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Important note: As this is not a course in probability, we will try to avoid developing the general theory of stochastic calculus (which includes such concepts as filtrations, martingales, and Ito calculus). This will unfortunately limit what we can actually prove rigorously, and so at some places the arguments will be somewhat informal in nature. A rigorous treatment of many of the topics here can be found for instance in Lawler’s Conformally Invariant Processes in the Plane, from which much of the material here is drawn.

In these notes, random variables will be denoted in boldface.

Definition 1 A real random variable {\mathbf{X}} is said to be normally distributed with mean {x_0 \in {\bf R}} and variance {\sigma^2 > 0} if one has

\displaystyle \mathop{\bf E} F(\mathbf{X}) = \frac{1}{\sqrt{2\pi} \sigma} \int_{\bf R} e^{-(x-x_0)^2/2\sigma^2} F(x)\ dx

for all test functions {F \in C_c({\bf R})}. Similarly, a complex random variable {\mathbf{Z}} is said to be normally distributed with mean {z_0 \in {\bf R}} and variance {\sigma^2>0} if one has

\displaystyle \mathop{\bf E} F(\mathbf{Z}) = \frac{1}{\pi \sigma^2} \int_{\bf C} e^{-|z-x_0|^2/\sigma^2} F(z)\ dx dy

for all test functions {F \in C_c({\bf C})}, where {dx dy} is the area element on {{\bf C}}.

A real Brownian motion with base point {x_0 \in {\bf R}} is a random, almost surely continuous function {\mathbf{B}^{x_0}: [0,+\infty) \rightarrow {\bf R}} (using the locally uniform topology on continuous functions) with the property that (almost surely) {\mathbf{B}^{x_0}(0) = x_0}, and for any sequence of times {0 \leq t_0 < t_1 < t_2 < \dots < t_n}, the increments {\mathbf{B}^{x_0}(t_i) - \mathbf{B}^{x_0}(t_{i-1})} for {i=1,\dots,n} are independent real random variables that are normally distributed with mean zero and variance {t_i - t_{i-1}}. Similarly, a complex Brownian motion with base point {z_0 \in {\bf R}} is a random, almost surely continuous function {\mathbf{B}^{z_0}: [0,+\infty) \rightarrow {\bf R}} with the property that {\mathbf{B}^{z_0}(0) = z_0} and for any sequence of times {0 \leq t_0 < t_1 < t_2 < \dots < t_n}, the increments {\mathbf{B}^{z_0}(t_i) - \mathbf{B}^{z_0}(t_{i-1})} for {i=1,\dots,n} are independent complex random variables that are normally distributed with mean zero and variance {t_i - t_{i-1}}.

Remark 2 Thanks to the central limit theorem, the hypothesis that the increments {\mathbf{B}^{x_0}(t_i) - \mathbf{B}^{x_0}(t_{i-1})} be normally distributed can be dropped from the definition of a Brownian motion, so long as one retains the independence and the normalisation of the mean and variance (technically one also needs some uniform integrability on the increments beyond the second moment, but we will not detail this here). A similar statement is also true for the complex Brownian motion (where now we need to normalise the variances and covariances of the real and imaginary parts of the increments).

Real and complex Brownian motions exist from any base point {x_0} or {z_0}; see e.g. this previous blog post for a construction. We have the following simple invariances:

Exercise 3

  • (i) (Translation invariance) If {\mathbf{B}^{x_0}} is a real Brownian motion with base point {x_0 \in {\bf R}}, and {h \in {\bf R}}, show that {\mathbf{B}^{x_0}+h} is a real Brownian motion with base point {x_0+h}. Similarly, if {\mathbf{B}^{z_0}} is a complex Brownian motion with base point {z_0 \in {\bf R}}, and {h \in {\bf C}}, show that {\mathbf{B}^{z_0}+c} is a complex Brownian motion with base point {z_0+h}.
  • (ii) (Dilation invariance) If {\mathbf{B}^{0}} is a real Brownian motion with base point {0}, and {\lambda \in {\bf R}} is non-zero, show that {t \mapsto \lambda \mathbf{B}^0(t / |\lambda|^{1/2})} is also a real Brownian motion with base point {0}. Similarly, if {\mathbf{B}^0} is a complex Brownian motion with base point {0}, and {\lambda \in {\bf C}} is non-zero, show that {t \mapsto \lambda \mathbf{B}^0(t / |\lambda|^{1/2})} is also a complex Brownian motion with base point {0}.
  • (iii) (Real and imaginary parts) If {\mathbf{B}^0} is a complex Brownian motion with base point {0}, show that {\sqrt{2} \mathrm{Re} \mathbf{B}^0} and {\sqrt{2} \mathrm{Im} \mathbf{B}^0} are independent real Brownian motions with base point {0}. Conversely, if {\mathbf{B}^0_1, \mathbf{B}^0_2} are independent real Brownian motions of base point {0}, show that {\frac{1}{\sqrt{2}} (\mathbf{B}^0_1 + i \mathbf{B}^0_2)} is a complex Brownian motion with base point {0}.

The next lemma is a special case of the optional stopping theorem.

Lemma 4 (Optional stopping identities)

  • (i) (Real case) Let {\mathbf{B}^{x_0}} be a real Brownian motion with base point {x_0 \in {\bf R}}. Let {\mathbf{t}} be a bounded stopping time – a bounded random variable with the property that for any time {t \geq 0}, the event that {\mathbf{t} \leq t} is determined by the values of the trajectory {\mathbf{B}^{x_0}} for times up to {t} (or more precisely, this event is measurable with respect to the {\sigma} algebra generated by this proprtion of the trajectory). Then

    \displaystyle \mathop{\bf E} \mathbf{B}^{x_0}(\mathbf{t}) = x_0


    \displaystyle \mathop{\bf E} (\mathbf{B}^{x_0}(\mathbf{t})-x_0)^2 - \mathbf{t} = 0


    \displaystyle \mathop{\bf E} (\mathbf{B}^{x_0}(\mathbf{t})-x_0)^4 = O( \mathop{\bf E} \mathbf{t}^2 ).

  • (ii) (Complex case) Let {\mathbf{B}^{z_0}} be a real Brownian motion with base point {z_0 \in {\bf R}}. Let {\mathbf{t}} be a bounded stopping time – a bounded random variable with the property that for any time {t \geq 0}, the event that {\mathbf{t} \leq t} is determined by the values of the trajectory {\mathbf{B}^{x_0}} for times up to {t}. Then

    \displaystyle \mathop{\bf E} \mathbf{B}^{z_0}(\mathbf{t}) = z_0

    \displaystyle \mathop{\bf E} (\mathrm{Re}(\mathbf{B}^{z_0}(\mathbf{t})-z_0))^2 - \frac{1}{2} \mathbf{t} = 0

    \displaystyle \mathop{\bf E} (\mathrm{Im}(\mathbf{B}^{z_0}(\mathbf{t})-z_0))^2 - \frac{1}{2} \mathbf{t} = 0

    \displaystyle \mathop{\bf E} \mathrm{Re}(\mathbf{B}^{z_0}(\mathbf{t})-z_0) \mathrm{Im}(\mathbf{B}^{z_0}(\mathbf{t})-z_0) = 0

    \displaystyle \mathop{\bf E} |\mathbf{B}^{x_0}(\mathbf{t})-z_0|^4 = O( \mathop{\bf E} \mathbf{t}^2 ).

Proof: (Slightly informal) We just prove (i) and leave (ii) as an exercise. By translation invariance we can take {x_0=0}. Let {T} be an upper bound for {\mathbf{t}}. Since {\mathbf{B}^0(T)} is a real normally distributed variable with mean zero and variance {T}, we have

\displaystyle \mathop{\bf E} \mathbf{B}^0( T ) = 0


\displaystyle \mathop{\bf E} \mathbf{B}^0( T )^2 = T


\displaystyle \mathop{\bf E} \mathbf{B}^0( T )^4 = 3T^2.

By the law of total expectation, we thus have

\displaystyle \mathop{\bf E} \mathop{\bf E}(\mathbf{B}^0( T ) | \mathbf{t}, \mathbf{B}^{z_0}(\mathbf{t}) ) = 0


\displaystyle \mathop{\bf E} \mathop{\bf E}((\mathbf{B}^0( T ))^2 | \mathbf{t}, \mathbf{B}^{z_0}(\mathbf{t}) ) = T


\displaystyle \mathop{\bf E} \mathop{\bf E}((\mathbf{B}^0( T ))^4 | \mathbf{t}, \mathbf{B}^{z_0}(\mathbf{t}) ) = 3T^2

where the inner conditional expectations are with respect to the event that {\mathbf{t}, \mathbf{B}^{0}(\mathbf{t})} attains a particular point in {S}. However, from the independent increment nature of Brownian motion, once one conditions {(\mathbf{t}, \mathbf{B}^{0}(\mathbf{t}))} to a fixed point {(t, x)}, the random variable {\mathbf{B}^0(T)} becomes a real normally distributed variable with mean {x} and variance {T-t}. Thus we have

\displaystyle \mathop{\bf E}(\mathbf{B}^0( T ) | \mathbf{t}, \mathbf{B}^{z_0}(\mathbf{t}) ) = \mathbf{B}^{z_0}(\mathbf{t})


\displaystyle \mathop{\bf E}( (\mathbf{B}^0( T ))^2 | \mathbf{t}, \mathbf{B}^{z_0}(\mathbf{t}) ) = \mathbf{B}^{z_0}(\mathbf{t})^2 + T - \mathbf{t}


\displaystyle \mathop{\bf E}( (\mathbf{B}^0( T ))^4 | \mathbf{t}, \mathbf{B}^{z_0}(\mathbf{t}) ) = \mathbf{B}^{z_0}(\mathbf{t})^4 + 6(T - \mathbf{t}) \mathbf{B}^{z_0}(\mathbf{t})^2 + 3(T - \mathbf{t})^2

which give the first two claims, and (after some algebra) the identity

\displaystyle \mathop{\bf E} \mathbf{B}^{z_0}(\mathbf{t})^4 - 6 \mathbf{t} \mathbf{B}^{z_0}(\mathbf{t})^2 + 3 \mathbf{t}^2 = 0

which then also gives the third claim. \Box

Exercise 5 Prove the second part of Lemma 4.

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We now approach conformal maps from yet another perspective. Given an open subset {U} of the complex numbers {{\bf C}}, define a univalent function on {U} to be a holomorphic function {f: U \rightarrow {\bf C}} that is also injective. We will primarily be studying this concept in the case when {U} is the unit disk {D(0,1) := \{ z \in {\bf C}: |z| < 1 \}}.

Clearly, a univalent function {f: D(0,1) \rightarrow {\bf C}} on the unit disk is a conformal map from {D(0,1)} to the image {f(D(0,1))}; in particular, {f(D(0,1))} is simply connected, and not all of {{\bf C}} (since otherwise the inverse map {f^{-1}: {\bf C} \rightarrow D(0,1)} would violate Liouville’s theorem). In the converse direction, the Riemann mapping theorem tells us that every open simply connected proper subset {V \subsetneq {\bf C}} of the complex numbers is the image of a univalent function on {D(0,1)}. Furthermore, if {V} contains the origin, then the univalent function {f: D(0,1) \rightarrow {\bf C}} with this image becomes unique once we normalise {f(0) = 0} and {f'(0) > 0}. Thus the Riemann mapping theorem provides a one-to-one correspondence between open simply connected proper subsets of the complex plane containing the origin, and univalent functions {f: D(0,1) \rightarrow {\bf C}} with {f(0)=0} and {f'(0)>0}. We will focus particular attention on the univalent functions {f: D(0,1) \rightarrow {\bf C}} with the normalisation {f(0)=0} and {f'(0)=1}; such functions will be called schlicht functions.

One basic example of a univalent function on {D(0,1)} is the Cayley transform {z \mapsto \frac{1+z}{1-z}}, which is a Möbius transformation from {D(0,1)} to the right half-plane {\{ \mathrm{Re}(z) > 0 \}}. (The slight variant {z \mapsto \frac{1-z}{1+z}} is also referred to as the Cayley transform, as is the closely related map {z \mapsto \frac{z-i}{z+i}}, which maps {D(0,1)} to the upper half-plane.) One can square this map to obtain a further univalent function {z \mapsto \left( \frac{1+z}{1-z} \right)^2}, which now maps {D(0,1)} to the complex numbers with the negative real axis {(-\infty,0]} removed. One can normalise this function to be schlicht to obtain the Koebe function

\displaystyle  f(z) := \frac{1}{4}\left( \left( \frac{1+z}{1-z} \right)^2 - 1\right) = \frac{z}{(1-z)^2}, \ \ \ \ \ (1)

which now maps {D(0,1)} to the complex numbers with the half-line {(-\infty,-1/4]} removed. A little more generally, for any {\theta \in {\bf R}} we have the rotated Koebe function

\displaystyle  f(z) := \frac{z}{(1 - e^{i\theta} z)^2} \ \ \ \ \ (2)

that is a schlicht function that maps {D(0,1)} to the complex numbers with the half-line {\{ -re^{-i\theta}: r \geq 1/4\}} removed.

Every schlicht function {f: D(0,1) \rightarrow {\bf C}} has a convergent Taylor expansion

\displaystyle  f(z) = a_1 z + a_2 z^2 + a_3 z^3 + \dots

for some complex coefficients {a_1,a_2,\dots} with {a_1=1}. For instance, the Koebe function has the expansion

\displaystyle  f(z) = z + 2 z^2 + 3 z^3 + \dots = \sum_{n=1}^\infty n z^n

and similarly the rotated Koebe function has the expansion

\displaystyle  f(z) = z + 2 e^{i\theta} z^2 + 3 e^{2i\theta} z^3 + \dots = \sum_{n=1}^\infty n e^{(n-1)\theta} z^n.

Intuitively, the Koebe function and its rotations should be the “largest” schlicht functions available. This is formalised by the famous Bieberbach conjecture, which asserts that for any schlicht function, the coefficients {a_n} should obey the bound {|a_n| \leq n} for all {n}. After a large number of partial results, this conjecture was eventually solved by de Branges; see for instance this survey of Korevaar or this survey of Koepf for a history.

It turns out that to resolve these sorts of questions, it is convenient to restrict attention to schlicht functions {g: D(0,1) \rightarrow {\bf C}} that are odd, thus {g(-z)=-g(z)} for all {z}, and the Taylor expansion now reads

\displaystyle  g(z) = b_1 z + b_3 z^3 + b_5 z^5 + \dots

for some complex coefficients {b_1,b_3,\dots} with {b_1=1}. One can transform a general schlicht function {f: D(0,1) \rightarrow {\bf C}} to an odd schlicht function {g: D(0,1) \rightarrow {\bf C}} by observing that the function {f(z^2)/z^2: D(0,1) \rightarrow {\bf C}}, after removing the singularity at zero, is a non-zero function that equals {1} at the origin, and thus (as {D(0,1)} is simply connected) has a unique holomorphic square root {(f(z^2)/z^2)^{1/2}} that also equals {1} at the origin. If one then sets

\displaystyle  g(z) := z (f(z^2)/z^2)^{1/2} \ \ \ \ \ (3)

it is not difficult to verify that {g} is an odd schlicht function which additionally obeys the equation

\displaystyle  f(z^2) = g(z)^2. \ \ \ \ \ (4)

Conversely, given an odd schlicht function {g}, the formula (4) uniquely determines a schlicht function {f}.

For instance, if {f} is the Koebe function (1), {g} becomes

\displaystyle  g(z) = \frac{z}{1-z^2} = z + z^3 + z^5 + \dots, \ \ \ \ \ (5)

which maps {D(0,1)} to the complex numbers with two slits {\{ \pm iy: y > 1/2 \}} removed, and if {f} is the rotated Koebe function (2), {g} becomes

\displaystyle  g(z) = \frac{z}{1- e^{i\theta} z^2} = z + e^{i\theta} z^3 + e^{2i\theta} z^5 + \dots. \ \ \ \ \ (6)

De Branges established the Bieberbach conjecture by first proving an analogous conjecture for odd schlicht functions known as Robertson’s conjecture. More precisely, we have

Theorem 1 (de Branges’ theorem) Let {n \geq 1} be a natural number.

  • (i) (Robertson conjecture) If {g(z) = b_1 z + b_3 z^3 + b_5 z^5 + \dots} is an odd schlicht function, then

    \displaystyle  \sum_{k=1}^n |b_{2k-1}|^2 \leq n.

  • (ii) (Bieberbach conjecture) If {f(z) = a_1 z + a_2 z^2 + a_3 z^3 + \dots} is a schlicht function, then

    \displaystyle  |a_n| \leq n.

It is easy to see that the Robertson conjecture for a given value of {n} implies the Bieberbach conjecture for the same value of {n}. Indeed, if {f(z) = a_1 z + a_2 z^2 + a_3 z^3 + \dots} is schlicht, and {g(z) = b_1 z + b_3 z^3 + b_5 z^5 + \dots} is the odd schlicht function given by (3), then from extracting the {z^{2n}} coefficient of (4) we obtain a formula

\displaystyle  a_n = \sum_{j=1}^n b_{2j-1} b_{2(n+1-j)-1}

for the coefficients of {f} in terms of the coefficients of {g}. Applying the Cauchy-Schwarz inequality, we derive the Bieberbach conjecture for this value of {n} from the Robertson conjecture for the same value of {n}. We remark that Littlewood and Paley had conjectured a stronger form {|b_{2k-1}| \leq 1} of Robertson’s conjecture, but this was disproved for {k=3} by Fekete and Szegö.

To prove the Robertson and Bieberbach conjectures, one first takes a logarithm and deduces both conjectures from a similar conjecture about the Taylor coefficients of {\log \frac{f(z)}{z}}, known as the Milin conjecture. Next, one continuously enlarges the image {f(D(0,1))} of the schlicht function to cover all of {{\bf C}}; done properly, this places the schlicht function {f} as the initial function {f = f_0} in a sequence {(f_t)_{t \geq 0}} of univalent maps {f_t: D(0,1) \rightarrow {\bf C}} known as a Loewner chain. The functions {f_t} obey a useful differential equation known as the Loewner equation, that involves an unspecified forcing term {\mu_t} (or {\theta(t)}, in the case that the image is a slit domain) coming from the boundary; this in turn gives useful differential equations for the Taylor coefficients of {f(z)}, {g(z)}, or {\log \frac{f(z)}{z}}. After some elementary calculus manipulations to “integrate” this equations, the Bieberbach, Robertson, and Milin conjectures are then reduced to establishing the non-negativity of a certain explicit hypergeometric function, which is non-trivial to prove (and will not be done here, except for small values of {n}) but for which several proofs exist in the literature.

The theory of Loewner chains subsequently became fundamental to a more recent topic in complex analysis, that of the Schramm-Loewner equation (SLE), which is the focus of the next and final set of notes.

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We now leave the topic of Riemann surfaces, and turn now to the (loosely related) topic of conformal mapping (and quasiconformal mapping). Recall that a conformal map {f: U \rightarrow V} from an open subset {U} of the complex plane to another open set {V} is a map that is holomorphic and bijective, which (by Rouché’s theorem) also forces the derivative of {f} to be nowhere vanishing. We then say that the two open sets {U,V} are conformally equivalent. From the Cauchy-Riemann equations we see that conformal maps are orientation-preserving and angle-preserving; from the Newton approximation {f( z_0 + \Delta z) \approx f(z_0) + f'(z_0) \Delta z + O( |\Delta z|^2)} we see that they almost preserve small circles, indeed for {\varepsilon} small the circle {\{ z: |z-z_0| = \varepsilon\}} will approximately map to {\{ w: |w - f(z_0)| = |f'(z_0)| \varepsilon \}}.

In previous quarters, we proved a fundamental theorem about this concept, the Riemann mapping theorem:

Theorem 1 (Riemann mapping theorem) Let {U} be a simply connected open subset of {{\bf C}} that is not all of {{\bf C}}. Then {U} is conformally equivalent to the unit disk {D(0,1)}.

This theorem was proven in these 246A lecture notes, using an argument of Koebe. At a very high level, one can sketch Koebe’s proof of the Riemann mapping theorem as follows: among all the injective holomorphic maps {f: U \rightarrow D(0,1)} from {U} to {D(0,1)} that map some fixed point {z_0 \in U} to {0}, pick one that maximises the magnitude {|f'(z_0)|} of the derivative (ignoring for this discussion the issue of proving that a maximiser exists). If {f(U)} avoids some point in {D(0,1)}, one can compose {f} with various holomorphic maps and use Schwarz’s lemma and the chain rule to increase {|f'(z_0)|} without destroying injectivity; see the previous lecture notes for details. The conformal map {\phi: U \rightarrow D(0,1)} is unique up to Möbius automorphisms of the disk; one can fix the map by picking two distinct points {z_0,z_1} in {U}, and requiring {\phi(z_0)} to be zero and {\phi(z_1)} to be positive real.

It is a beautiful observation of Thurston that the concept of a conformal mapping has a discrete counterpart, namely the mapping of one circle packing to another. Furthermore, one can run a version of Koebe’s argument (using now a discrete version of Perron’s method) to prove the Riemann mapping theorem through circle packings. In principle, this leads to a mostly elementary approach to conformal geometry, based on extremely classical mathematics that goes all the way back to Apollonius. However, in order to prove the basic existence and uniqueness theorems of circle packing, as well as the convergence to conformal maps in the continuous limit, it seems to be necessary (or at least highly convenient) to use much more modern machinery, including the theory of quasiconformal mapping, and also the Riemann mapping theorem itself (so in particular we are not structuring these notes to provide a completely independent proof of that theorem, though this may well be possible).

To make the above discussion more precise we need some notation.

Definition 2 (Circle packing) A (finite) circle packing is a finite collection {(C_j)_{j \in J}} of circles {C_j = \{ z \in {\bf C}: |z-z_j| = r_j\}} in the complex numbers indexed by some finite set {J}, whose interiors are all disjoint (but which are allowed to be tangent to each other), and whose union is connected. The nerve of a circle packing is the finite graph whose vertices {\{z_j: j \in J \}} are the centres of the circle packing, with two such centres connected by an edge if the circles are tangent. (In these notes all graphs are undirected, finite and simple, unless otherwise specified.)

It is clear that the nerve of a circle packing is connected and planar, since one can draw the nerve by placing each vertex (tautologically) in its location in the complex plane, and drawing each edge by the line segment between the centres of the circles it connects (this line segment will pass through the point of tangency of the two circles). Later in these notes we will also have to consider some infinite circle packings, most notably the infinite regular hexagonal circle packing.

The first basic theorem in the subject is the following converse statement:

Theorem 3 (Circle packing theorem) Every connected planar graph is the nerve of a circle packing.

Of course, there can be multiple circle packings associated to a given connected planar graph; indeed, since reflections across a line and Möbius transformations map circles to circles (or lines), they will map circle packings to circle packings (unless one or more of the circles is sent to a line). It turns out that once one adds enough edges to the planar graph, the circle packing is otherwise rigid:

Theorem 4 (Koebe-Andreev-Thurston theorem) If a connected planar graph is maximal (i.e., no further edge can be added to it without destroying planarity), then the circle packing given by the above theorem is unique up to reflections and Möbius transformations.

Exercise 5 Let {G} be a connected planar graph with {n \geq 3} vertices. Show that the following are equivalent:

  • (i) {G} is a maximal planar graph.
  • (ii) {G} has {3n-6} edges.
  • (iii) Every drawing {D} of {G} divides the plane into faces that have three edges each. (This includes one unbounded face.)
  • (iv) At least one drawing {D} of {G} divides the plane into faces that have three edges each.

(Hint: use Euler’s formula {V-E+F=2}, where {F} is the number of faces including the unbounded face.)

Thurston conjectured that circle packings can be used to approximate the conformal map arising in the Riemann mapping theorem. Here is an informal statement:

Conjecture 6 (Informal Thurston conjecture) Let {U} be a simply connected domain, with two distinct points {z_0,z_1}. Let {\phi: U \rightarrow D(0,1)} be the conformal map from {U} to {D(0,1)} that maps {z_0} to the origin and {z_1} to a positive real. For any small {\varepsilon>0}, let {{\mathcal C}_\varepsilon} be the portion of the regular hexagonal circle packing by circles of radius {\varepsilon} that are contained in {U}, and let {{\mathcal C}'_\varepsilon} be an circle packing of {D(0,1)} with all “boundary circles” tangent to {D(0,1)}, giving rise to an “approximate map” {\phi_\varepsilon: U_\varepsilon \rightarrow D(0,1)} defined on the subset {U_\varepsilon} of {U} consisting of the circles of {{\mathcal C}_\varepsilon}, their interiors, and the interstitial regions between triples of mutually tangent circles. Normalise this map so that {\phi_\varepsilon(z_0)} is zero and {\phi_\varepsilon(z_1)} is a positive real. Then {\phi_\varepsilon} converges to {\phi} as {\varepsilon \rightarrow 0}.

A rigorous version of this conjecture was proven by Rodin and Sullivan. Besides some elementary geometric lemmas (regarding the relative sizes of various configurations of tangent circles), the main ingredients are a rigidity result for the regular hexagonal circle packing, and the theory of quasiconformal maps. Quasiconformal maps are what seem on the surface to be a very broad generalisation of the notion of a conformal map. Informally, conformal maps take infinitesimal circles to infinitesimal circles, whereas quasiconformal maps take infinitesimal circles to infinitesimal ellipses of bounded eccentricity. In terms of Wirtinger derivatives, conformal maps obey the Cauchy-Riemann equation {\frac{\partial \phi}{\partial \overline{z}} = 0}, while (sufficiently smooth) quasiconformal maps only obey an inequality {|\frac{\partial \phi}{\partial \overline{z}}| \leq \frac{K-1}{K+1} |\frac{\partial \phi}{\partial z}|}. As such, quasiconformal maps are considerably more plentiful than conformal maps, and in particular it is possible to create piecewise smooth quasiconformal maps by gluing together various simple maps such as affine maps or Möbius transformations; such piecewise maps will naturally arise when trying to rigorously build the map {\phi_\varepsilon} alluded to in the above conjecture. On the other hand, it turns out that quasiconformal maps still have many vestiges of the rigidity properties enjoyed by conformal maps; for instance, there are quasiconformal analogues of fundamental theorems in conformal mapping such as the Schwarz reflection principle, Liouville’s theorem, or Hurwitz’s theorem. Among other things, these quasiconformal rigidity theorems allow one to create conformal maps from the limit of quasiconformal maps in many circumstances, and this will be how the Thurston conjecture will be proven. A key technical tool in establishing these sorts of rigidity theorems will be the theory of an important quasiconformal (quasi-)invariant, the conformal modulus (or, equivalently, the extremal length, which is the reciprocal of the modulus).

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The fundamental object of study in real differential geometry are the real manifolds: Hausdorff topological spaces {M = M^n} that locally look like open subsets of a Euclidean space {{\bf R}^n}, and which can be equipped with an atlas {(\phi_\alpha: U_\alpha \rightarrow V_\alpha)_{\alpha \in A}} of coordinate charts {\phi_\alpha: U_\alpha \rightarrow V_\alpha} from open subsets {U_\alpha} covering {M} to open subsets {V_\alpha} in {{\bf R}^n}, which are homeomorphisms; in particular, the transition maps {\tau_{\alpha,\beta}: \phi_\alpha( U_\alpha \cap U_\beta ) \rightarrow \phi_\beta( U_\alpha \cap U_\beta )} defined by {\tau_{\alpha,\beta}: \phi_\beta \circ \phi_\alpha^{-1}} are all continuous. (It is also common to impose the requirement that the manifold {M} be second countable, though this will not be important for the current discussion.) A smooth real manifold is a real manifold in which the transition maps are all smooth.

In a similar fashion, the fundamental object of study in complex differential geometry are the complex manifolds, in which the model space is {{\bf C}^n} rather than {{\bf R}^n}, and the transition maps {\tau_{\alpha\beta}} are required to be holomorphic (and not merely smooth or continuous). In the real case, the one-dimensional manifolds (curves) are quite simple to understand, particularly if one requires the manifold to be connected; for instance, all compact connected one-dimensional real manifolds are homeomorphic to the unit circle (why?). However, in the complex case, the connected one-dimensional manifolds – the ones that look locally like subsets of {{\bf C}} – are much richer, and are known as Riemann surfaces. For sake of completeness we give the (somewhat lengthy) formal definition:

Definition 1 (Riemann surface) If {M} is a Hausdorff connected topological space, a (one-dimensional complex) atlas is a collection {(\phi_\alpha: U_\alpha \rightarrow V_\alpha)_{\alpha \in A}} of homeomorphisms from open subsets {(U_\alpha)_{\alpha \in A}} of {M} that cover {M} to open subsets {V_\alpha} of the complex numbers {{\bf C}}, such that the transition maps {\tau_{\alpha,\beta}: \phi_\alpha( U_\alpha \cap U_\beta ) \rightarrow \phi_\beta( U_\alpha \cap U_\beta )} defined by {\tau_{\alpha,\beta}: \phi_\beta \circ \phi_\alpha^{-1}} are all holomorphic. Here {A} is an arbitrary index set. Two atlases {(\phi_\alpha: U_\alpha \rightarrow V_\alpha)_{\alpha \in A}}, {(\phi'_\beta: U'_\beta \rightarrow V'_\beta)_{\beta \in B}} on {M} are said to be equivalent if their union is also an atlas, thus the transition maps {\phi'_\beta \circ \phi_\alpha^{-1}: \phi_\alpha(U_\alpha \cap U'_\beta) \rightarrow \phi'_\beta(U_\alpha \cap U'_\beta)} and their inverses are all holomorphic. A Riemann surface is a Hausdorff connected topological space {M} equipped with an equivalence class of one-dimensional complex atlases.

A map {f: M \rightarrow M'} from one Riemann surface {M} to another {M'} is holomorphic if the maps {\phi'_\beta \circ f \circ \phi_\alpha^{-1}: \phi_\alpha(U_\alpha \cap f^{-1}(U'_\beta)) \rightarrow {\bf C}} are holomorphic for any charts {\phi_\alpha: U_\alpha \rightarrow V_\alpha}, {\phi'_\beta: U'_\beta \rightarrow V'_\beta} of an atlas of {M} and {M'} respectively; it is not hard to see that this definition does not depend on the choice of atlas. It is also clear that the composition of two holomorphic maps is holomorphic (and in fact the class of Riemann surfaces with their holomorphic maps forms a category).

Here are some basic examples of Riemann surfaces.

Example 2 (Quotients of {{\bf C}}) The complex numbers {{\bf C}} clearly form a Riemann surface (using the identity map {\phi: {\bf C} \rightarrow {\bf C}} as the single chart for an atlas). Of course, maps {f: {\bf C} \rightarrow {\bf C}} that are holomorphic in the usual sense will also be holomorphic in the sense of the above definition, and vice versa, so the notion of holomorphicity for Riemann surfaces is compatible with that of holomorphicity for complex maps. More generally, given any discrete additive subgroup {\Lambda} of {{\bf C}}, the quotient {{\bf C}/\Lambda} is a Riemann surface. There are an infinite number of possible atlases to use here; one such is to pick a sufficiently small neighbourhood {U} of the origin in {{\bf C}} and take the atlas {(\phi_\alpha: U_\alpha \rightarrow U)_{\alpha \in {\bf C}/\Lambda}} where {U_\alpha := \alpha+U} and {\phi_\alpha(\alpha+z) := z} for all {z \in U}. In particular, given any non-real complex number {\omega}, the complex torus {{\bf C} / \langle 1, \omega \rangle} formed by quotienting {{\bf C}} by the lattice {\langle 1, \omega \rangle := \{ n + m \omega: n,m \in {\bf Z}\}} is a Riemann surface.

Example 3 Any open connected subset {U} of {{\bf C}} is a Riemann surface. By the Riemann mapping theorem, all simply connected open {U \subset {\bf C}}, other than {{\bf C}} itself, are isomorphic (as Riemann surfaces) to the unit disk (or, equivalently, to the upper half-plane).

Example 4 (Riemann sphere) The Riemann sphere {{\bf C} \cup \{\infty\}}, as a topological manifold, is the one-point compactification of {{\bf C}}. Topologically, this is a sphere and is in particular connected. One can cover the Riemann sphere by the two open sets {U_1 := {\bf C}} and {U_2 := {\bf C} \cup \{\infty\} \backslash \{0\}}, and give these two open sets the charts {\phi_1: U_1 \rightarrow {\bf C}} and {\phi_2: U_2 \rightarrow {\bf C}} defined by {\phi_1(z) := z} for {z \in {\bf C}}, {\phi_2(z) := 1/z} for {z \in {\bf C} \backslash \{0\}}, and {\phi_2(\infty) := 0}. This is a complex atlas since the {1/z} is holomorphic on {{\bf C} \backslash \{0\}}.

An alternate way of viewing the Riemann sphere is as the projective line {\mathbf{CP}^1}. Topologically, this is the punctured complex plane {{\bf C}^2 \backslash \{(0,0)\}} quotiented out by non-zero complex dilations, thus elements of this space are equivalence classes {[z,w] := \{ (\lambda z, \lambda w): \lambda \in {\bf C} \backslash \{0\}\}} with the usual quotient topology. One can cover this space by two open sets {U_1 := \{ [z,1]: z \in {\bf C} \}} and {U_2: \{ [1,w]: w \in {\bf C} \}} and give these two open sets the charts {\phi: U_1 \rightarrow {\bf C}} and {\phi_2: U_2 \rightarrow {\bf C}} defined by {\phi_1([z,1]) := z} for {z \in {\bf C}}, {\phi_2([1,w]) := w}. This is a complex atlas, basically because {[z,1] = [1,1/z]} for {z \in {\bf C} \backslash \{0\}} and {1/z} is holomorphic on {{\bf C} \backslash \{0\}}.

Exercise 5 Verify that the Riemann sphere is isomorphic (as a Riemann surface) to the projective line.

Example 6 (Smooth algebraic plane curves) Let {P(z_1,z_2,z_3)} be a complex polynomial in three variables which is homogeneous of some degree {d \geq 1}, thus

\displaystyle P( \lambda z_1, \lambda z_2, \lambda z_3) = \lambda^d P( z_1, z_2, z_3). \ \ \ \ \ (1)


Define the complex projective plane {\mathbf{CP}^2} to be the punctured space {{\bf C}^3 \backslash \{0\}} quotiented out by non-zero complex dilations, with the usual quotient topology. (There is another important topology to place here of fundamental importance in algebraic geometry, namely the Zariski topology, but we will ignore this topology here.) This is a compact space, whose elements are equivalence classes {[z_1,z_2,z_3] := \{ (\lambda z_1, \lambda z_2, \lambda z_3)\}}. Inside this plane we can define the (projective, degree {d}) algebraic curve

\displaystyle Z(P) := \{ [z_1,z_2,z_3] \in \mathbf{CP}^2: P(z_1,z_2,z_3) = 0 \};

this is well defined thanks to (1). It is easy to verify that {Z(P)} is a closed subset of {\mathbf{CP}^2} and hence compact; it is non-empty thanks to the fundamental theorem of algebra.

Suppose that {P} is irreducible, which means that it is not the product of polynomials of smaller degree. As we shall show in the appendix, this makes the algebraic curve connected. (Actually, algebraic curves remain connected even in the reducible case, thanks to Bezout’s theorem, but we will not prove that theorem here.) We will in fact make the stronger nonsingularity hypothesis: there is no triple {(z_1,z_2,z_3) \in {\bf C}^3 \backslash \{(0,0,0)\}} such that the four numbers {P(z_1,z_2,z_3), \frac{\partial}{\partial z_j} P(z_1,z_2,z_3)} simultaneously vanish for {j=1,2,3}. (This looks like four constraints, but is in fact essentially just three, due to the Euler identity

\displaystyle \sum_{j=1}^3 z_j \frac{\partial}{\partial z_j} P(z_1,z_2,z_3) = d P(z_1,z_2,z_3)

that arises from differentiating (1) in {\lambda}. The fact that nonsingularity implies irreducibility is another consequence of Bezout’s theorem, which is not proven here.) For instance, the polynomial {z_1^2 z_3 - z_2^3} is irreducible but singular (there is a “cusp” singularity at {[0,0,1]}). With this hypothesis, we call the curve {Z(P)} smooth.

Now suppose {[z_1,z_2,z_3]} is a point in {Z(P)}; without loss of generality we may take {z_3} non-zero, and then we can normalise {z_3=1}. Now one can think of {P(z_1,z_2,1)} as an inhomogeneous polynomial in just two variables {z_1,z_2}, and by nondegeneracy we see that the gradient {(\frac{\partial}{\partial z_1} P(z_1,z_2,1), \frac{\partial}{\partial z_2} P(z_1,z_2,1))} is non-zero whenever {P(z_1,z_2,1)=0}. By the (complexified) implicit function theorem, this ensures that the affine algebraic curve

\displaystyle Z(P)_{aff} := \{ (z_1,z_2) \in {\bf C}^2: P(z_1,z_2,1) = 0 \}

is a Riemann surface in a neighbourhood of {(z_1,z_2,1)}; we leave this as an exercise. This can be used to give a coordinate chart for {Z(P)} in a neighbourhood of {[z_1,z_2,z_3]} when {z_3 \neq 0}. Similarly when {z_1,z_2} is non-zero. This can be shown to give an atlas on {Z(P)}, which (assuming the connectedness claim that we will prove later) gives {Z(P)} the structure of a Riemann surface.

Exercise 7 State and prove a complex version of the implicit function theorem that justifies the above claim that the charts in the above example form an atlas, and an algebraic curve associated to a non-singular polynomial is a Riemann surface.

Exercise 8

  • (i) Show that all (irreducible plane projective) algebraic curves of degree {1} are isomorphic to the Riemann sphere. (Hint: reduce to an explicit linear polynomial such as {z_3}.)
  • (ii) Show that all (irreducible plane projective) algebraic curves of degree {2} are isomorphic to the Riemann sphere. (Hint: to reduce computation, first use some linear algebra to reduce the homogeneous quadratic polynomial to a standard form, such as {z_1^2+z_2^2+z_3^2} or {z_2 z_3 - z_1^2}.)

Exercise 9 If {a,b} are complex numbers, show that the projective cubic curve

\displaystyle \{ [z_1, z_2, z_3]: z_2^2 z_3 = z_1^3 + a z_1 z_3^2 + b z_3^3 \}

is nonsingular if and only if the discriminant {-16 (4a^3 + 27b^2)} is non-zero. (When this occurs, the curve is called an elliptic curve (in Weierstrass form), which is a fundamentally important example of a Riemann surface in many areas of mathematics, and number theory in particular. One can also define the discriminant for polynomials of higher degree, but we will not do so here.)

A recurring theme in mathematics is that an object {X} is often best studied by understanding spaces of “good” functions on {X}. In complex analysis, there are two basic types of good functions:

Definition 10 Let {X} be a Riemann surface. A holomorphic function on {X} is a holomorphic map from {X} to {{\bf C}}; the space of all such functions will be denoted {{\mathcal O}(X)}. A meromorphic function on {X} is a holomorphic map from {X} to the Riemann sphere {{\bf C} \cup \{\infty\}}, that is not identically equal to {\infty}; the space of all such functions will be denoted {M(X)}.

One can also define holomorphicity and meromorphicity in terms of charts: a function {f: X \rightarrow {\bf C}} is holomorphic if and only if, for any chart {\phi_\alpha: U_\alpha \rightarrow {\bf C}}, the map {f \circ \phi^{-1}_\alpha: \phi_\alpha(U_\alpha) \rightarrow {\bf C}} is holomorphic in the usual complex analysis sense; similarly, a function {f: X \rightarrow {\bf C} \cup \{\infty\}} is meromorphic if and only if the preimage {f^{-1}(\{\infty\})} is discrete (otherwise, by analytic continuation and the connectedness of {X}, {f} will be identically equal to {\infty}) and for any chart {\phi_\alpha: U_\alpha \rightarrow X}, the map {f \circ \phi_\alpha^{-1}: \phi_\alpha(U_\alpha) \rightarrow {\bf C} \cup \{\infty\}} becomes a meromorphic function in the usual complex analysis sense, after removing the discrete set of complex numbers where this map is infinite. One consequence of this alternate definition is that the space {{\mathcal O}(X)} of holomorphic functions is a commutative complex algebra (a complex vector space closed under pointwise multiplication), while the space {M(X)} of meromorphic functions is a complex field (a commutative complex algebra where every non-zero element has an inverse). Another consequence is that one can define the notion of a zero of given order {k}, or a pole of order {k}, for a holomorphic or meromorphic function, by composing with a chart map and using the usual complex analysis notions there, noting (from the holomorphicity of transition maps and their inverses) that this does not depend on the choice of chart. (However, one cannot similarly define the residue of a meromorphic function on {X} this way, as the residue turns out to be chart-dependent thanks to the chain rule. Residues should instead be applied to meromorphic {1}-forms, a concept we will introduce later.) A third consequence is analytic continuation: if two holomorphic or meromorphic functions on {X} agree on a non-empty open set, then they agree everywhere.

On the complex numbers {{\bf C}}, there are of course many holomorphic functions and meromorphic functions; for instance any power series with an infinite radius of convergence will give a holomorphic function, and the quotient of any two such functions (with non-zero denominator) will give a meromorphic function. Furthermore, we have extremely wide latitude in how to specify the zeroes of the holomorphic function, or the zeroes and poles of the meromorphic function, thanks to tools such as the Weierstrass factorisation theorem or the Mittag-Leffler theorem (covered in previous quarters).

It turns out, however, that the situation changes dramatically when the Riemann surface {X} is compact, with the holomorphic and meromorphic functions becoming much more rigid. First of all, compactness eliminates all holomorphic functions except for the constants:

Lemma 11 Let {f \in \mathcal{O}(X)} be a holomorphic function on a compact Riemann surface {X}. Then {f} is constant.

This result should be seen as a close sibling of Liouville’s theorem that all bounded entire functions are constant. (Indeed, in the case of a complex torus, this lemma is a corollary of Liouville’s theorem.)

Proof: As {f} is continuous and {X} is compact, {|f(z_0)|} must attain a maximum at some point {z_0 \in X}. Working in a chart around {z_0} and applying the maximum principle, we conclude that {f} is constant in a neighbourhood of {z_0}, and hence is constant everywhere by analytic continuation. \Box

This dramatically cuts down the number of possible meromorphic functions – indeed, for an abstract Riemann surface, it is not immediately obvious that there are any non-constant meromorphic functions at all! As the poles are isolated and the surface is compact, a meromorphic function can only have finitely many poles, and if one prescribes the location of the poles and the maximum order at each pole, then we shall see that the space of meromorphic functions is now finite dimensional. The precise dimensions of these spaces are in fact rather interesting, and obey a basic duality law known as the Riemann-Roch theorem. We will give a mostly self-contained proof of the Riemann-Roch theorem in these notes, omitting only some facts about genus and Euler characteristic, as well as construction of certain meromorphic {1}-forms (also known as Abelian differentials).

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Next quarter (starting Monday, April 2) I will be teaching Math 246C (complex analysis) here at UCLA.  This is the third in a three-series graduate course on complex analysis; a few years ago I taught the first course in this series (246A), so this course can be thought of in some sense as a sequel to that one (and would certainly assume knowledge of the material in that course as a prerequisite), although it also assumes knowledge of material from the second course 246B (which covers such topics as Weierstrass factorization and the theory of harmonic functions).

246C is primarily a topics course, and tends to be a somewhat miscellaneous collection of complex analysis subjects that were not covered in the previous two installments of the series.  The initial topics I have in mind to cover are

As usual, I will be posting lecture notes on this blog as the course progresses.

[Update: Mar 13: removed elliptic functions, as I have just learned that this was already covered in the prior 246B course.]

Suppose one has a bounded sequence {(a_n)_{n=1}^\infty = (a_1, a_2, \dots)} of real numbers. What kinds of limits can one form from this sequence?

Of course, we have the usual notion of limit {\lim_{n \rightarrow \infty} a_n}, which in this post I will refer to as the classical limit to distinguish from the other limits discussed in this post. The classical limit, if it exists, is the unique real number {L} such that for every {\varepsilon>0}, one has {|a_n-L| \leq \varepsilon} for all sufficiently large {n}. We say that a sequence is (classically) convergent if its classical limit exists. The classical limit obeys many useful limit laws when applied to classically convergent sequences. Firstly, it is linear: if {(a_n)_{n=1}^\infty} and {(b_n)_{n=1}^\infty} are classically convergent sequences, then {(a_n+b_n)_{n=1}^\infty} is also classically convergent with

\displaystyle  \lim_{n \rightarrow \infty} (a_n + b_n) = (\lim_{n \rightarrow \infty} a_n) + (\lim_{n \rightarrow \infty} b_n) \ \ \ \ \ (1)

and similarly for any scalar {c}, {(ca_n)_{n=1}^\infty} is classically convergent with

\displaystyle  \lim_{n \rightarrow \infty} (ca_n) = c \lim_{n \rightarrow \infty} a_n. \ \ \ \ \ (2)

It is also an algebra homomorphism: {(a_n b_n)_{n=1}^\infty} is also classically convergent with

\displaystyle  \lim_{n \rightarrow \infty} (a_n b_n) = (\lim_{n \rightarrow \infty} a_n) (\lim_{n \rightarrow \infty} b_n). \ \ \ \ \ (3)

We also have shift invariance: if {(a_n)_{n=1}^\infty} is classically convergent, then so is {(a_{n+1})_{n=1}^\infty} with

\displaystyle  \lim_{n \rightarrow \infty} a_{n+1} = \lim_{n \rightarrow \infty} a_n \ \ \ \ \ (4)

and more generally in fact for any injection {\phi: {\bf N} \rightarrow {\bf N}}, {(a_{\phi(n)})_{n=1}^\infty} is classically convergent with

\displaystyle  \lim_{n \rightarrow \infty} a_{\phi(n)} = \lim_{n \rightarrow \infty} a_n. \ \ \ \ \ (5)

The classical limit of a sequence is unchanged if one modifies any finite number of elements of the sequence. Finally, we have boundedness: for any classically convergent sequence {(a_n)_{n=1}^\infty}, one has

\displaystyle  \inf_n a_n \leq \lim_{n \rightarrow \infty} a_n \leq \sup_n a_n. \ \ \ \ \ (6)

One can in fact show without much difficulty that these laws uniquely determine the classical limit functional on convergent sequences.

One would like to extend the classical limit notion to more general bounded sequences; however, when doing so one must give up one or more of the desirable limit laws that were listed above. Consider for instance the sequence {a_n = (-1)^n}. On the one hand, one has {a_n^2 = 1} for all {n}, so if one wishes to retain the homomorphism property (3), any “limit” of this sequence {a_n} would have to necessarily square to {1}, that is to say it must equal {+1} or {-1}. On the other hand, if one wished to retain the shift invariance property (4) as well as the homogeneity property (2), any “limit” of this sequence would have to equal its own negation and thus be zero.

Nevertheless there are a number of useful generalisations and variants of the classical limit concept for non-convergent sequences that obey a significant portion of the above limit laws. For instance, we have the limit superior

\displaystyle  \limsup_{n \rightarrow \infty} a_n := \inf_N \sup_{n \geq N} a_n

and limit inferior

\displaystyle  \liminf_{n \rightarrow \infty} a_n := \sup_N \inf_{n \geq N} a_n

which are well-defined real numbers for any bounded sequence {(a_n)_{n=1}^\infty}; they agree with the classical limit when the sequence is convergent, but disagree otherwise. They enjoy the shift-invariance property (4), and the boundedness property (6), but do not in general obey the homomorphism property (3) or the linearity property (1); indeed, we only have the subadditivity property

\displaystyle  \limsup_{n \rightarrow \infty} (a_n + b_n) \leq (\limsup_{n \rightarrow \infty} a_n) + (\limsup_{n \rightarrow \infty} b_n)

for the limit superior, and the superadditivity property

\displaystyle  \liminf_{n \rightarrow \infty} (a_n + b_n) \geq (\liminf_{n \rightarrow \infty} a_n) + (\liminf_{n \rightarrow \infty} b_n)

for the limit inferior. The homogeneity property (2) is only obeyed by the limits superior and inferior for non-negative {c}; for negative {c}, one must have the limit inferior on one side of (2) and the limit superior on the other, thus for instance

\displaystyle  \limsup_{n \rightarrow \infty} (-a_n) = - \liminf_{n \rightarrow \infty} a_n.

The limit superior and limit inferior are examples of limit points of the sequence, which can for instance be defined as points that are limits of at least one subsequence of the original sequence. Indeed, the limit superior is always the largest limit point of the sequence, and the limit inferior is always the smallest limit point. However, limit points can be highly non-unique (indeed they are unique if and only if the sequence is classically convergent), and so it is difficult to sensibly interpret most of the usual limit laws in this setting, with the exception of the homogeneity property (2) and the boundedness property (6) that are easy to state for limit points.

Another notion of limit are the Césaro limits

\displaystyle  \mathrm{C}\!-\!\lim_{n \rightarrow \infty} a_n := \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N a_n;

if this limit exists, we say that the sequence is Césaro convergent. If the sequence {(a_n)_{n=1}^\infty} already has a classical limit, then it also has a Césaro limit that agrees with the classical limit; but there are additional sequences that have a Césaro limit but not a classical one. For instance, the non-classically convergent sequence {a_n= (-1)^n} discussed above is Césaro convergent, with a Césaro limit of {0}. However, there are still bounded sequences that do not have Césaro limit, such as {a_n := \sin( \log n )} (exercise!). The Césaro limit is linear, bounded, and shift invariant, but not an algebra homomorphism and also does not obey the rearrangement property (5).

Using the Hahn-Banach theorem, one can extend the classical limit functional to generalised limit functionals {\mathop{\widetilde \lim}_{n \rightarrow \infty} a_n}, defined to be bounded linear functionals from the space {\ell^\infty({\bf N})} of bounded real sequences to the real numbers {{\bf R}} that extend the classical limit functional (defined on the space {c_0({\bf N}) + {\bf R}} of convergent sequences) without any increase in the operator norm. (In some of my past writings I made the slight error of referring to these generalised limit functionals as Banach limits, though as discussed below, the latter actually refers to a subclass of generalised limit functionals.) It is not difficult to see that such generalised limit functionals will range between the limit inferior and limit superior. In fact, for any specific sequence {(a_n)_{n=1}^\infty} and any number {L} lying in the closed interval {[\liminf_{n \rightarrow \infty} a_n, \limsup_{n \rightarrow \infty} a_n]}, there exists at least one generalised limit functional {\mathop{\widetilde \lim}_{n \rightarrow \infty}} that takes the value {L} when applied to {a_n}; for instance, for any number {\theta} in {[-1,1]}, there exists a generalised limit functional that assigns that number {\theta} as the “limit” of the sequence {a_n = (-1)^n}. This claim can be seen by first designing such a limit functional on the vector space spanned by the convergent sequences and by {(a_n)_{n=1}^\infty}, and then appealing to the Hahn-Banach theorem to extend to all sequences. This observation also gives a necessary and sufficient criterion for a bounded sequence {(a_n)_{n=1}^\infty} to classically converge to a limit {L}, namely that all generalised limits of this sequence must equal {L}.

Because of the reliance on the Hahn-Banach theorem, the existence of generalised limits requires the axiom of choice (or some weakened version thereof); there are presumably models of set theory without the axiom of choice in which no generalised limits exist, but I do not know of an explicit reference for this.

Generalised limits can obey the shift-invariance property (4) or the algebra homomorphism property (3), but as the above analysis of the sequence {a_n = (-1)^n} shows, they cannot do both. Generalised limits that obey the shift-invariance property (4) are known as Banach limits; one can for instance construct them by applying the Hahn-Banach theorem to the Césaro limit functional; alternatively, if {\mathop{\widetilde \lim}} is any generalised limit, then the Césaro-type functional {(a_n)_{n=1}^\infty \mapsto \mathop{\widetilde \lim}_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N a_n} will be a Banach limit. The existence of Banach limits can be viewed as a demonstration of the amenability of the natural numbers (or integers); see this previous blog post for further discussion.

Generalised limits that obey the algebra homomorphism property (3) are known as ultrafilter limits. If one is given a generalised limit functional {p\!-\!\lim_{n \rightarrow \infty}} that obeys (2), then for any subset {A} of the natural numbers {{\bf N}}, the generalised limit {p\!-\!\lim_{n \rightarrow \infty} 1_A(n)} must equal its own square (since {1_A(n)^2 = 1_A(n)}) and is thus either {0} or {1}. If one defines {p \subset 2^{2^{\bf N}}} to be the collection of all subsets {A} of {{\bf N}} for which {p\!-\!\lim_{n \rightarrow \infty} 1_A(n) = 1}, one can verify that {p} obeys the axioms of a non-principal ultrafilter. Conversely, if {p} is a non-principal ultrafilter, one can define the associated generalised limit {p\!-\!\lim_{n \rightarrow \infty} a_n} of any bounded sequence {(a_n)_{n=1}^\infty} to be the unique real number {L} such that the sets {\{ n \in {\bf N}: |a_n - L| \leq \varepsilon \}} lie in {p} for all {\varepsilon>0}; one can check that this does indeed give a well-defined generalised limit that obeys (2). Non-principal ultrafilters can be constructed using Zorn’s lemma. In fact, they do not quite need the full strength of the axiom of choice; see the Wikipedia article on the ultrafilter lemma for examples.

We have previously noted that generalised limits of a sequence can converge to any point between the limit inferior and limit superior. The same is not true if one restricts to Banach limits or ultrafilter limits. For instance, by the arguments already given, the only possible Banach limit for the sequence {a_n = (-1)^n} is zero. Meanwhile, an ultrafilter limit must converge to a limit point of the original sequence, but conversely every limit point can be attained by at least one ultrafilter limit; we leave these assertions as an exercise to the interested reader. In particular, a bounded sequence converges classically to a limit {L} if and only if all ultrafilter limits converge to {L}.

There is no generalisation of the classical limit functional to any space that includes non-classically convergent sequences that obeys the subsequence property (5), since any non-classically-convergent sequence will have one subsequence that converges to the limit superior, and another subsequence that converges to the limit inferior, and one of these will have to violate (5) since the limit superior and limit inferior are distinct. So the above limit notions come close to the best generalisations of limit that one can use in practice.

We summarise the above discussion in the following table:

Limit Always defined Linear Shift-invariant Homomorphism Constructive
Classical No Yes Yes Yes Yes
Superior Yes No Yes No Yes
Inferior Yes No Yes No Yes
Césaro No Yes Yes No Yes
Generalised Yes Yes Depends Depends No
Banach Yes Yes Yes No No
Ultrafilter Yes Yes No Yes No

By an odd coincidence, I stumbled upon a second question in as many weeks about power series, and once again the only way I know how to prove the result is by complex methods; once again, I am leaving it here as a challenge to any interested readers, and I would be particularly interested in knowing of a proof that was not based on complex analysis (or thinly disguised versions thereof), or for a reference to previous literature where something like this identity has occured. (I suspect for instance that something like this may have shown up before in free probability, based on the answer to part (ii) of the problem.)

Here is a purely algebraic form of the problem:

Problem 1 Let {F = F(z)} be a formal function of one variable {z}. Suppose that {G = G(z)} is the formal function defined by

\displaystyle G := \sum_{n=1}^\infty \left( \frac{F^n}{n!} \right)^{(n-1)}

\displaystyle = F + \left(\frac{F^2}{2}\right)' + \left(\frac{F^3}{6}\right)'' + \dots

\displaystyle = F + FF' + (F (F')^2 + \frac{1}{2} F^2 F'') + \dots,

where we use {f^{(k)}} to denote the {k}-fold derivative of {f} with respect to the variable {z}.

  • (i) Show that {F} can be formally recovered from {G} by the formula

    \displaystyle F = \sum_{n=1}^\infty (-1)^{n-1} \left( \frac{G^n}{n!} \right)^{(n-1)}

    \displaystyle = G - \left(\frac{G^2}{2}\right)' + \left(\frac{G^3}{6}\right)'' - \dots

    \displaystyle = G - GG' + (G (G')^2 + \frac{1}{2} G^2 G'') - \dots.

  • (ii) There is a remarkable further formal identity relating {F(z)} with {G(z)} that does not explicitly involve any infinite summation. What is this identity?

To rigorously formulate part (i) of this problem, one could work in the commutative differential ring of formal infinite series generated by polynomial combinations of {F} and its derivatives (with no constant term). Part (ii) is a bit trickier to formulate in this abstract ring; the identity in question is easier to state if {F, G} are formal power series, or (even better) convergent power series, as it involves operations such as composition or inversion that can be more easily defined in those latter settings.

To illustrate Problem 1(i), let us compute up to third order in {F}, using {{\mathcal O}(F^4)} to denote any quantity involving four or more factors of {F} and its derivatives, and similarly for other exponents than {4}. Then we have

\displaystyle G = F + FF' + (F (F')^2 + \frac{1}{2} F^2 F'') + {\mathcal O}(F^4)

and hence

\displaystyle G' = F' + (F')^2 + FF'' + {\mathcal O}(F^3)

\displaystyle G'' = F'' + {\mathcal O}(F^2);

multiplying, we have

\displaystyle GG' = FF' + F (F')^2 + F^2 F'' + F (F')^2 + {\mathcal O}(F^4)


\displaystyle G (G')^2 + \frac{1}{2} G^2 G'' = F (F')^2 + \frac{1}{2} F^2 F'' + {\mathcal O}(F^4)

and hence after a lot of canceling

\displaystyle G - GG' + (G (G')^2 + \frac{1}{2} G^2 G'') = F + {\mathcal O}(F^4).

Thus Problem 1(i) holds up to errors of {{\mathcal O}(F^4)} at least. In principle one can continue verifying Problem 1(i) to increasingly high order in {F}, but the computations rapidly become quite lengthy, and I do not know of a direct way to ensure that one always obtains the required cancellation at the end of the computation.

Problem 1(i) can also be posed in formal power series: if

\displaystyle F(z) = a_1 z + a_2 z^2 + a_3 z^3 + \dots

is a formal power series with no constant term with complex coefficients {a_1, a_2, \dots} with {|a_1|<1}, then one can verify that the series

\displaystyle G := \sum_{n=1}^\infty \left( \frac{F^n}{n!} \right)^{(n-1)}

makes sense as a formal power series with no constant term, thus

\displaystyle G(z) = b_1 z + b_2 z^2 + b_3 z^3 + \dots.

For instance it is not difficult to show that {b_1 = \frac{a_1}{1-a_1}}. If one further has {|b_1| < 1}, then it turns out that

\displaystyle F = \sum_{n=1}^\infty (-1)^{n-1} \left( \frac{G^n}{n!} \right)^{(n-1)}

as formal power series. Currently the only way I know how to show this is by first proving the claim for power series with a positive radius of convergence using the Cauchy integral formula, but even this is a bit tricky unless one has managed to guess the identity in (ii) first. (In fact, the way I discovered this problem was by first trying to solve (a variant of) the identity in (ii) by Taylor expansion in the course of attacking another problem, and obtaining the transform in Problem 1 as a consequence.)

The transform that takes {F} to {G} resembles both the exponential function

\displaystyle \exp(F) = \sum_{n=0}^\infty \frac{F^n}{n!}

and Taylor’s formula

\displaystyle F(z) = \sum_{n=0}^\infty \frac{F^{(n)}(0)}{n!} z^n

but does not seem to be directly connected to either (this is more apparent once one knows the identity in (ii)).

In the previous set of notes we introduced the notion of a complex diffeomorphism {f: U \rightarrow V} between two open subsets {U,V} of the complex plane {{\bf C}} (or more generally, two Riemann surfaces): an invertible holomorphic map whose inverse was also holomorphic. (Actually, the last part is automatic, thanks to Exercise 40 of Notes 4.) Such maps are also known as biholomorphic maps or conformal maps (although in some literature the notion of “conformal map” is expanded to permit maps such as the complex conjugation map {z \mapsto \overline{z}} that are angle-preserving but not orientation-preserving, as well as maps such as the exponential map {z \mapsto \exp(z)} from {{\bf C}} to {{\bf C} \backslash \{0\}} that are only locally injective rather than globally injective). Such complex diffeomorphisms can be used in complex analysis (or in the analysis of harmonic functions) to change the underlying domain {U} to a domain that may be more convenient for calculations, thanks to the following basic lemma:

Lemma 1 (Holomorphicity and harmonicity are conformal invariants) Let {\phi: U \rightarrow V} be a complex diffeomorphism between two Riemann surfaces {U,V}.

  • (i) If {f: V \rightarrow W} is a function to another Riemann surface {W}, then {f} is holomorphic if and only if {f \circ \phi: U \rightarrow W} is holomorphic.
  • (ii) If {U,V} are open subsets of {{\bf C}} and {u: V \rightarrow {\bf R}} is a function, then {u} is harmonic if and only if {u \circ \phi: U \rightarrow {\bf R}} is harmonic.

Proof: Part (i) is immediate since the composition of two holomorphic functions is holomorphic. For part (ii), observe that if {u: V \rightarrow {\bf R}} is harmonic then on any ball {B(z_0,r)} in {V}, {u} is the real part of some holomorphic function {f: B(z_0,r) \rightarrow {\bf C}} thanks to Exercise 62 of Notes 3. By part (i), {f \circ \phi: B(z_0,r) \rightarrow {\bf C}} is also holomorphic. Taking real parts we see that {u \circ \phi} is harmonic on each ball {B(z_0,r)} in {V}, and hence harmonic on all of {V}, giving one direction of (ii); the other direction is proven similarly. \Box

Exercise 2 Establish Lemma 1(ii) by direct calculation, avoiding the use of holomorphic functions. (Hint: the calculations are cleanest if one uses Wirtinger derivatives, as per Exercise 27 of Notes 1.)

Exercise 3 Let {\phi: U \rightarrow V} be a complex diffeomorphism between two open subsets {U,V} of {{\bf C}}, let {z_0} be a point in {U}, let {m} be a natural number, and let {f: V \rightarrow {\bf C} \cup \{\infty\}} be holomorphic. Show that {f: V \rightarrow {\bf C} \cup \{\infty\}} has a zero (resp. a pole) of order {m} at {\phi(z_0)} if and only if {f \circ \phi: U \rightarrow {\bf C} \cup \{\infty\}} has a zero (resp. a pole) of order {m} at {z_0}.

From Lemma 1(ii) we can now define the notion of a harmonic function {u: M \rightarrow {\bf R}} on a Riemann surface {M}; such a function {u} is harmonic if, for every coordinate chart {\phi_\alpha: U_\alpha \rightarrow V_\alpha} in some atlas, the map {u \circ \phi_\alpha^{-1}: V_\alpha \rightarrow {\bf R}} is harmonic. Lemma 1(ii) ensures that this definition of harmonicity does not depend on the choice of atlas. Similarly, using Exercise 3 one can define what it means for a holomorphic map {f: M \rightarrow {\bf C} \cup \{\infty\}} on a Riemann surface {M} to have a pole or zero of a given order at a point {p_0 \in M}, with the definition being independent of the choice of atlas.

In view of Lemma 1, it is thus natural to ask which Riemann surfaces are complex diffeomorphic to each other, and more generally to understand the space of holomorphic maps from one given Riemann surface to another. We will initially focus attention on three important model Riemann surfaces:

  • (i) (Elliptic model) The Riemann sphere {{\bf C} \cup \{\infty\}};
  • (ii) (Parabolic model) The complex plane {{\bf C}}; and
  • (iii) (Hyperbolic model) The unit disk {D(0,1)}.

The designation of these model Riemann surfaces as elliptic, parabolic, and hyperbolic comes from Riemannian geometry, where it is natural to endow each of these surfaces with a constant curvature Riemannian metric which is positive, zero, or negative in the elliptic, parabolic, and hyperbolic cases respectively. However, we will not discuss Riemannian geometry further here.

All three model Riemann surfaces are simply connected, but none of them are complex diffeomorphic to any other; indeed, there are no non-constant holomorphic maps from the Riemann sphere to the plane or the disk, nor are there any non-constant holomorphic maps from the plane to the disk (although there are plenty of holomorphic maps going in the opposite directions). The complex automorphisms (that is, the complex diffeomorphisms from a surface to itself) of each of the three surfaces can be classified explicitly. The automorphisms of the Riemann sphere turn out to be the Möbius transformations {z \mapsto \frac{az+b}{cz+d}} with {ad-bc \neq 0}, also known as fractional linear transformations. The automorphisms of the complex plane are the linear transformations {z \mapsto az+b} with {a \neq 0}, and the automorphisms of the disk are the fractional linear transformations of the form {z \mapsto e^{i\theta} \frac{\alpha - z}{1 - \overline{\alpha} z}} for {\theta \in {\bf R}} and {\alpha \in D(0,1)}. Holomorphic maps {f: D(0,1) \rightarrow D(0,1)} from the disk {D(0,1)} to itself that fix the origin obey a basic but incredibly important estimate known as the Schwarz lemma: they are “dominated” by the identity function {z \mapsto z} in the sense that {|f(z)| \leq |z|} for all {z \in D(0,1)}. Among other things, this lemma gives guidance to determine when a given Riemann surface is complex diffeomorphic to a disk; we shall discuss this point further below.

It is a beautiful and fundamental fact in complex analysis that these three model Riemann surfaces are in fact an exhaustive list of the simply connected Riemann surfaces, up to complex diffeomorphism. More precisely, we have the Riemann mapping theorem and the uniformisation theorem:

Theorem 4 (Riemann mapping theorem) Let {U} be a simply connected open subset of {{\bf C}} that is not all of {{\bf C}}. Then {U} is complex diffeomorphic to {D(0,1)}.

Theorem 5 (Uniformisation theorem) Let {M} be a simply connected Riemann surface. Then {M} is complex diffeomorphic to {{\bf C} \cup \{\infty\}}, {{\bf C}}, or {D(0,1)}.

As we shall see, every connected Riemann surface can be viewed as the quotient of its simply connected universal cover by a discrete group of automorphisms known as deck transformations. This in principle gives a complete classification of Riemann surfaces up to complex diffeomorphism, although the situation is still somewhat complicated in the hyperbolic case because of the wide variety of discrete groups of automorphisms available in that case.

We will prove the Riemann mapping theorem in these notes, using the elegant argument of Koebe that is based on the Schwarz lemma and Montel’s theorem (Exercise 57 of Notes 4). The uniformisation theorem is however more difficult to establish; we discuss some components of a proof (based on the Perron method of subharmonic functions) here, but stop short of providing a complete proof.

The above theorems show that it is in principle possible to conformally map various domains into model domains such as the unit disk, but the proofs of these theorems do not readily produce explicit conformal maps for this purpose. For some domains we can just write down a suitable such map. For instance:

Exercise 6 (Cayley transform) Let {{\bf H} := \{ z \in {\bf C}: \mathrm{Im} z > 0 \}} be the upper half-plane. Show that the Cayley transform {\phi: {\bf H} \rightarrow D(0,1)}, defined by

\displaystyle  \phi(z) := \frac{z-i}{z+i},

is a complex diffeomorphism from the upper half-plane {{\bf H}} to the disk {D(0,1)}, with inverse map {\phi^{-1}: D(0,1) \rightarrow {\bf H}} given by

\displaystyle  \phi^{-1}(w) := i \frac{1+w}{1-w}.

Exercise 7 Show that for any real numbers {a<b}, the strip {\{ z \in {\bf C}: a < \mathrm{Re}(z) < b \}} is complex diffeomorphic to the disk {D(0,1)}. (Hint: use the complex exponential and a linear transformation to map the strip onto the half-plane {{\bf H}}.)

Exercise 8 Show that for any real numbers $latex {a<b0, a < \theta < b \}}&fg=000000$ is complex diffeomorphic to the disk {D(0,1)}. (Hint: use a branch of either the complex logarithm, or of a complex power {z \mapsto z^\alpha}.)

We will discuss some other explicit conformal maps in this set of notes, such as the Schwarz-Christoffel maps that transform the upper half-plane {{\bf H}} to polygonal regions. Further examples of conformal mapping can be found in the text of Stein-Shakarchi.

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My colleague Tom Liggett recently posed to me the following problem about power series in one real variable {x}. Observe that the power series

\displaystyle  \sum_{n=0}^\infty (-1)^n\frac{x^n}{n!}

has very rapidly decaying coefficients (of order {O(1/n!)}), leading to an infinite radius of convergence; also, as the series converges to {e^{-x}}, the series decays very rapidly as {x} approaches {+\infty}. The problem is whether this is essentially the only example of this type. More precisely:

Problem 1 Let {a_0, a_1, \dots} be a bounded sequence of real numbers, and suppose that the power series

\displaystyle  f(x) := \sum_{n=0}^\infty a_n\frac{x^n}{n!}

(which has an infinite radius of convergence) decays like {O(e^{-x})} as {x \rightarrow +\infty}, in the sense that the function {e^x f(x)} remains bounded as {x \rightarrow +\infty}. Must the sequence {a_n} be of the form {a_n = C (-1)^n} for some constant {C}?

As it turns out, the problem has a very nice solution using complex analysis methods, which by coincidence I happen to be teaching right now. I am therefore posing as a challenge to my complex analysis students and to other readers of this blog to answer the above problem by complex methods; feel free to post solutions in the comments below (and in particular, if you don’t want to be spoiled, you should probably refrain from reading the comments). In fact, the only way I know how to solve this problem currently is by complex methods; I would be interested in seeing a purely real-variable solution that is not simply a thinly disguised version of a complex-variable argument.

(To be fair to my students, the complex variable argument does require one additional tool that is not directly covered in my notes. That tool can be found here.)

In the previous set of notes we saw that functions {f: U \rightarrow {\bf C}} that were holomorphic on an open set {U} enjoyed a large number of useful properties, particularly if the domain {U} was simply connected. In many situations, though, we need to consider functions {f} that are only holomorphic (or even well-defined) on most of a domain {U}, thus they are actually functions {f: U \backslash S \rightarrow {\bf C}} outside of some small singular set {S} inside {U}. (In this set of notes we only consider interior singularities; one can also discuss singular behaviour at the boundary of {U}, 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 {S} to be closed, so that the domain {U \backslash S} on which {f} remains holomorphic is still open. A typical class of examples are the functions of the form {\frac{f(z)}{z-z_0}} that were already encountered in the Cauchy integral formula; if {f: U \rightarrow {\bf C}} is holomorphic and {z_0 \in U}, such a function would be holomorphic save for a singularity at {z_0}. Another basic class of examples are the rational functions {P(z)/Q(z)}, which are holomorphic outside of the zeroes of the denominator {Q}.

Singularities come in varying levels of “badness” in complex analysis. The least harmful type of singularity is the removable singularity – a point {z_0} which is an isolated singularity (i.e., an isolated point of the singular set {S}) where the function {f} 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 {\frac{\sin(z)}{z}}, which has a removable singularity at the origin {0}, which can be removed by declaring the sinc function to equal {1} at {0}. 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 {f: U \backslash S \rightarrow {\bf C}} is bounded near an isolated singularity {z_0 \in S}, then the singularity at {z_0} may be removed.

After removable singularities, the mildest form of singularity one can encounter is that of a pole – an isolated singularity {z_0} such that {f(z)} can be factored as {f(z) = \frac{g(z)}{(z-z_0)^m}} for some {m \geq 1} (known as the order of the pole), where {g} has a removable singularity at {z_0} (and is non-zero at {z_0} once the singularity is removed). Such functions have already made a frequent appearance in previous notes, particularly the case of simple poles when {m=1}. The behaviour near {z_0} of function {f} with a pole of order {m} is well understood: for instance, {|f(z)|} goes to infinity as {z} approaches {z_0} (at a rate comparable to {|z-z_0|^{-m}}). These singularities are not, strictly speaking, removable; but if one compactifies the range {{\bf C}} of the holomorphic function {f: U \backslash S \rightarrow {\bf C}} to a slightly larger space {{\bf C} \cup \{\infty\}} known as the Riemann sphere, then the singularity can be removed. In particular, functions {f: U \backslash S \rightarrow {\bf C}} which only have isolated singularities that are either poles or removable can be extended to holomorphic functions {f: U \rightarrow {\bf C} \cup \{\infty\}} 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 {U} 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 {f(z) = e^{1/z}}, which turns out to have an essential singularity at {z=0}. The behaviour of such essential singularities is quite wild; we will show here the Casorati-Weierstrass theorem, which shows that the image of {f} 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) {z_0}, it is possible to expand {f} as a variant of a Taylor series known as a Laurent series {\sum_{n=-\infty}^\infty a_n (z-z_0)^n}. The {\frac{1}{z-z_0}} coefficient {a_{-1}} of this series is particularly important for contour integration purposes, and is known as the residue of {f} at the isolated singularity {z_0}. 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 {z \mapsto \exp(z)} or a power function {z \mapsto z^n}, leading to branches of multivalued functions such as the complex logarithm {z \mapsto \log(z)} or the {n^{th}} root function {z \mapsto z^{1/n}} 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 {U} 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).

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