviour of in a neighbourhood of ; in particular, it does not depend on the domain so long as remains inside of that domain. By convention, we also set if is holomorphic at (i.e., if ).

We then have

Theorem 21 (Residue theorem)Let be a simply connected open set, and let be holomorphic outside of a closed discrete singular set (thus all singularities in are isolated singularities). Let be a closed curve in . Thenwhere only finitely many of the terms on the right-hand side are non-zero.

*Proof:* The image of is contained in some large ball; restricting and to this ball, we may assume without loss of generality that is both discrete and compact, and thus finite (by the Bolzano-Weierstrass theorem).

Next, we reduce to the case where all the residues vanish. We introduce the rational function defined by

From Laurent expansion around each singularity we see that for all , thus . Also, from the definition of winding number (see Definition 38 of Notes 3) we have

Setting , it thus suffices to show that

As is simply connected, is homotopic in (as closed contours) to a point. Let denote the homotopy. We would like to mimic the proof of Cauchy’s theorem (Theorem 4 of Notes 3) to conclude (9). The difficulty is that the homotopy may pass through points in . However, note from the vanishing of the residue that one has a Laurent expansion of the form

for some coefficients , in some punctured disk , with both series being absolutely convergent in this punctured disk. From term by term differentiation (see Theorem 15 of Notes 1) we see that has an antiderivative in this punctured disk, namely

(note how crucial it is that the term is absent in order to form this antiderivative). The absolute convergence of the series on the right-hand side in can be seen from the comparison test. From the fundamental theorem of calculus, we thus conclude that is conservative on . Also, for any that is *not* in , we see from Cauchy’s theorem that is conservative on for some radius . Putting this together using a compactness argument, we conclude that there exists a radius , such that for all in the image of the homotopy , the function is conservative in .

Now we repeat the proof of Cauchy’s theorem (Theorem 4 of Notes 3), discretising the homotopy into short closed polygonal paths (each of diameter less than ) around which the integral of is zero, to conclude (9). The argument is completely analogous, save for the technicality that the paths may occasionally pass through one of the points in . But this can be easily rectified by perturbing each of the paths by adding a short detour around any point of that is passed through; we leave the details to the interested reader.

Combining the residue theorem with the Jordan curve theorem, we obtain the following special case, which is already enough for many applications:

Corollary 22 (Residue theorem for simple closed contours)Let be a simple closed anticlockwise contour in . Suppose that is holomorphic on an open set containing the image and interior of , outside of a closed discrete that does not intersect the image of . Then we haveIf is oriented clockwise instead of anticlockwise, then we instead have

Exercise 23 (Homology version of residue theorem)Show that the residue theorem continues to hold when the closed curve is replaced by a -cycle (as in Exercise 63 of Notes 3) that avoids all the singularities in , and the requirement that be simply connected is replaced by the requirement that contains all the points outside of the image of where .

Exercise 24 (Exterior version of residue theorem)Let be a simple closed anticlockwise contour in . Suppose that is holomorphic on an open set containing the image andexteriorof , outside of a finite that does not intersect the image of . Suppose also that converges to a finite limit in the limit . Show thatIf is oriented clockwise instead of anticlockwise, show instead that

In order to use the residue theorem effectively, one of course needs some tools to compute the residue at a given point. The Fourier inversion formula (4) expresses such residues as a contour integral, but this is not so useful in practice as often the best way to compute such integrals is via the residue theorem, leaving one back where one started! But if the singularity is not an essential one, we have some useful formulae:

Exercise 25Let be holomorphic on an open set outside of a singular set , and let be an isolated point of .

- (i) If has a removable singularity at , show that .
- (ii) If has a simple pole at , show that .
- (iii) If has a pole of order at most at for some , show that
In particular, if near for some that is holomorphic at , then

Using these facts, show that Cauchy’s theorem (Theorem 14 from Notes 3), the Cauchy integral formula (Theorem 39 from Notes 3), and the higher order Cauchy integral formula (Exercise 40 from Notes 3) can be derived from the residue theorem. (Of course, this is not an independent proof of these theorems, as they were used in the

proofof the residue theorem!)

The residue theorem can be applied in countless ways; we give only a small sample of them below.

Exercise 26Use the residue theorem to give an alternate proof of the fundamental theorem of algebra, by considering the integral for a polynomial of degree and some large radius .

Exercise 27Let be a Dirichlet polynomial of the formfor some sequence of complex numbers, with only finitely many of the non-zero. Establish Perron’s formula

for any real numbers with not an integer. What happens if is an integer? Generalisations and variants of this formula, particularly with the Dirichlet polynomial replaced by more general Dirichlet series in which infinitely many of the are allowed to be non-zero, are of particular use in analytic number theory; see for instance this previous blog post.

Exercise 28 (Spectral theorem for matrices)This exercise presumes some familiarity with linear algebra. Let be a positive integer, and let denote the ring of complex matrices. Let be a matrix in . The characteristic polynomial , where is the identity matrix, is a polynomial of degree in with leading coefficient ; we let be the distinct zeroes of this polynomial, and let be the multiplicities; thus by the fundamental theorem of algebra we haveWe refer to the set as the spectrum of . Let be any closed anticlockwise curve that contains the spectrum of in its interior, and let be an open subset of that contains and its interior.

- (i) Show that the resolvent is a meromorphic function on with poles at the spectrum of , where we call a matrix-valued function meromorphic if each of its components are meromorphic. (
Hint:use the adjugate matrix.)- (ii) For any holomorphic , we define the matrix by the formula
(cf. the Cauchy integral formula). We refer to as the holomorphic functional calculus for applied to . Show that the matrix does not depend on the choice of , depends linearly on , and equals the identity matrix when is the constant function . Furthermore, if is the function , show that

Conclude in particular that if is a polynomial

with complex coefficients , then the function (as defined by the holomorphic functional calculus) matches how one would define algebraically, in the sense that

- (iii) Prove the Cayley-Hamilton theorem . (Note from (ii) that it does not matter whether one interprets algebraically, or via the holomorphic functional calculus.)
- (iv) If is holomorphic, show that the matrix-valued function has only removable singularities in .
- (v) If are holomorphic, establish the identity
- (vi) Show that there exist matrices that are idempotent (thus for all ), commute with each other and with , sum to the identity (thus ), annihilate each other (thus for all distinct ) and are such that for each , one has the nilpotency property
In particular, we have the

spectral decompositionwhere each is a nilpotent matrix with . Finally, show that the range of (viewed as a linear operator from to itself) has dimension . Find a way to interpret each as the (negative of the) “residue” of the resolvent operator at .

Under some additional hypotheses, it is possible to extend the analysis in the above exercise to infinite-dimensional matrices or other linear operators, but we will not do so here.

** — 3. The argument principle — **

We have not yet defined the complex logarithm of a complex number , but one of the properties we would expect of this logarithm is that its derivative should be the reciprocal function: . In particular, by the chain rule we would expect the formula

for a holomorphic function , at least away from the zeroes of . Inspired by this formal calculation, we refer to the function as the log-derivative of . Observe the product rule and quotient rule, when applied to complex differentiable functions that are non-zero at some point , gives the formulae

which are of course consistent with the formal calculation (10), given how we expect the logarithm to act on products and quotients. Thus, for instance, if , are polynomials that are factored as

and

for some non-zero complex numbers , distinct complex numbers , and positive integers , then the log-derivative of the rational function is given by

In particular, the log-derivative of is meromorphic with poles at , with a residue of at each zero of , and a residue of at each pole of .

A general rule of thumb in complex analysis is that holomorphic functions behave like generalisations of polynomials, and meromorphic functions behave like generalisations of rational functions. In view of this rule of thumb and the above calculation, the following lemma should thus not be surprising:

- (i) If is holomorphic and non-zero at , then the log-derivative is also holomorphic at .
- (ii) If is holomorphic at with a zero of order , then the log-derivative has a simple pole at with residue .
- (iii) If has a removable singularity at , and is non-zero once the singularity is removed, then the log-derivative has a removable singularity at .
- (iv) If has a removable singularity at , and has a zero of order once the singularity is removed, then the log-derivative has a simple pole at with residue .
- (v) If has a pole of order at , then the log-derivative has a simple pole at with residue .

*Proof:* The claim (i) is obvious. For (ii), we use Taylor expansion to factor for some holomorphic and non-zero near , and then from (11) we have

Since is holomorphic at , the claim (ii) follows. The claim (v) is proven similarly using a factorisation , and using (12) in place of (11). The claims (iii), (iv) then follow from (i), (ii) respectively after removing the singularity.

Remark 30Note that the lemma does not cover all possible singularity and zero scenarios. For instance, could be identically zero, in which case the log-derivative is nowhere defined. If has an essential singularity then the log-derivative can be a pole (as seen for instance by the example for some ) or another essential singularity (as can be seen for instance by the example ). Finally, if has a non-isolated singularity, then the log-derivative could exhibit a wide range of behaviour (but probably will be quite wild as one approaches the singular set).

By combining the above lemma with the residue theorem, we obtain the argument principle:

Theorem 31 (Argument principle)Let be a simple closed anticlockwise contour. Let be an open set containing and its interior. Let be a meromorphic function on that is holomorphic and non-zero on the image of . Suppose that after removing all the removable singularities of , has zeroes in the interior of (of orders respectively), and poles in the interior of (of orders respectively). ( is also allowed to have zeroes and poles in the exterior of .) Then we havewhere is the closed contour .

*Proof:* The first equality of (13) follows from the residue theorem and Lemma 29. From the change of variables formula (Exercise 16(ix) of Notes 2) we have

and the second identity also follows.

We isolate the special case of the argument principle when there are no poles for special mention:

Corollary 32 (Special case of argument principle)Let be a simple closed anticlockwise contour, let be an open set containing the image of and its interior, and let be holomorphic. Suppose that has no zeroes on the image of . Then the number of zeroes of (counting multiplicity) in the interior of is equal to the winding number of around the origin.

Recalling that the winding number is a homotopy invariant (Lemma 41 of Notes 3), we conclude that the number of zeroes of a holomorphic function in the interior of a simple closed anticlockwise contour is also invariant with respect to continuous perturbations, so long as zeroes never cross the contour itself. More precisely:

*Proof:* By Corollary 32, it suffices to show that

But the curves and are homotopic as closed curves in , using the homotopy defined by

(note that this avoids the origin by hypothesis). The claim then follows from Lemma 41 of Notes 3.

Informally, the above corollary asserts that zeroes of holomorphic functions cannot be created or destroyed, as long as they are confined within a closed contour.

Example 34Let be the unit circle . The polynomial has a double zero at , so (counting multiplicity) has two zeroes in the interior of . If we consider instead the perturbation for some , this has simple zeroes at and respectively, so as long as , the holomorphic function also has two zeroes in the interior of ; but as crosses , the zeroes of pass through , and one no longer has any zeroes of in the interior of . The situation can be contrasted with the real case: the function has a double zero at the origin when , but as soon as becomes positive, the zeroes immediately disappear from the real line. Note that the stability of zeroes fails if we do not count zeroes with multiplicity; thus, as a general rule of thumb, one should always try to count zeroes with multiplicity when doing complex analysis. (Heuristically, one can think of a zero of order as simple zeroes that are “infinitesimally close together".)

Example 35When one considers meromorphic functions instead of holomorphic ones, then the number of zeroes inside a region need not be stable any more, but the number of zeroesminusthe number of poles will be stable. Consider for instance the meromorphic function , which has a removable singularity at but no zeroes or poles. If we perturb it to for some , then we suddenly have a double pole at , but this is balanced by two simple zeroes at and ; in the limit as we see that the two zeroes “collide” with the double pole, annihilating both the zeroes and the poles.

A particularly useful special case of the stability of zeroes is Rouche’s theorem:

Theorem 36 (Rouche’s theorem)Let be a simple closed contour, and let be an open set containing the image of and its interior. Let be holomorphic. If one has for all in the image of , then and have the same number of zeroes (counting multiplicity) in the interior of .

*Proof:* We may assume without loss of generality that is anticlockwise. By hypothesis, and cannot have zeroes on the image of . The claim then follows from Corollary 33 with , , , , and .

Rouche’s theorem has many consequences for complex analysis. One basic consequence is the open mapping theorem:

Theorem 37 (Open mapping theorem)Let be an open connected non-empty subset of , and let be holomorphic and not constant. Then is also open.

*Proof:* Let . As is not constant, the zeroes of are isolated (Corollary 24 of Notes 3). Thus, for sufficiently small, is nonvanishing on the image of the circle . Clearly has at least one zero in the interior of this circle. Thus, by Rouche’s theorem, if is sufficiently close to , then will also have at least one zero in the interior of this circle. In particular, contains a neighbourhood of , and the claim follows.

Exercise 38Use Rouche’s theorem to obtain another proof of the fundamental theorem of algebra, by showing that a polynomial with and has exactly zeroes (counting multiplicity) in the complex plane. (Hint:compare with inside some large circle .)

Exercise 39 (Inverse function theorem)Let be an open subset of , let , and let be a holomorphic function such that . Show that there exists a neighbourhood of in such that the map is a complex diffeomorphism; that is to say, it is holomorphic, invertible, and the inverse is also holomorphic. Finally, show thatfor all . (

Hint:one can either mimic the real-variable proof of the inverse function theorem using the contraction mapping theorem, or one can use Rouche’s theorem and the open mapping theorem to construct the inverse.)

Exercise 40Let be an open subset of , and be a map. Show that the following are equivalent:

- (i) is a local complex diffeomorphism. That is to say, for every there is a neighbourhood of in such that is open and is a complex diffeomorphism (as defined in the preceding exercise).
- (ii) is holomorphic on and is a local homeomorphism. That is to say, for every there is a neighbourhood of in such that is open and is a homeomorphism.
- (iii) is holomorphic on and is a local injection. That is to say, for every there is a neighbourhood of in such that is injective.
- (iv) is holomorphic on , and the derivative is nowhere vanishing.

Exercise 41 (Hurwitz’s theorem)Let be an open connected non-empty subset of , and let be a sequence of holomorphic functions that converge uniformly on compact sets to a limit (which is then necessarily also holomorphic, thanks to Theorem 34 of Notes 3). Prove the following two versions of Hurwitz’s theorem:

- (i) If none of the have any zeroes in , show that either also has no zeroes in , or is identically zero.
- (ii) If all of the are univalent (that is to say, they are
injectiveholomorphic functions), show that either is also univalent, or is constant.

Exercise 42 (Bloch’s theorem)The purpose of this exercise is to establish a more quantitative variant of the open mapping theorem, due to Bloch; this will be useful later in this notes for proving the Picard and Montel theorems. Let be a holomorphic function on a disk , and suppose that is non-zero

- (i) Suppose that for all . Show that there is an absolute constant such that contains the disk . (
Hint:one can normalise , , . Use the higher order Cauchy integral formula to get some bound on for near the origin, and use this to approximate by near the origin. Then apply Rouche’s theorem.)- (ii) Without the hypothesis in (i), show that there is an absolute constant such that contains a disk of radius . (
Hint:if one has for all , then we can apply (i) with replaced by . If not, pick with , and start over with replaced by and replaced by . One cannot iterate this process indefinitely as it will create a singularity of in .)

** — 4. Branches of the complex logarithm — **

We have refrained until now from discussing one of the most basic transcendental functions in complex analysis, the complex logarithm. In real analysis, the real logarithm can be defined as the inverse of the exponential function ; it can also be equivalently defined as the antiderivative of the function , with the initial condition . (We use here for the real logarithm in order to distinguish it from the complex logarithm below.)

Let’s see what happens when one tries to extend these definitions to the complex domain. We begin with the inversion of the complex exponential. From Euler’s formula we have that ; more generally, we have whenever for some integer . In particular, the exponential function is not injective. Indeed, for any non-zero , we have a *multi-valued* logarithm

which, by Euler’s formula, can be written as

where

denotes all the possible arguments of in polar form. These arguments are a coset of the group , and so the complex logarithm is a coset of the group . For instance, if , then

and

The complex exponential never vanishes, so by our definitions we see that is the empty set. As such, we will usually omit the origin from the domain when discussing the complex exponential.

Of course, one also encounters multi-valued functions in real analysis, starting when one tries to invert the squaring function , as any given positive number has two square roots. In the real case, one can eliminate this multi-valuedness by picking a branch of the square root function – a function which selects one of the multiple choices for that function at each point in the domain. In particular, we have the positive branch of the square root function on , as well as the negative branch . One could also create more discontinuous branches of the square root function, for instance the function that sends to for , and to for .

Suppose now that we have a branch of the logarithm function, thus

for any . If is complex differentiable at some point , then by differentiating (14) at using the chain rule, we see that

and hence by (14) again we have

(which is of course consistent with the real-variable formula ). If now is a closed contour in , and is differentiable on the entire image of , then the fundamental theorem of calculus then tells us that

On the other hand, is equal to . We thus conclude that for any branch of the complex logarithm, the set on which is complex differentiable cannot contain any closed curve that winds non-trivially around the origin. Thus for instance one cannot find a branch of that is holomorphic on all of , or even on a neighbourhood of the unit circle (or any other curve going around the origin).

On the other hand, if is a *simply connected* open subset of , then from Cauchy’s theorem the function is conservative on . If we pick a point in and arbitrarily select a logarithm of , we can then use the fundamental theorem of calculus to find an antiderivative of on with . By definition, is holomorphic, and from the chain rule we have for all that

and hence by the quotient rule

As is connected, must therefore be constant; by construction we have , and thus

for all . In other words, is a branch of the complex logarithm.

Thus, for instance, the region formed by excluding the negative real axis from the complex plane is simply connected (it is star-shaped around ), and so must admit a holomorphic branch of the complex logarithm. One such branch is the *standard branch* of the complex logarithm, defined as

where is the *standard branch* of the argument, defined as the unique argument in in the interval . This branch of the logarithm is continuous on , and hence (by the exercise below) is holomorphic on this region, and is thus an antiderivative of here. Similarly if one replaces the negative real axis by other rays emenating from the origin (or indeed from arbitrary simple curves from zero to infinity, see Exercise 44 below.)

Exercise 43Let be a connected non-empty open subset of .

- (i) If and are continuous branches of the complex logarithm, show that there exists a natural number such that for all .
- (ii) Show that any continuous branch of the complex logarithm is holomorphic.
- (iii) Show that there is a continuous branch of the logarithm if and only if and lie in the same connected component of . (
Hint:for the “if” direction, use a continuity argument to show that the winding number of any closed curve in around vanishes. For the “only if”, encircle the connected component of in (which is a compact subset of by hypothesis) by a simple polygonal path in .)

Exercise 44Let be a continuous injective map with and as .

- (i) Show that is not all of . (
Hint:modify the construction in Section 4 of Notes 3 that showed that a simple closed curve admitted at least one point with non-zero winding number.)- (ii) Show that the complement is simply connected. (
Hint:modify the remaining arguments in Section 4 of Notes 3). In particular, by the preceding discussion, there is a branch of the complex logarithm that is holomorphic outside of .

It is instructive to view the identity

, through the lens of branches of the complex logarithm such as the standard branch . From the fundamental theorem of calculus, one has

for any curve that avoids the negative real axis. Of course, the contour does not avoid this negative axis, but it can be approximated by (non-closed) contours that do. More precisely, one has

where is the map . As each avoids the negative real axis, we thus have

We observe that has a jump discontinuity of on the negative real axis, and specifically

and

which gives an alternate derivation of the identity (15). More generally, the identity

for any closed curve avoiding the origin can be interpreted using the standard branch of the logarithm as a version of the Alexander numbering rule (Exercise 55 of Notes 3): each crossing of across the branch cut triggers a jump up or down in the count towards the winding number, depending on whether the crossing was in the anticlockwise or clockwise direction.

One can use branches of the complex logarithm to create branches of the root functions for natural numbers . As with the complex exponential, the function is not injective, and so is multivalued (see Exercise 15 of Notes 0). One cannot form a continuous branch of this function on for any , as the corresponding branch of would then contradict the quantisation of order of singularities (Exercise 13). However, on any domain where there is a holomorphic branch of the complex logarithm, one can define a holomorphic branch of the function by the formula

It is easy to see that is indeed holomorphic with for all . Thus for instance we have the standard branch of the root function, which is holomorphic away from the negative real axis. More generally, one can define a “standard branch of ” for any complex by the formula , for instance the standard branch of can be computed to be .

The presence of branch cuts can prevent one from directly applying the residue theorem to calculate integrals involving branches of multi-valued functions. But in some cases, the presence of the branch cut can actually be *exploited* to compute an integral. The following exercise provides an example:

Exercise 45Compute the improper integralby applying the residue theorem to the function for some branch of with branch cut on the

positivereal axis, and using a “keyhole” contour that is a perturbation ofthe key point is that the branch cut makes the contribution of (the perturbations) of and fail to cancel each other.

The construction of holomorphic branches of can be extended to other logarithms:

Exercise 46Let be a simply connected subset of , and let be a holomorphic function with no zeroes on .

- (i) Show that there exists a holomorphic branch of the complex logarithm , thus .
- (ii) Show that for any natural number , there exists a holomorphic branch of the root function , thus .

Actually, one can invert other non-injective holomorphic functions than the complex exponential, provided that these functions are a covering map. We recall this topological concept:

Definition 47 (Covering map)Let be a continuous map between two connected topological spaces . We say that is acovering mapif, for each , there exists an open neighbourhood of in such that the preimage is the disjoint union of open subsets of , such that for each , the map is a homeomorphism. In this situation, we call acovering spaceof .

In complex analysis, one specialises to the situation in which are Riemann surfaces (e.g. they could be open subsets of ), and is a holomorphic map. In that case, the homeomorphisms are in fact complex diffeomorphisms, thanks to Exercise 40.

Example 48The exponential map is a covering map, because for any element of written in polar form as , one can pick (say) the neighbourhoodof , and observe that the preimage of is the disjoint union of the open sets

for , and that the exponential map is a diffeomorphism. A similar calculation shows that for any natural number , the map is a covering map from to . However, the map is

nota covering map from to , because it fails to be a local diffeomorphism at zero due to the vanishing derivative (here we use Exercise 40). One final (non-)example: the map isnota covering map from the upper half-plane to , because the preimage of any small disk around splits into two disconnected regions, and only one of them is homeomorphic to via the map .

From topology we have the following lifting property:

Lemma 49 (Lifting lemma)Let be a continuous covering map between two path-connected and locally path-connected topological spaces . Let be a simply connected and path connected topological space, and let be continuous. Let , and let be such that . Then there exists a unique continuous map such that and , which we call aliftof by .

*Proof:* We first verify uniqueness. If we have two continuous functions with and , then the set is clearly closed in and contains . From the covering map property we also see that is open, and hence by connectedness we have on all of , giving the claim.

To verify existence of the lift, we first prove the existence of monodromy. More precisely, given any curve with we show that there exists a unique curve such that and (the reader is encouraged to draw a picture to describe this situation). Uniqueness follows from the connectedness argument used to prove uniqueness of the lift , so we turn to existence. As in previous notes, we rely on a continuity argument. Let be the set of all for which there exists a curve such that , where is the restriction of to . Clearly is closed in and contains ; using the covering map property it is not difficult to show that is also open in . Thus is all of , giving the claim.

Now let , be homotopic curves with fixed endpoints, with initial point and some terminal point , and let be a homotopy. For each , we have a curve given by , and by the preceding paragraph we can associate a curve such that and . Another application of the continuity method shows that for all , the map is continuous; in particular, the map . On the other hand, lies in , which is a discrete set thanks to the covering map property. We conclude that is constant in , and in particular that .

Since is simply connected, any two curves with fixed endpoints are homotopic. We can thus define a function by declaring for any to be the point , where is any curve from to , and is constructed as before. By construction we have , and from the local path connectedness of and the covering map property of we can check that is continuous. The claim follows.

We can specialise this to the complex case and obtain

Corollary 50 (Holomorphic lifting lemma)Let be a holomorphic covering map between two path-connected Riemann surfaces . Let be a simply connected and path connected Riemann surface, and let be holomorphic. Let , and let be such that . Then there exists a unique holomorphic map such that and , which we call aliftof by .

*Proof:* A Riemann surface is automatically locally path-connected, and a connected Riemann surface is automatically path connected (observe that the set of all points on the surface that can be path-connected to a reference point is open, closed, and non-empty). Applying Lemma 49, we obtain all the required claims, except that the lift produced is only known to be continuous rather than holomorphic. But then we can locally express as the composition of one of the local inverses of with . Applying Exercise 40, these local inverses are holomorphic, and so is holomorphic also.

Remark 51It is also possible to establish the above corollary using the monodromy theorem and analytic continuation.

Exercise 53Let be simply connected, and let be holomorphic and avoid taking the values . Show that there exists a holomorphic function such that . (This can be proven either through Corollary 50, or by using the quadratic formula to solve for and then applying Exercise 46.)

In some cases it is also possible to obtain lifts in non-simply connected domains:

Exercise 54Show that there exists a holomorphic function such that for all . (Hint:use the Schwartz reflection principle, see Exercise 37 of Notes 3.)

As an illustration of what one can do with all this machinery, let us now prove the Picard theorems. We begin with the easier “little” Picard theorem.

Theorem 55 (Little Picard theorem)Let be entire and non-constant. Then omits at most one point of .

The example of the exponential function , whose range omits the origin, shows that one cannot make any stronger conclusion about .

*Proof:* Suppose for contradiction that we have an entire non-constant function such that omits at least two points. After applying a linear transformation, we may assume that avoids and , thus takes values in .

At this point, the most natural thing to do from a Riemann surface point of view would be to cover by a bounded region, so that Liouville’s theorem may be applied. This can be done easily once one has the machinery of elliptic functions; but as we do not have this machinery yet, we will instead use a more *ad hoc* covering of using the exponential and trigonometric functions to achieve a passable substitute for this strategy.

We turn to the details. Since avoids , we may apply Exercise 46 to write for some entire . As avoids , must avoid the integers .

Next, we apply Exercise 53 to write for some entire . The set must now avoid all complex numbers of the form for natural numbers and integers . In particular, if is large enough, we see that does not contain any disk of the form . Applying Bloch’s theorem (Exercise 42(ii)) in the contrapositive, we conclude that for any disk in , one has for some absolute constant . Sending to infinity and using the fundamental theorem of calculus, we conclude that is constant, hence and are also constant, a contradiction.

Now we prove the more difficult “great” Picard theorem.

Theorem 56 (Great Picard theorem)Let be holomorphic on a disk outside of a singularity at . If this singularity is essential, then omits at most one point of .

Note that if one only has a pole at , e.g. if for some natural number , then the conclusion of the great Picard theorem fails. This result easily implies both the little Picard theorem (because if is entire and non-polynomial, then has an essential singularity at the origin) and the Casorati-Weierstrass theorem (Theorem 11(iii)). By repeatedly passing to smaller neighbourhoods, one in fact sees that with at most one exception, every complex number is attained infinitely often by a function holomorphic in a punctured disk around an essential singularity.

*Proof:* This will be a variant of the proof of the little Picard theorem; it would again be more natural to use elliptic functions, but we will use some passable substitutes for such functions concocted in an *ad hoc* fashion out of exponential and trigonometric functions.

Assume for contradiction that has an essential singularity at and avoids at least two points in . Applying linear transformations to both the domain and range of , we may normalise , , and assume that avoids and , thus we have a holomorphic map with an essential singularity at .

The domain is not simply connected, so we work instead with the function

defined by

Clearly is holomorphic on the right-half plane and avoids . We also observe that obeys the periodicity property

As the right-half plane is simply connected, we may (as before) express for some holomorphic function , and then write for some holomorphic function that avoids all numbers of the form for natural numbers and integers . Using Bloch’s theorem as before, we see that for any disk in the right-half plane , we have for some absolute constant . We cannot set to infinity any more, but we can make as large as the real part of , giving the bound

In particular, on integrating along a line segment from to , and using the boundedness of on the compact set , we obtain a bound of the form

for some , and all and . Taking cosines using the formula , we obtain a polynomial type bound

On the other hand, from (16) one has

and hence

for all in the right half-plane. The set is discrete, the function is continuous, and the right half-plane is connected, so this function must in fact be constant. That is to say, there exists an integer such that

for all in the upper half plane. Equivalently, the function is periodic with period . From (17) and the triangle inequality we conclude that

We now upgrade this bound on (18) by exploiting the quantisation of pole orders (Exercise 13). As the function is periodic with period on the right half-plane, we may write

for some function , which is holomorphic thanks to the chain rule. From (18) we have

when . Applying Exercise 13 (with, say, and ), we conclude that has a removable singularity and is thus in particular bounded on (say) the disk . From (19) we conclude that is bounded on the region ; taking exponentials, we conclude that is also bounded on this region. Since , we conclude that is bounded on , and thus by Riemann’s theorem (Exercise 35 from Notes 3) has a removable singularity at the origin. But by taking Laurent series, this implies that has a pole of order at most at the origin, contradicting the hypothesis that the singularity of at the origin was essential.

Exercise 57 (Montel’s theorem)Let be an open subset of the complex plane. Define a holomorphic normal family on to be a collection of holomorphic functions with the following property: given any sequence in , there exists a subsequence which is uniformly convergent on compact sets (i.e., for every compact subset of , the sequence converges uniformly on to some limit). Similarly, define ameromorphic normal familyto be a collection of meromorphic functions such that for any sequence in , there exists a subsequence that are uniformly convergent on compact sets, using the metric on the Riemann sphere induced by the identification with the geometric sphere . (More succinctly, normal families are those families of holomorphic or meromorphic functions that are precompact in the locally uniform topology.)

- (i) (Little Montel theorem) Suppose that is a collection of holomorphic functions that are uniformly bounded on compact sets (i.e., for each compact there exists a constant such that for all and ). Show that is a holomorphic normal family. (
Hint:use the higher order Cauchy integral formula to establish some equicontinuity on this family on compact sets, then use the Arzelá-Ascoli theorem.- (ii) (Great Montel theorem) Let be three distinct elements of the Riemann sphere, and suppose that is a family of meromorphic functions which avoid the three points . Show that is a meromorphic normal family. (
Hint:use some elementary transformations to reduce to the case . Then, as in the proof of the Picard theorems, express each element of locally in the form and use Bloch’s theorem to get some uniform bounds on .)

Exercise 58 (Harnack principle)Let be an open connected subset of , and let be a sequence of harmonic functions which is pointwise nondecreasing (thus for all and ). Show that is either infinite everywhere on , or is harmonic. (Hint:work locally in a disk. Write each on this disk as the real part of a holomorphic function , and apply Montel’s theorem followed by the Hurwitz theorem to .) This result is known as Harnack’s principle.

Exercise 59

- (i) Show that the function is harmonic on but has no harmonic conjugate.
- (ii) Let , and let be a harmonic function obeying the bounds
for all and some constants . Show that there exists a real number and a harmonic function such that

for all . (

Hint:one can find a conjugate of outside of some branch cut, say the negative real axis restricted to . Adjust by a multiple of until the conjugate becomes continuous on this branch cut.)

Exercise 60 (Local description of holomorphic maps)Let be a holomorphic function on an open subset of , let be a point in , and suppose that has a zero of order at for some . Show that there exists a neighbourhood of in on which one has the factorisation , where is holomorphic with a simple zero at (and hence a complex diffeomorphism from a sufficiently small neighbourhood of to a neighbourhood of ). Use this to give an alternate proof of the open mapping theorem (Theorem 37).

Exercise 61 (Winding number and lifting)Let , let be a closed curve avoiding , and let be an integer. Show that the following are equivalent:

- (i) .
- (ii) There exists a complex number and a curve from to such that for all .
- (iii) is homotopic up to reparameterisation as closed curves in to the curve that maps to for some .

Filed under: 246A - complex analysis, math.AT, math.CA, math.CV, math.DG Tagged: argument principle, branch cut, complex logarithm, Riemann sphere, Rouche's theorem, singularity

Funny as it is always the usual suspects. ALEPH had a reputation for anomaly chasing in the past, and apparently members of the collaboration, although nowadays dispersed in other experiments, continue to show the clear signs of bump-happiness that distinguished them in the LEP era.

What am I talking about ? ALEPH is one of the four experiments that instrumented the Large Electron-Positron collider, LEP, a machine constructed at CERN in the late eighties to study the properties of the Z boson.

What am I talking about ? ALEPH is one of the four experiments that instrumented the Large Electron-Positron collider, LEP, a machine constructed at CERN in the late eighties to study the properties of the Z boson.

This summer there was a blog post from Sabine Hossenfelder claiming that “The LHC `nightmare scenario’ has come true” — implying that the Large Hadron Collider [LHC] has found nothing but a Standard Model Higgs particle (the simplest possible type), and will … Continue reading

This summer there was a blog post from Sabine Hossenfelder claiming that “The LHC `nightmare scenario’ has come true” — implying that the Large Hadron Collider [LHC] has found nothing but a Standard Model Higgs particle (the simplest possible type), and will find nothing more of great importance. With all due respect for the considerable intelligence and technical ability of the author of that post, I could not disagree more; not only are we not in a nightmare, it isn’t even night-time yet, and hardly time for sleep or even daydreaming. There’s a tremendous amount of work to do, and there may be many hidden discoveries yet to be made, lurking in existing LHC data. Or elsewhere.

I can defend this claim (and have done so as recently as this month; here are my slides). But there’s evidence from another quarter that it is far too early for such pessimism. It has appeared in a new paper (a preprint, so not yet peer-reviewed) by an experimentalist named Arno Heister, who is evaluating 20-year old data from the experiment known as ALEPH.

In the early 1990s the Large Electron-Positron (LEP) collider at CERN, in the same tunnel that now houses the LHC, produced nearly 4 million Z particles at the center of ALEPH; the Z’s decayed immediately into other particles, and ALEPH was used to observe those decays. Of course the data was studied in great detail, and you might think there couldn’t possibly be anything still left to find in that data, after over 20 years. But a hidden gem wouldn’t surprise those of us who have worked in this subject for a long time — especially those of us who have worked on hidden valleys. *(Hidden Valleys are theories with a set of new forces and low-mass particles, which, because they aren’t affected by the known forces excepting gravity, interact very weakly with the known particles. They are also often called “dark sectors” if they have something to do with dark matter.)*

For some reason most experimenters in particle physics don’t tend to look for things just because they can; they stick to signals that theorists have already predicted. Since hidden valleys only hit the market in a 2006 paper I wrote with then-student Kathryn Zurek, long after the experimenters at ALEPH had moved on to other experiments, nobody went back to look in ALEPH or other LEP data for hidden valley phenomena (with one exception.) I didn’t expect anyone to ever do so; it’s a lot of work to dig up and recommission old computer files.

This wouldn’t have been a problem if the big LHC experiments (ATLAS, CMS and LHCb) had looked extensively for the sorts of particles expected in hidden valleys. ATLAS and CMS especially have many advantages; for instance, the LHC has made over a hundred times more Z particles than LEP ever did. But despite specific proposals for what to look for (and a decade of pleading), only a few limited searches have been carried out, mostly for very long-lived particles, for particles with mass of a few GeV/c² or less, and for particles produced in unexpected Higgs decays. And that means that, yes, hidden physics could certainly still be found in old ALEPH data, and in other old experiments. Kudos to Dr. Heister for taking a look.

Now, has he actually **found** something hidden at ALEPH? It’s far too early to say. Dr. Heister is careful not to make a strong claim: his paper refers to an observed excess, not to the discovery of or even evidence for anything. But his analysis can be interpreted as showing a hint of a new particle (let’s call it the **V particle**, just to have a name for it) decaying sometimes to a muon and an anti-muon, and probably also sometimes to an electron and an anti-electron, with a rest mass about 1/3 of that of the Z particle — about 30 GeV/c². Here’s one of the plots from his paper, showing the invariant mass of the muon and anti-muon in Z decays that also have evidence of a bottom quark and a bottom anti-quark (each one giving a jet of hadrons that has been “b-tagged”). There’s an excess at about 30 GeV.

The simplest physical effect that would produce such a bump is a new particle; indeed this is how the Z particle itself was identified, over three decades ago.

However, the statistical significance of the bump is still only (after look-elsewhere effect) at most 3 standard deviations, according to the paper. So this bump could just be a fluke; we’ve seen similar ones disappear with more data, for example this one. There are also a couple of serious issues that will give experts pause *(the width of the bump is surprisingly large; the angular correlations seem consistent with background rather than a new signal; etc.)* So the data itself is not enough to convince anyone, including Dr. Heister, though it is certainly interesting.

Conversely it is intriguing that the bump in the plot above is observed in events with bottom quarks. It is common for hidden valleys *(including everything from a simple abelian Higgs models to more complex confining models)* to contain

- at least one spin-one particle V (which can decay to muon/anti-muon or electron/positron) and
- at least one spin-zero particle S (which can decay to bottom/anti-bottom preferentially, with occasional decays to tau/anti-tau.)

For example, in such models, a rare decay such as Z ⇒ V + S, producing a muon/anti-muon pair plus two bottom quark/anti-quark jets, would often be a possibility.*

**[In this case the bottom and anti-bottom jets would themselves show a peak in their invariant mass, but unfortunately their distribution in the presence of a candidate V was not shown. One other obvious prediction of such a model is a handful of striking Z ⇒ V + S ⇒ muon/anti-muon + tau/anti-tau events; but the expected number is very small and somewhat model-dependent.]*

Another possibility (also common in hidden valleys) is that the bottom-tagged jets aren’t actually from real bottom quarks, and are instead fake bottom jets generated by one or two new long-lived hidden valley particles.

But clearly, before anyone gets excited, far more evidence is required. We’ll need to see similar studies done at one or more of the three other experiments that ran concurrently with ALEPH — L3, OPAL, and DELPHI. And of course ATLAS, CMS, and LHCb will surely take a look in their own data; for instance, ATLAS and CMS could search for a dilepton resonance in events with at least two bottom-tagged jets, where the whole system of bottom-tagged jets and dileptons has a invariant mass not greater than about 100 GeV/c². They should also look for the V particle in other ways — perhaps following the methods I’ve suggested repeatedly (see for example pages 40-45 of this 2008 talk) — since the V might not only appear in Z particle decays. *[That is: look for boosted V’s; look for V’s in high-energy events or high missing-energy events; look for V’s in events with many jets, possibly with bottom-tags; etc.]* In any case, if anything like the V particle really exists, several (and perhaps all) of the experiments should see some evidence for it, and in more than just a single context.

Though we should be skeptical that today’s paper on ALEPH data is the first step toward a major discovery, at minimum it is important for what it indirectly confirms: that searches at the LHC are far from complete, and that discoveries might lie hidden, for example in rare Z decays (and in rare decays of other particles, such as the top quark.) Neither ATLAS, CMS nor LHCb have ever done a search for rare but spectacular Z particle decays, but they certainly could, as they recently did for the Higgs particle; and if Heister’s excess turns out to be a real signal, they will be seen to have missed a huge opportunity. So I hope that Heister’s paper, at a minimum, will encourage the LHC experiments to undertake a broader and more comprehensive program of searches for low-mass particles with very weak interactions. Otherwise, my own nightmare, in which the diamonds hidden in the rough might remain undetected — perhaps for decades — might come true.

Filed under: Other Collider News, Particle Physics Tagged: ALEPH, atlas, cms, dilepton, HiddenValleys, LEP, LHC, LHCb

Two years ago, when I was the target of an online shaming campaign, what helped me through it were hundreds of messages of support from friends, slight acquaintances, and strangers of every background. I vowed then to return the favor, by standing up when I saw decent people unfairly shamed. Today I have an opportunity to make good.

Some time ago I had the privilege of interacting a bit with Sam Altman, president of the famed startup incubator Y Combinator (and a guy who’s thanked in pretty much everything Paul Graham writes). By way of our mutual friend, the renowned former quantum computing researcher Michael Nielsen, Sam got in touch with me to solicit suggestions for “outside-the-box” scientists and writers, for a new grant program that Y Combinator was starting. I found Sam eager to delve into the merits of any suggestion, however outlandish, and was delighted to be able to make a difference for a few talented people who needed support.

Sam has also been one of the Silicon Valley leaders who’s written most clearly and openly about the threat to America posed by Donald Trump and the need to stop him, and he’s donated tens of thousands of dollars to anti-Trump causes. Needless to say, I supported Sam on that as well.

Now Sam is under attack on social media, and there are even calls for him to resign as the president of Y Combinator. Like me two years ago, Sam has instantly become the corporeal embodiment of the “nerd privilege” that keeps the marginalized out of Silicon Valley.

Why? Because, despite his own emphatic anti-Trump views, Sam rejected demands to fire Peter Thiel (who has an advisory role at Y Combinator) because of Thiel’s support for Trump. Sam explained his reasoning at some length:

[A]s repugnant as Trump is to many of us, we are not going to fire someone over his or her support of a political candidate. As far as we know, that would be unprecedented for supporting a major party nominee, and a dangerous path to start down (of course, if Peter said some of the things Trump says himself, he would no longer be part of Y Combinator) … The way we got into a situation with Trump as a major party nominee in the first place was by not talking to people who are very different than we are … I don’t understand how 43% of the country supports Trump. But I’d like to find out, because we have to include everyone in our path forward. If our best ideas are to stop talking to or fire anyone who disagrees with us, we’ll be facing this whole situation again in 2020.

The usual criticism of nerds is that we might have narrow technical abilities, but we lack wisdom about human affairs. It’s ironic, then, that it appears to have fallen to Silicon Valley nerds to guard some of the most important human wisdom our sorry species ever came across—namely, the liberal ideals of the Enlightenment. Like Sam, I despise pretty much everything Trump stands for, and I’ve been far from silent about it: I’ve blogged, donated money, advocated vote swapping, endured anonymous comments like “kill yourself kike”—whatever seemed like it might help even infinitesimally to ensure the richly-deserved electoral thrashing that Trump mercifully seems to be headed for in a few weeks.

But I also, I confess, oppose the forces that apparently see Trump less as a global calamity to be averted, than as a golden opportunity to take down anything they don’t like that’s ever been spotted within a thousand-mile radius of Trump Tower. (Where does this Kevin Bacon game end, anyway? Do “six degrees of Trump” suffice to contaminate you?)

And not only do I not feel a shadow of a hint of a moral conflict here, but it seems to me that precisely the same liberal Enlightenment principles are behind both of these stances.

But I’d go yet further. It sort of flabbergasts me when social-justice activists don’t understand that, if we condemn not only Trump, not only his supporters, but even *vociferous Trump opponents who associate with Trump supporters (!)*, all we’ll do is to feed the narrative that got Trumpism as far as it has—namely, that of a smug, bubble-encased, virtue-signalling leftist elite subject to runaway political correctness spirals. Like, a hundred million Americans’ worldviews revolve around the fear of liberal persecution, and we’re going to change their minds by firing those who refuse to fire *them*? As a recent *Washington Post* story illustrates, the opposite approach is harder but can bear spectacular results.

Now, as for Peter Thiel: three years ago, he funded a small interdisciplinary workshop on the coast of France that I attended. With me there were a bunch of honest-to-goodness conservative Christians, a Freudian psychoanalyst, a novelist, a right-wing radio host, some scientists and Silicon Valley executives, and of course Thiel himself. Each, I found, offered tons to disagree about but also some morsels to learn.

Thiel’s worldview, focused on the technological and organizational greatness that (in his view) Western civilization used to have and has subsequently lost, was a bit too dark and pessimistic for me, and I’m a pretty dark and pessimistic person. Thiel gave a complicated, meandering lecture that involved comparing modern narratives about Silicon Valley entrepreneurs against myths of gods, heroes, and martyrs throughout history, such as Romulus and Remus (the legendary founders of Rome). The talk might have made more sense to Thiel than to his listeners.

At the same time, Thiel’s range of knowledge and curiosity was pretty awesome. He avidly followed all the talks (including mine, on P vs. NP and quantum complexity theory) and asked pertinent questions. When the conversation turned to D-Wave, and Thiel’s own decision not to invest in it, he laid out the conclusions he’d come to from an extremely quick look at the question, then quizzed me as to whether he’d gotten anything wrong. He hadn’t.

From that conversation among others, I formed the impression that Thiel’s success as an investor is, at least in part, down neither to luck nor to connections, but to a module in his brain that most people lack, which makes blazingly fast and accurate judgments about tech startups. No wonder Y Combinator would want to keep him as an adviser.

But, OK, I’m so used to the same person being spectacularly right on some things and spectacularly wrong on others, that it no longer causes even slight cognitive dissonance. You just take the issues one by one.

I was happy, on balance, when it came out that Thiel had financed the lawsuit that brought down Gawker Media. Gawker really *had* used its power to bully the innocent, and it had broken the law to do it. And if it’s an unaccountable, anti-egalitarian, billionaire Godzilla against a vicious, privacy-violating, nerd-baiting King Kong—well then, I guess I’m with Godzilla.

More recently, I was appalled when Thiel spoke at the Republican convention, pandering to the crowd with Fox-News-style attack lines that were unworthy of a mind of his caliber. I lost a lot of respect for Thiel that day. But that’s the thing: unlike with literally every other speaker at the GOP convention, my respect for Thiel had started from a point that *made a decrease possible.*

I reject huge parts of Thiel’s worldview. I also reject any worldview that would threaten me with ostracism for talking to Thiel, attending a workshop he sponsors, or saying anything good about him. This is not actually a difficult balance.

Today, when it sometimes seems like much of the world has united in salivating for a cataclysmic showdown between whites and non-whites, Christians and Muslims, “dudebros” and feminists, etc., and that the salivators differ mostly just in who they want to see victorious in the coming battle and who humiliated, it can feel lonely to stick up for naïve, outdated values like the free exchange of ideas, friendly disagreement, the presumption of innocence, and the primacy of the individual over the tribe. But those are the values that took us all the way from a bronze spear through the enemy’s heart to a snarky rebuttal on the arXiv, and they’ll continue to build anything worth building.

And now to watch the third debate (I’ll check the comments afterward)…

**Update (Oct. 20):** See also this post from a blog called TheMoneyIllusion. My favorite excerpt:

So let’s see. Not only should Trump be shunned for his appalling political views, an otherwise highly respected Silicon Valley entrepreneur who just happens to support Trump (along with 80 million other Americans) should also be shunned. And a person who despises Trump and works against him but who defends Thiel’s right to his own political views should also resign. Does that mean I should be shunned too? After all, I’m a guy who hates Trump, writing a post that defends a guy who hates Trump, who wrote a post defending a guy’s freedom to support Trump, who in turn supports Trump. And suppose my mother sticks up for me? Should she also be shunned?

It’s almost enough to make me vote . . . no, just kidding.

Question … Which people on the left are beyond the pale? Suppose Thiel had supported Hugo Chavez? How about Castro? Mao? Pol Pot? Perhaps the degrees of separation could be calibrated to the awfulness of the left-winger:

Chavez: One degree of separation. (Corbyn, Sean Penn, etc.)

Castro: Two degrees of separation is still toxic.

Lenin: Three degrees of separation.

Mao: Four degrees of separation.

Pol Pot: Five degrees of separation.

No, I'm not here to knock on the door of the Big Five*.

I was a couple of doors down at the Simons Foundation....

-cvj

*P.S. But I do hope to have exciting news to report on the publishing front soon... Click to continue reading this post

The post Flatiron appeared first on Asymptotia.

As I’ve mentioned several times, I just finished a two-month sentence on a grand jury in Schenectady County (well, technically, I have to go back for one more day, because they didn’t finish everything). I’m not allowed to talk about the details of the cases we heard, but I have some general thoughts about the process that I think are blog-safe.

Several people I’ve talked to about this who also did grand jury service at one point reported finding the experience more interesting than annoying; sadly, I can’t say the same. I have an extremely low tolerance for people inconsiderately wasting my time, and large amounts of the process made me feel like I was being jerked around, which always makes my blood boil.

The fundamental issue, to me, has to do with respect for the people who are giving up their time to serve on the jury, and the court was frankly awful on this count. We were asked to report at 10am every day we were called in, but never once started before 10:15, and they never had any clear schedule for the day. On one occasion, we sat waiting in the jury room past 10:30 because two different attorneys each thought the other was presenting first, so neither one came in. On another occasion, an assistant DA rolled in at about 10:30 to start, and got a little huffy when told that he had to wait until some jurors came back from the bathroom.

Cases came and went with no explanation, sometimes with a week or more passing between parts of the same case– on our final regular day last week, we couldn’t come up with a solid count of how many open cases we have yet to wrap up. On a couple of occasions, they didn’t even tell us they were done with a case for the day– the ADA finished a witness, left the room as if to get the next witness, and a different ADA came in to start a different case.

It’s remarkable how little effort it would take to fix that, too. The one ADA I ended up actually liking left a positive impression because he treated us like adults. When he started late, he apologized, and explained the delay (he’d had a meeting with a judge, or a witness showed up late), and when he took a break or was done for the day, he told us that, and explained why, and roughly what to expect in the future. It’s not much more than basic courtesy, but its absence from the rest of the proceedings made it really stand out.

As for the process itself, I described it to some other people as “like being stuck in an all-day faculty meeting.” In the same way that faculty meetings are often bogged down in silly procedural details, or derailed into old and long-running arguments, much of what goes on in the presentation of cases has absolutely nothing to do with the actual topic at hand and the people who are present.

For those not familiar with the quirks of the American legal system, “grand jury” is an intermediate step between investigation of a crime and the sort of jury trial that you see in movies and tv. The grand jury hears evidence only from the prosecution, not the defense, and the end result is not a conviction but an indictment, which indicates that the prosecution has enough evidence to justify proceeding to a regular jury trial. The standard is not “guilty beyond a reasonable doubt” but “reasonable cause to believe the offense was committed,” and the vote of the grand jury does not need to be unanimous.

That’s a really low bar to clear– there’s a lot of truth to the lawyer joke that a DA who wanted to could get a grand jury to indict a ham sandwich– but despite that huge amounts of time were wasted on anticipating a defense that wasn’t going to be presented to us. And a few of the cases involved *massive* overkill in establishing points that really weren’t in any doubt. This is done not because it’s necessary for the task the grand jury has, but (as far as I can tell) so they can impress the defense with the big list of witness and evidence they showed the grand jury when they sit down to make a plea deal.

There are also a bunch of ridiculous inefficiencies involved in the testimony, for basically historical reasons. The official record is produced by a stenographer who transcribes everything that’s said, which means that even when they have photographs or documents, they have to go through this stupid dance of making the witness describe the picture in words so it ends up in the transcript. When photographs were difficult and expensive to duplicate that might’ve made sense, but these days, they’re all digitized, and could easily be stored together with the electronic transcript.

The transcript format also forces a bunch of stupid redundancy, as each case needs to have a self-contained record. Which means that we had to re-establish the professional qualifications of the same handful of police officers over and over again. On one occasion, the same two officers testified in two different cases on the same day, and we had to go through the “How long have you been a police officer?” dance again, despite the fact that the officer in question had gone through the same story for the same grand jury fifteen minutes earlier.

I was also bothered a bit by the fact that the process explicitly emphasizes the *least* reliable forms of evidence available. Everything that comes in has to come from personal testimony of witnesses, often at a great remove from the events in question. We regularly had police witnesses admit they couldn’t remember some detail of the case, whereupon they– but not the jury– would be shown the report they wrote at the time, to “refresh their memory.” Which usually wound up with them essentially reading the report to us, taking five times as long as it would’ve if they just gave us the report to read.

This is not a knock on those officers, by the way– these are busy people, who handle a lot of cases, and it’s not unreasonable to be confused about the exact address of a call from months or years earlier. Those officers who confidently rattled off the exact details of the cases probably weren’t doing so because they had exceptional memory ability, but because they had reviewed those very same reports *before* coming in to testify. But given the vast amount of research showing how memories shift over time, it’s kind of farcical to go through this process at all– if you ask me to pick which I trust more, I’m going with the report written at the time the events happened, not the personal recollection of the witness a long time later.

The ostensible reason for this emphasis on personal testimony is that having the witnesses there in person allows you to assess their credibility, but that’s undermined by the process. In the absence of any kind of defense or cross-examination, *everybody* looks credible, particularly to the low standard needed to hand down an indictment. One of the few interesting ways to pass time during the duller bits of testimony was playing “If I were the defense attorney, how would I counter this?” And while I could see plenty of ways one might raise a “reasonable doubt” about the guilt of the accused, there was never anything that made me question whether there was “reasonable cause to believe” that the case being presented should go to trial.

So, in terms of the process, the net effect was probably a slight increase in my cynicism about the legal system. I suspect a trial jury would be a different and maybe more interesting experience, but this did not leave me with a particularly positive impression of the DA’s office, or the general notion of grand juries. And it really emphasizes the awfulness of those high-profile cases (generally involving police misconduct, as in Ferguson, etc.) where a grand jury does *not* return an indictment.

As for the cases themselves, I can’t discuss details, but they were mostly just depressing. We caught some fairly awful stuff, but for the most part, I was just reminded of the bit from Donald Westlake’s *Bad News*, where a judge reflects that “It was his task in this life to acknowledge and then punish stupidity.” The cases we heard were mostly sad stories about sad people making utterly terrible decisions. Some of them were bad enough to be kind of hilarious, but the cumulative effect was just sad.

A week or so into the term, I jokingly said on Twitter that my advice to anyone receiving a grand jury summons was to postpone it for the maximum period allowed, and during that period move to another state. That’s exaggerated, of course– I met some interesting people, and had some enjoyable conversations during the breaks– but on the whole, I really can’t recommend the experience.

“As the universe expands and dark energy remains constant (negative pressure) then where does the ever increasing amount of dark energy come from? Is this genuinely creating something from nothing (bit of lay man’s hype here), do conservation laws not apply? Puzzled over this for ages now.”

-- pete best

“When speaking of the Einstein equation, is it the case that the contribution of dark matter is always included in the stress energy tensor (source term) and that dark energy is included in the cosmological constant term? If so, is this the main reason to distinguish between these two forms of ‘darkness’? I ask because I don’t normally read about dark energy being ‘composed of particles’ in the way dark matter is discussed phenomenologically.”

-- CGT

Dear Pete, CGT:

Dark energy is often portrayed as very mysterious. But when you look at the math, it’s really the simplest aspect of general relativity.

Ahead, allow me to clarify that your questions refer to “dark energy” but are specifically about the cosmological constant which is a certain type of dark energy. For all we know, the cosmological constant fits all existing observations. Dark energy could be more complicated than that, but let’s start with the cosmological constant.

Einstein’s field equations can be derived from very few assumptions. First, there’s the equivalence principle, which can be formulated mathematically as the requirement that the equations be tensor-equations. Second, the equations should describe the curvature of space-time. Third, the source of gravity is the stress-energy tensor and it’s locally conserved.

If you write down the simplest equations which fulfill these criteria you get Einstein’s field equations with two free constants. One constant can be fixed by deriving the Newtonian limit and it turns out to be Newton’s constant, G. The other constant is the cosmological constant, usually denoted Λ. You can make the equations more complicated by adding higher order terms, but at low energies these two constants are the only relevant ones.

If the cosmological constant is not zero, then flat space-time is no longer a solution of the equations. If the constant is positive-valued in particular, space will undergo accelerated expansion if there are no other matter sources, or these are negligible in comparison to Λ. Our universe presently seems to be in a phase that is dominated by a positive cosmological constant – that’s the easiest way to explain the observations which were awarded the 2011 Nobel Prize in physics.

Things get difficult if one tries to find an interpretation of the rather unambiguous mathematics. You can for example take the term with the cosmological constant and not think of it as geometrical, but instead move it to the other side of the equation and think of it as some stuff that causes curvature. If you do that, you might be tempted to read the entries of the cosmological constant term as if it was a kind of fluid. It would then correspond to a fluid with constant density and with constant, negative pressure. That’s something one can write down. But does this interpretation make any sense? I don’t know. There isn’t any known fluid with such behavior.

Since the cosmological constant is also present if matter sources are absent, it can be interpreted as the energy-density and pressure of the vacuum. Indeed, one can calculate such a term in quantum field theory, just that the result is infamously 120 orders of magnitude too large. But that’s a different story and shall be told another time. The cosmological constant term is therefore often referred to as the “vacuum energy,” but that’s sloppy. It’s an energy-density, not an energy, and that’s an important difference.

How can it possibly be that an energy density remains constant as the universe expands, you ask. Doesn’t this mean you need to create more energy from somewhere? No, you don’t need to create anything. This is a confusion which comes about because you interpret the density which has been assigned to the cosmological constant like a density of matter, but that’s not what it is. If it was some kind of stuff we know, then, yes, you would expect the density to dilute as space expands. But the cosmological constant is a property of space-time itself. As space expands, there’s more space, and that space still has the same vacuum energy density – it’s constant!

The cosmological constant term is indeed conserved in general relativity, and it’s conserved separately from that of the other energy and matter sources. It’s just that conservation of stress-energy in general relativity works differently than you might be used to from flat space.

According to Noether’s theorem there’s a conserved quantity for every (continuous) symmetry. A flat space-time is the same at every place and at every moment of time. We say it has a translational invariance in space and time. These are symmetries, and they come with conserved quantities: Translational invariance of space conserves momentum, translational invariance in time conserves energy.

In a curved space-time generically neither symmetry is fulfilled, hence neither energy nor momentum are conserved. So, if you take the vacuum energy density and you integrate it over some volume to get an energy, then the total energy grows with the volume indeed. It’s just not conserved. How strange! But that makes perfect sense: It’s not conserved because space expands and hence we have no invariance in time. Consequently, there’s no conserved quantity for invariance in time.

But General Relativity has a more complicated type of symmetry to which Noether’s theorem can be applied. This gives rise to a local conservation of stress-momentum when coupled to gravity (the stress-momentum tensor is covariantly conserved).

The conservation law for the density of a pressureless fluid, for example, works as you expect it to work: As space expands, the density goes down with the volume. For radiation – which has pressure – the energy density falls faster than that of matter because wavelengths also redshift. And if you put the cosmological constant term with its negative pressure into the conservation law, both energy and pressure remain the same. It’s all consistent: They are conserved if they are constant.

Dark energy now is a generalization of the cosmological constant, in which one invents some fields which give rise to a similar term. There are various fields that theoretical physicists have played with: chameleon fields and phantom fields and quintessence and such. The difference to the cosmological constant is that these fields’ densities do change with time, albeit slowly. There is however presently no evidence that this is the case.

As to the question which dark stuff to include in which term. Dark matter is usually assumed to be pressureless, which means that for what its gravitational pull is concerned it behaves just like normal matter. Dark energy, in contrast, has negative pressure and does odd things. That’s why they are usually collected in different terms.

Why don’t you normally read about dark energy being made of particles? Because you need some really strange stuff to get something that behaves like dark energy. You can’t make it out of any kind of particle that we know – this would either give you a matter term or a radiation term, neither of which does what dark energy needs to do.

If dark energy was some kind of field, or some kind of condensate, then it would be made of something else. In that case its density might indeed also vary from one place to the next and we might be able to detect the presence of that field in some way. Again though, there isn’t presently any evidence for that.

Thanks for your interesting questions!

Ahead, allow me to clarify that your questions refer to “dark energy” but are specifically about the cosmological constant which is a certain type of dark energy. For all we know, the cosmological constant fits all existing observations. Dark energy could be more complicated than that, but let’s start with the cosmological constant.

Einstein’s field equations can be derived from very few assumptions. First, there’s the equivalence principle, which can be formulated mathematically as the requirement that the equations be tensor-equations. Second, the equations should describe the curvature of space-time. Third, the source of gravity is the stress-energy tensor and it’s locally conserved.

If you write down the simplest equations which fulfill these criteria you get Einstein’s field equations with two free constants. One constant can be fixed by deriving the Newtonian limit and it turns out to be Newton’s constant, G. The other constant is the cosmological constant, usually denoted Λ. You can make the equations more complicated by adding higher order terms, but at low energies these two constants are the only relevant ones.

Einstein's field equations. [Image Source] |

Things get difficult if one tries to find an interpretation of the rather unambiguous mathematics. You can for example take the term with the cosmological constant and not think of it as geometrical, but instead move it to the other side of the equation and think of it as some stuff that causes curvature. If you do that, you might be tempted to read the entries of the cosmological constant term as if it was a kind of fluid. It would then correspond to a fluid with constant density and with constant, negative pressure. That’s something one can write down. But does this interpretation make any sense? I don’t know. There isn’t any known fluid with such behavior.

Since the cosmological constant is also present if matter sources are absent, it can be interpreted as the energy-density and pressure of the vacuum. Indeed, one can calculate such a term in quantum field theory, just that the result is infamously 120 orders of magnitude too large. But that’s a different story and shall be told another time. The cosmological constant term is therefore often referred to as the “vacuum energy,” but that’s sloppy. It’s an energy-density, not an energy, and that’s an important difference.

How can it possibly be that an energy density remains constant as the universe expands, you ask. Doesn’t this mean you need to create more energy from somewhere? No, you don’t need to create anything. This is a confusion which comes about because you interpret the density which has been assigned to the cosmological constant like a density of matter, but that’s not what it is. If it was some kind of stuff we know, then, yes, you would expect the density to dilute as space expands. But the cosmological constant is a property of space-time itself. As space expands, there’s more space, and that space still has the same vacuum energy density – it’s constant!

The cosmological constant term is indeed conserved in general relativity, and it’s conserved separately from that of the other energy and matter sources. It’s just that conservation of stress-energy in general relativity works differently than you might be used to from flat space.

According to Noether’s theorem there’s a conserved quantity for every (continuous) symmetry. A flat space-time is the same at every place and at every moment of time. We say it has a translational invariance in space and time. These are symmetries, and they come with conserved quantities: Translational invariance of space conserves momentum, translational invariance in time conserves energy.

In a curved space-time generically neither symmetry is fulfilled, hence neither energy nor momentum are conserved. So, if you take the vacuum energy density and you integrate it over some volume to get an energy, then the total energy grows with the volume indeed. It’s just not conserved. How strange! But that makes perfect sense: It’s not conserved because space expands and hence we have no invariance in time. Consequently, there’s no conserved quantity for invariance in time.

But General Relativity has a more complicated type of symmetry to which Noether’s theorem can be applied. This gives rise to a local conservation of stress-momentum when coupled to gravity (the stress-momentum tensor is covariantly conserved).

The conservation law for the density of a pressureless fluid, for example, works as you expect it to work: As space expands, the density goes down with the volume. For radiation – which has pressure – the energy density falls faster than that of matter because wavelengths also redshift. And if you put the cosmological constant term with its negative pressure into the conservation law, both energy and pressure remain the same. It’s all consistent: They are conserved if they are constant.

Dark energy now is a generalization of the cosmological constant, in which one invents some fields which give rise to a similar term. There are various fields that theoretical physicists have played with: chameleon fields and phantom fields and quintessence and such. The difference to the cosmological constant is that these fields’ densities do change with time, albeit slowly. There is however presently no evidence that this is the case.

As to the question which dark stuff to include in which term. Dark matter is usually assumed to be pressureless, which means that for what its gravitational pull is concerned it behaves just like normal matter. Dark energy, in contrast, has negative pressure and does odd things. That’s why they are usually collected in different terms.

Why don’t you normally read about dark energy being made of particles? Because you need some really strange stuff to get something that behaves like dark energy. You can’t make it out of any kind of particle that we know – this would either give you a matter term or a radiation term, neither of which does what dark energy needs to do.

If dark energy was some kind of field, or some kind of condensate, then it would be made of something else. In that case its density might indeed also vary from one place to the next and we might be able to detect the presence of that field in some way. Again though, there isn’t presently any evidence for that.

Thanks for your interesting questions!