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Consider a disk ${D(z_0,r) := \{ z: |z-z_0| < r \}}$ in the complex plane. If one applies an affine-linear map ${f(z) = az+b}$ to this disk, one obtains

$\displaystyle f(D(z_0,r)) = D(f(z_0), |f'(z_0)| r).$

For maps that are merely holomorphic instead of affine-linear, one has some variants of this assertion, which I am recording here mostly for my own reference:

Theorem 1 (Holomorphic images of disks) Let ${D(z_0,r)}$ be a disk in the complex plane, and ${f: D(z_0,r) \rightarrow {\bf C}}$ be a holomorphic function with ${f'(z_0) \neq 0}$.
• (i) (Open mapping theorem or inverse function theorem) ${f(D(z_0,r))}$ contains a disk ${D(f(z_0),\varepsilon)}$ for some ${\varepsilon>0}$. (In fact there is even a holomorphic right inverse of ${f}$ from ${D(f(z_0), \varepsilon)}$ to ${D(z_0,r)}$.)
• (ii) (Bloch theorem) ${f(D(z_0,r))}$ contains a disk ${D(w, c |f'(z_0)| r)}$ for some absolute constant ${c>0}$ and some ${w \in {\bf C}}$. (In fact there is even a holomorphic right inverse of ${f}$ from ${D(w, c |f'(z_0)| r)}$ to ${D(z_0,r)}$.)
• (iii) (Koebe quarter theorem) If ${f}$ is injective, then ${f(D(z_0,r))}$ contains the disk ${D(f(z_0), \frac{1}{4} |f'(z_0)| r)}$.
• (iv) If ${f}$ is a polynomial of degree ${n}$, then ${f(D(z_0,r))}$ contains the disk ${D(f(z_0), \frac{1}{n} |f'(z_0)| r)}$.
• (v) If one has a bound of the form ${|f'(z)| \leq A |f'(z_0)|}$ for all ${z \in D(z_0,r)}$ and some ${A>1}$, then ${f(D(z_0,r))}$ contains the disk ${D(f(z_0), \frac{c}{A} |f'(z_0)| r)}$ for some absolute constant ${c>0}$. (In fact there is holomorphic right inverse of ${f}$ from ${D(f(z_0), \frac{c}{A} |f'(z_0)| r)}$ to ${D(z_0,r)}$.)

Parts (i), (ii), (iii) of this theorem are standard, as indicated by the given links. I found part (iv) as (a consequence of) Theorem 2 of this paper of Degot, who remarks that it “seems not already known in spite of its simplicity”; an equivalent form of this result also appears in Lemma 4 of this paper of Miller. The proof is simple:

Proof: (Proof of (iv)) Let ${w \in D(f(z_0), \frac{1}{n} |f'(z_0)| r)}$, then we have a lower bound for the log-derivative of ${f(z)-w}$ at ${z_0}$:

$\displaystyle \frac{|f'(z_0)|}{|f(z_0)-w|} > \frac{n}{r}$

(with the convention that the left-hand side is infinite when ${f(z_0)=w}$). But by the fundamental theorem of algebra we have

$\displaystyle \frac{f'(z_0)}{f(z_0)-w} = \sum_{j=1}^n \frac{1}{z_0-\zeta_j}$

where ${\zeta_1,\dots,\zeta_n}$ are the roots of the polynomial ${f(z)-w}$ (counting multiplicity). By the pigeonhole principle, there must therefore exist a root ${\zeta_j}$ of ${f(z) - w}$ such that

$\displaystyle \frac{1}{|z_0-\zeta_j|} > \frac{1}{r}$

and hence ${\zeta_j \in D(z_0,r)}$. Thus ${f(D(z_0,r))}$ contains ${w}$, and the claim follows. $\Box$

The constant ${\frac{1}{n}}$ in (iv) is completely sharp: if ${f(z) = z^n}$ and ${z_0}$ is non-zero then ${f(D(z_0,|z_0|))}$ contains the disk

$\displaystyle D(f(z_0), \frac{1}{n} |f'(z_0)| r) = D( z_0^n, |z_0|^n)$

but avoids the origin, thus does not contain any disk of the form ${D( z_0^n, |z_0|^n+\varepsilon)}$. This example also shows that despite parts (ii), (iii) of the theorem, one cannot hope for a general inclusion of the form

$\displaystyle f(D(z_0,r)) \supset D(f(z_0), c |f'(z_0)| r )$

for an absolute constant ${c>0}$.

Part (v) is implicit in the standard proof of Bloch’s theorem (part (ii)), and is easy to establish:

Proof: (Proof of (v)) From the Cauchy inequalities one has ${f''(z) = O(\frac{A}{r} |f'(z_0)|)}$ for ${z \in D(z_0,r/2)}$, hence by Taylor’s theorem with remainder ${f(z) = f(z_0) + f'(z_0) (z-z_0) (1 + O( A \frac{|z-z_0|}{r} ) )}$ for ${z \in D(z_0, r/2)}$. By Rouche’s theorem, this implies that the function ${f(z)-w}$ has a unique zero in ${D(z_0, 2cr/A)}$ for any ${w \in D(f(z_0), cr|f'(z_0)|/A)}$, if ${c>0}$ is a sufficiently small absolute constant. The claim follows. $\Box$

Note that part (v) implies part (i). A standard point picking argument also lets one deduce part (ii) from part (v):

Proof: (Proof of (ii)) By shrinking ${r}$ slightly if necessary we may assume that ${f}$ extends analytically to the closure of the disk ${D(z_0,r)}$. Let ${c}$ be the constant in (v) with ${A=2}$; we will prove (iii) with ${c}$ replaced by ${c/2}$. If we have ${|f'(z)| \leq 2 |f'(z_0)|}$ for all ${z \in D(z_0,r/2)}$ then we are done by (v), so we may assume without loss of generality that there is ${z_1 \in D(z_0,r/2)}$ such that ${|f'(z_1)| > 2 |f'(z_0)|}$. If ${|f'(z)| \leq 2 |f'(z_1)|}$ for all ${z \in D(z_1,r/4)}$ then by (v) we have

$\displaystyle f( D(z_0, r) ) \supset f( D(z_1,r/2) ) \supset D( f(z_1), \frac{c}{2} |f'(z_1)| \frac{r}{2} )$

$\displaystyle \supset D( f(z_1), \frac{c}{2} |f'(z_0)| r )$

and we are again done. Hence we may assume without loss of generality that there is ${z_2 \in D(z_1,r/4)}$ such that ${|f'(z_2)| > 2 |f'(z_1)|}$. Iterating this procedure in the obvious fashion we either are done, or obtain a Cauchy sequence ${z_0, z_1, \dots}$ in ${D(z_0,r)}$ such that ${f'(z_j)}$ goes to infinity as ${j \rightarrow \infty}$, which contradicts the analytic nature of ${f}$ (and hence continuous nature of ${f'}$) on the closure of ${D(z_0,r)}$. This gives the claim. $\Box$

Here is another classical result stated by Alexander (and then proven by Kakeya and by Szego, but also implied to a classical theorem of Grace and Heawood) that is broadly compatible with parts (iii), (iv) of the above theorem:

Proposition 2 Let ${D(z_0,r)}$ be a disk in the complex plane, and ${f: D(z_0,r) \rightarrow {\bf C}}$ be a polynomial of degree ${n \geq 1}$ with ${f'(z) \neq 0}$ for all ${z \in D(z_0,r)}$. Then ${f}$ is injective on ${D(z_0, \sin\frac{\pi}{n})}$.

The radius ${\sin \frac{\pi}{n}}$ is best possible, for the polynomial ${f(z) = z^n}$ has ${f'}$ non-vanishing on ${D(1,1)}$, but one has ${f(\cos(\pi/n) e^{i \pi/n}) = f(\cos(\pi/n) e^{-i\pi/n})}$, and ${\cos(\pi/n) e^{i \pi/n}, \cos(\pi/n) e^{-i\pi/n}}$ lie on the boundary of ${D(1,\sin \frac{\pi}{n})}$.

If one narrows ${\sin \frac{\pi}{n}}$ slightly to ${\sin \frac{\pi}{2n}}$ then one can quickly prove this proposition as follows. Suppose for contradiction that there exist distinct ${z_1, z_2 \in D(z_0, \sin\frac{\pi}{n})}$ with ${f(z_1)=f(z_2)}$, thus if we let ${\gamma}$ be the line segment contour from ${z_1}$ to ${z_2}$ then ${\int_\gamma f'(z)\ dz}$. However, by assumption we may factor ${f'(z) = c (z-\zeta_1) \dots (z-\zeta_{n-1})}$ where all the ${\zeta_j}$ lie outside of ${D(z_0,r)}$. Elementary trigonometry then tells us that the argument of ${z-\zeta_j}$ only varies by less than ${\frac{\pi}{n}}$ as ${z}$ traverses ${\gamma}$, hence the argument of ${f'(z)}$ only varies by less than ${\pi}$. Thus ${f'(z)}$ takes values in an open half-plane avoiding the origin and so it is not possible for ${\int_\gamma f'(z)\ dz}$ to vanish.

To recover the best constant of ${\sin \frac{\pi}{n}}$ requires some effort. By taking contrapositives and applying an affine rescaling and some trigonometry, the proposition can be deduced from the following result, known variously as the Grace-Heawood theorem or the complex Rolle theorem.

Proposition 3 (Grace-Heawood theorem) Let ${f: {\bf C} \rightarrow {\bf C}}$ be a polynomial of degree ${n \geq 1}$ such that ${f(1)=f(-1)}$. Then ${f'}$ contains a zero in the closure of ${D( 0, \cot \frac{\pi}{n} )}$.

This is in turn implied by a remarkable and powerful theorem of Grace (which we shall prove shortly). Given two polynomials ${f,g}$ of degree at most ${n}$, define the apolar form ${(f,g)_n}$ by

$\displaystyle (f,g)_n := \sum_{k=0}^n (-1)^k f^{(k)}(0) g^{(n-k)}(0). \ \ \ \ \ (1)$

Theorem 4 (Grace’s theorem) Let ${C}$ be a circle or line in ${{\bf C}}$, dividing ${{\bf C} \backslash C}$ into two open connected regions ${\Omega_1, \Omega_2}$. Let ${f,g}$ be two polynomials of degree at most ${n \geq 1}$, with all the zeroes of ${f}$ lying in ${\Omega_1}$ and all the zeroes of ${g}$ lying in ${\Omega_2}$. Then ${(f,g)_n \neq 0}$.

(Contrapositively: if ${(f,g)_n=0}$, then the zeroes of ${f}$ cannot be separated from the zeroes of ${g}$ by a circle or line.)

Indeed, a brief calculation reveals the identity

$\displaystyle f(1) - f(-1) = (f', g)_{n-1}$

where ${g}$ is the degree ${n-1}$ polynomial

$\displaystyle g(z) := \frac{1}{n!} ((z+1)^n - (z-1)^n).$

The zeroes of ${g}$ are ${i \cot \frac{\pi j}{n}}$ for ${j=1,\dots,n-1}$, so the Grace-Heawood theorem follows by applying Grace’s theorem with ${C}$ equal to the boundary of ${D(0, \cot \frac{\pi}{n})}$.

The same method of proof gives the following nice consequence:

Theorem 5 (Perpendicular bisector theorem) Let ${f: {\bf C} \rightarrow C}$ be a polynomial such that ${f(z_1)=f(z_2)}$ for some distinct ${z_1,z_2}$. Then the zeroes of ${f'}$ cannot all lie on one side of the perpendicular bisector of ${z_1,z_2}$. For instance, if ${f(1)=f(-1)}$, then the zeroes of ${f'}$ cannot all lie in the halfplane ${\{ z: \mathrm{Re} z > 0 \}}$ or the halfplane ${\{ z: \mathrm{Re} z < 0 \}}$.

I’d be interested in seeing a proof of this latter theorem that did not proceed via Grace’s theorem.

Now we give a proof of Grace’s theorem. The case ${n=1}$ can be established by direct computation, so suppose inductively that ${n>1}$ and that the claim has already been established for ${n-1}$. Given the involvement of circles and lines it is natural to suspect that a Möbius transformation symmetry is involved. This is indeed the case and can be made precise as follows. Let ${V_n}$ denote the vector space of polynomials ${f}$ of degree at most ${n}$, then the apolar form is a bilinear form ${(,)_n: V_n \times V_n \rightarrow {\bf C}}$. Each translation ${z \mapsto z+a}$ on the complex plane induces a corresponding map on ${V_n}$, mapping each polynomial ${f}$ to its shift ${\tau_a f(z) := f(z-a)}$. We claim that the apolar form is invariant with respect to these translations:

$\displaystyle ( \tau_a f, \tau_a g )_n = (f,g)_n.$

Taking derivatives in ${a}$, it suffices to establish the skew-adjointness relation

$\displaystyle (f', g)_n + (f,g')_n = 0$

but this is clear from the alternating form of (1).

Next, we see that the inversion map ${z \mapsto 1/z}$ also induces a corresponding map on ${V_n}$, mapping each polynomial ${f \in V_n}$ to its inversion ${\iota f(z) := z^n f(1/z)}$. From (1) we see that this map also (projectively) preserves the apolar form:

$\displaystyle (\iota f, \iota g)_n = (-1)^n (f,g)_n.$

More generally, the group of Möbius transformations on the Riemann sphere acts projectively on ${V_n}$, with each Möbius transformation ${T: {\bf C} \rightarrow {\bf C}}$ mapping each ${f \in V_n}$ to ${Tf(z) := g_T(z) f(T^{-1} z)}$, where ${g_T}$ is the unique (up to constants) rational function that maps this a map from ${V_n}$ to ${V_n}$ (its divisor is ${n(T \infty) - n(\infty)}$). Since the Möbius transformations are generated by translations and inversion, we see that the action of Möbius transformations projectively preserves the apolar form; also, we see this action of ${T}$ on ${V_n}$ also moves the zeroes of each ${f \in V_n}$ by ${T}$ (viewing polynomials of degree less than ${n}$ in ${V_n}$ as having zeroes at infinity). In particular, the hypotheses and conclusions of Grace’s theorem are preserved by this Möbius action. We can then apply such a transformation to move one of the zeroes of ${f}$ to infinity (thus making ${f}$ a polynomial of degree ${n-1}$), so that ${C}$ must now be a circle, with the zeroes of ${g}$ inside the circle and the remaining zeroes of ${f}$ outside the circle. But then

$\displaystyle (f,g)_n = (f, g')_{n-1}.$

By the Gauss-Lucas theorem, the zeroes of ${g'}$ are also inside ${C}$. The claim now follows from the induction hypothesis.

Starting on Oct 2, I will be teaching Math 246A, the first course in the three-quarter graduate complex analysis sequence at the math department here at UCLA.  This first course covers much of the same ground as an honours undergraduate complex analysis course, in particular focusing on the basic properties of holomorphic functions such as the Cauchy and residue theorems, the classification of singularities, and the maximum principle, but there will be more of an emphasis on rigour, generalisation and abstraction, and connections with other parts of mathematics.  The main text I will be using for this course is Stein-Shakarchi (with Ahlfors as a secondary text), but I will also be using the blog lecture notes I wrote the last time I taught this course in 2016. At this time I do not expect to significantly deviate from my past lecture notes, though I do not know at present how different the pace will be this quarter when the course is taught remotely. As with my 247B course last spring, the lectures will be open to the public, though other coursework components will be restricted to enrolled students.

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

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

and

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

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)).

Previous set of notes: Notes 4. Next set of notes: 246B Notes 1.

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 41 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), ${f \circ \phi: \phi^{-1}(B(z_0,r)) \rightarrow {\bf C}}$ is also holomorphic. Taking real parts we see that ${u \circ \phi}$ is harmonic on each ball preimage ${\phi^{-1}(B(z_0,r))}$ 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; we can also identify such functions as equivalence classes of meromorphic functions ${f: M \backslash S \rightarrow {\bf C}}$ in complete analogy with the case of meromorphic functions on domains ${U}$. Finally, we can define the notion of an essential singularity of a holomorphic function ${f: M \backslash S \rightarrow {\bf C}}$ at some isolated singularity ${p \in S}$ in a Riemann surface as one that cannot be extended to a holomorphic function ${f: (M \backslash S) \cup \{p\} \rightarrow{\bf C} \cup \{\infty\}}$.

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 58 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, 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 ${a, the strip ${\{ re^{i\theta}: r>0, a < \theta < b \}}$ 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.

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.)

Previous set of notes: Notes 3. Next set of notes: Notes 5.

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 37 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 this topic will be postponed to the next course in this sequence.

Previous set of notes: Notes 2. Next set of notes: Notes 4.

[Warning: these notes have been substantially edited on Nov 9, 2021. Any references to theorem or exercise numbers before this date may now be inaccurate.]

We now come to perhaps the most central theorem in complex analysis (save possibly for the fundamental theorem of calculus), namely Cauchy’s theorem, which allows one to compute a large number of contour integrals ${\int_\gamma f(z)\ dz}$ even without knowing any explicit antiderivative of ${f}$. There are many forms and variants of Cauchy’s theorem. To give one such version, we need the basic topological notion of a homotopy:

Definition 1 (Homotopy) Let ${U}$ be an open subset of ${{\bf C}}$, and let ${\gamma_0: [a,b] \rightarrow U}$, ${\gamma_1: [a,b] \rightarrow U}$ be two curves in ${U}$.

• (i) If ${\gamma_0, \gamma_1}$ have the same initial point ${z_0}$ and terminal point ${z_1}$, we say that ${\gamma_0}$ and ${\gamma_1}$ are homotopic with fixed endpoints in ${U}$ if there exists a continuous map ${\gamma: [0,1] \times [a,b] \rightarrow U}$ such that ${\gamma(0,t) = \gamma_0(t)}$ and ${\gamma(1,t) = \gamma_1(t)}$ for all ${t \in [a,b]}$, and such that ${\gamma(s,a) = z_0}$ and ${\gamma(s,b) = z_1}$ for all ${s \in [0,1]}$.
• (ii) If ${\gamma_0, \gamma_1}$ are closed (but possibly with different initial points), we say that ${\gamma_0}$ and ${\gamma_1}$ are homotopic as closed curves in ${U}$ if there exists a continuous map ${\gamma: [0,1] \times [a,b] \rightarrow U}$ such that ${\gamma(0,t) = \gamma_0(t)}$ and ${\gamma(1,t) = \gamma_1(t)}$ for all ${t \in [a,b]}$, and such that ${\gamma(s,a) = \gamma(s,b)}$ for all ${s \in [0,1]}$.
• (iii) If ${\gamma_2: [c,d] \rightarrow U}$ and ${\gamma_3: [e,f] \rightarrow U}$ are curves with the same initial point and same terminal point, we say that ${\gamma_2}$ and ${\gamma_3}$ are homotopic with fixed endpoints up to reparameterisation in ${U}$ if there is a reparameterisation ${\tilde \gamma_2: [a,b] \rightarrow U}$ of ${\gamma_2}$ which is homotopic with fixed endpoints in ${U}$ to a reparameterisation ${\tilde \gamma_3: [a,b] \rightarrow U}$ of ${\gamma_3}$.
• (iv) If ${\gamma_2: [c,d] \rightarrow U}$ and ${\gamma_3: [e,f] \rightarrow U}$ are closed curves, we say that ${\gamma_2}$ and ${\gamma_3}$ are homotopic as closed curves up to reparameterisation in ${U}$ if there is a reparameterisation ${\tilde \gamma_2: [a,b] \rightarrow U}$ of ${\gamma_2}$ which is homotopic as closed curves in ${U}$ to a reparameterisation ${\tilde \gamma_3: [a,b] \rightarrow U}$ of ${\gamma_3}$.

In the first two cases, the map ${\gamma}$ will be referred to as a homotopy from ${\gamma_0}$ to ${\gamma_1}$, and we will also say that ${\gamma_0}$ can be continously deformed to ${\gamma_1}$ (either with fixed endpoints, or as closed curves).

Example 2 If ${U}$ is a convex set, that is to say that ${(1-s) z_0 + s z_1 \in U}$ whenever ${z_0,z_1 \in U}$ and ${0 \leq s \leq 1}$, then any two curves ${\gamma_0, \gamma_1: [0,1] \rightarrow U}$ from one point ${z_0}$ to another ${z_1}$ are homotopic, by using the homotopy

$\displaystyle \gamma(s,t) := (1-s) \gamma_0(t) + s \gamma_1(t).$

For a similar reason, in a convex open set ${U}$, any two closed curves will be homotopic to each other as closed curves.

Exercise 3 Let ${U}$ be an open subset of ${{\bf C}}$.

• (i) Prove that the property of being homotopic with fixed endpoints in ${U}$ is an equivalence relation.
• (ii) Prove that the property of being homotopic as closed curves in ${U}$ is an equivalence relation.
• (iii) If ${\gamma_0: [a,b] \rightarrow U}$, ${\gamma_1: [c,d] \rightarrow U}$ are closed curves with the same initial point, show that ${\gamma_0}$ is homotopic to ${\gamma_1}$ as closed curves up to reparameterisation if and only if ${\gamma_0}$ is homotopic to ${\gamma_2 + \gamma_1 + (-\gamma_2)}$ with fixed endpoints for some closed curve ${\gamma_2}$ with the same initial point as ${\gamma_0}$ or ${\gamma_1}$ up to reparameterisation.
• (iv) Define a point in ${U}$ to be a curve ${\gamma_1: [a,b] \rightarrow U}$ of the form ${\gamma_1(t) = z_0}$ for some ${z_0 \in U}$ and all ${t \in [a,b]}$. Let ${\gamma_0: [a,b] \rightarrow U}$ be a closed curve in ${U}$. Show that ${\gamma_0}$ is homotopic with fixed endpoints to a point in ${U}$ if and only if ${\gamma_0}$ is homotopic as a closed curve to a point in ${U}$. (In either case, we will call ${\gamma_0}$ homotopic to a point, null-homotopic, or contractible to a point in ${U}$.)
• (v) If ${\gamma_0, \gamma_1: [a,b] \rightarrow U}$ are curves with the same initial point and the same terminal point, show that ${\gamma_0}$ is homotopic to ${\gamma_1}$ with fixed endpoints in ${U}$ if and only if ${\gamma_0 + (-\gamma_1)}$ is homotopic to a point in ${U}$.
• (vi) If ${U}$ is connected, and ${\gamma_0, \gamma_1: [a,b] \rightarrow U}$ are any two curves in ${U}$, show that there exists a continuous map ${\gamma: [0,1] \times [a,b] \rightarrow U}$ such that ${\gamma(0,t) = \gamma_0(t)}$ and ${\gamma(1,t) = \gamma_1(t)}$ for all ${t \in [a,b]}$. Thus the notion of homotopy becomes rather trivial if one does not fix the endpoints or require the curve to be closed.
• (vii) Show that if ${\gamma_1: [a,b] \rightarrow U}$ is a reparameterisation of ${\gamma_0: [a,b] \rightarrow U}$, then ${\gamma_0}$ and ${\gamma_1}$ are homotopic with fixed endpoints in U.
• (viii) Prove that the property of being homotopic with fixed endpoints in ${U}$ up to reparameterisation is an equivalence relation.
• (ix) Prove that the property of being homotopic as closed curves in ${U}$ up to reparameterisation is an equivalence relation.

We can then phrase Cauchy’s theorem as an assertion that contour integration on holomorphic functions is a homotopy invariant. More precisely:

Theorem 4 (Cauchy’s theorem) Let ${U}$ be an open subset of ${{\bf C}}$, and let ${f: U \rightarrow {\bf C}}$ be holomorphic.

• (i) If ${\gamma_0: [a,b] \rightarrow U}$ and ${\gamma_1: [c,d] \rightarrow U}$ are rectifiable curves that are homotopic in ${U}$ with fixed endpoints up to reparameterisation, then

$\displaystyle \int_{\gamma_0} f(z)\ dz = \int_{\gamma_1} f(z)\ dz.$

• (ii) If ${\gamma_0: [a,b] \rightarrow U}$ and ${\gamma_1: [c,d] \rightarrow U}$ are closed rectifiable curves that are homotopic in ${U}$ as closed curves up to reparameterisation, then

$\displaystyle \int_{\gamma_0} f(z)\ dz = \int_{\gamma_1} f(z)\ dz.$

This version of Cauchy’s theorem is particularly useful for applications, as it explicitly brings into play the powerful technique of contour shifting, which allows one to compute a contour integral by replacing the contour with a homotopic contour on which the integral is easier to either compute or integrate. This formulation of Cauchy’s theorem also highlights the close relationship between contour integrals and the algebraic topology of the complex plane (and open subsets ${U}$ thereof). Setting ${\gamma_1}$ to be a point, we obtain an important special case of Cauchy’s theorem (which is in fact equivalent to the full theorem):

Corollary 5 (Cauchy’s theorem, again) Let ${U}$ be an open subset of ${{\bf C}}$, and let ${f: U \rightarrow {\bf C}}$ be holomorphic. Then for any closed rectifiable curve ${\gamma}$ in ${U}$ that is contractible in ${U}$ to a point, one has ${\int_\gamma f(z)\ dz = 0}$.

Exercise 6 Show that Theorem 4 and Corollary 5 are logically equivalent.

An important feature to note about Cauchy’s theorem is the global nature of its hypothesis on ${f}$. The conclusion of Cauchy’s theorem only involves the values of a function ${f}$ on the images of the two curves ${\gamma_0, \gamma_1}$. However, in order for the hypotheses of Cauchy’s theorem to apply, the function ${f}$ must be holomorphic not only on the images on ${\gamma_0, \gamma_1}$, but on an open set ${U}$ that is large enough (and sufficiently free of “holes”) to support a homotopy between the two curves. This point can be emphasised through the following fundamental near-counterexample to Cauchy’s theorem:

Example 7 (Key example) Let ${U := {\bf C} \backslash \{0\}}$, and let ${f: U \rightarrow {\bf C}}$ be the holomorphic function ${f(z) := \frac{1}{z}}$. Let ${\gamma_{0,1,\circlearrowleft}: [0,2\pi] \rightarrow {\bf C}}$ be the closed unit circle contour ${\gamma_{0,1,\circlearrowleft}(t) := e^{it}}$. Direct calculation shows that

$\displaystyle \int_{\gamma_{0,1,\circlearrowleft}} f(z)\ dz = 2\pi i \neq 0.$

As a consequence of this and Cauchy’s theorem, we conclude that the contour ${\gamma_{0,1,\circlearrowleft}}$ is not contractible to a point in ${U}$; note that this does not contradict Example 2 because ${U}$ is not convex. Thus we see that the lack of holomorphicity (or singularity) of ${f}$ at the origin can be “blamed” for the non-vanishing of the integral of ${f}$ on the closed contour ${\gamma_{0,1,\circlearrowleft}}$, even though this contour does not come anywhere near the origin. Thus we see that the global behaviour of ${f}$, not just the behaviour in the local neighbourhood of ${\gamma_{0,1,\circlearrowleft}}$, has an impact on the contour integral.
One can of course rewrite this example to involve non-closed contours instead of closed ones. For instance, if we let ${\gamma_0, \gamma_1: [0,\pi] \rightarrow U}$ denote the half-circle contours ${\gamma_0(t) := e^{it}}$ and ${\gamma_1(t) := e^{-it}}$, then ${\gamma_0,\gamma_1}$ are both contours in ${U}$ from ${+1}$ to ${-1}$, but one has

$\displaystyle \int_{\gamma_0} f(z)\ dz = +\pi i$

whereas

$\displaystyle \int_{\gamma_1} f(z)\ dz = -\pi i.$

In order for this to be consistent with Cauchy’s theorem, we conclude that ${\gamma_0}$ and ${\gamma_1}$ are not homotopic in ${U}$ (even after reparameterisation).

In the specific case of functions of the form ${\frac{1}{z}}$, or more generally ${\frac{f(z)}{z-z_0}}$ for some point ${z_0}$ and some ${f}$ that is holomorphic in some neighbourhood of ${z_0}$, we can quantify the precise failure of Cauchy’s theorem through the Cauchy integral formula, and through the concept of a winding number. These turn out to be extremely powerful tools for understanding both the nature of holomorphic functions and the topology of open subsets of the complex plane, as we shall see in this and later notes.

Previous set of notes: Notes 1. Next set of notes: Notes 3.

Having discussed differentiation of complex mappings in the preceding notes, we now turn to the integration of complex maps. We first briefly review the situation of integration of (suitably regular) real functions ${f: [a,b] \rightarrow {\bf R}}$ of one variable. Actually there are three closely related concepts of integration that arise in this setting:

• (i) The signed definite integral ${\int_a^b f(x)\ dx}$, which is usually interpreted as the Riemann integral (or equivalently, the Darboux integral), which can be defined as the limit (if it exists) of the Riemann sums

$\displaystyle \sum_{j=1}^n f(x_j^*) (x_j - x_{j-1}) \ \ \ \ \ (1)$

where ${a = x_0 < x_1 < \dots < x_n = b}$ is some partition of ${[a,b]}$, ${x_j^*}$ is an element of the interval ${[x_{j-1},x_j]}$, and the limit is taken as the maximum mesh size ${\max_{1 \leq j \leq n} |x_j - x_{j-1}|}$ goes to zero (this can be formalised using the concept of a net). It is convenient to adopt the convention that ${\int_b^a f(x)\ dx := - \int_a^b f(x)\ dx}$ for ${a < b}$; alternatively one can interpret ${\int_b^a f(x)\ dx}$ as the limit of the Riemann sums (1), where now the (reversed) partition ${b = x_0 > x_1 > \dots > x_n = a}$ goes leftwards from ${b}$ to ${a}$, rather than rightwards from ${a}$ to ${b}$.

• (ii) The unsigned definite integral ${\int_{[a,b]} f(x)\ dx}$, usually interpreted as the Lebesgue integral. The precise definition of this integral is a little complicated (see e.g. this previous post), but roughly speaking the idea is to approximate ${f}$ by simple functions ${\sum_{i=1}^n c_i 1_{E_i}}$ for some coefficients ${c_i \in {\bf R}}$ and sets ${E_i \subset [a,b]}$, and then approximate the integral ${\int_{[a,b]} f(x)\ dx}$ by the quantities ${\sum_{i=1}^n c_i m(E_i)}$, where ${m(E_i)}$ is the Lebesgue measure of ${E_i}$. In contrast to the signed definite integral, no orientation is imposed or used on the underlying domain of integration, which is viewed as an “undirected” set ${[a,b]}$.
• (iii) The indefinite integral or antiderivative ${\int f(x)\ dx}$, defined as any function ${F: [a,b] \rightarrow {\bf R}}$ whose derivative ${F'}$ exists and is equal to ${f}$ on ${[a,b]}$. Famously, the antiderivative is only defined up to the addition of an arbitrary constant ${C}$, thus for instance ${\int x\ dx = \frac{1}{2} x^2 + C}$.

There are some other variants of the above integrals (e.g. the Henstock-Kurzweil integral, discussed for instance in this previous post), which can handle slightly different classes of functions and have slightly different properties than the standard integrals listed here, but we will not need to discuss such alternative integrals in this course (with the exception of some improper and principal value integrals, which we will encounter in later notes).

The above three notions of integration are closely related to each other. For instance, if ${f: [a,b] \rightarrow {\bf R}}$ is a Riemann integrable function, then the signed definite integral and unsigned definite integral coincide (when the former is oriented correctly), thus

$\displaystyle \int_a^b f(x)\ dx = \int_{[a,b]} f(x)\ dx$

and

$\displaystyle \int_b^a f(x)\ dx = -\int_{[a,b]} f(x)\ dx$

If ${f: [a,b] \rightarrow {\bf R}}$ is continuous, then by the fundamental theorem of calculus, it possesses an antiderivative ${F = \int f(x)\ dx}$, which is well defined up to an additive constant ${C}$, and

$\displaystyle \int_c^d f(x)\ dx = F(d) - F(c)$

for any ${c,d \in [a,b]}$, thus for instance ${\int_a^b F(x)\ dx = F(b) - F(a)}$ and ${\int_b^a F(x)\ dx = F(a) - F(b)}$.
All three of the above integration concepts have analogues in complex analysis. By far the most important notion will be the complex analogue of the signed definite integral, namely the contour integral ${\int_\gamma f(z)\ dz}$, in which the directed line segment from one real number ${a}$ to another ${b}$ is now replaced by a type of curve in the complex plane known as a contour. The contour integral can be viewed as the special case of the more general line integral ${\int_\gamma f(z) dx + g(z) dy}$, that is of particular relevance in complex analysis. There are also analogues of the Lebesgue integral, namely the arclength measure integrals ${\int_\gamma f(z)\ |dz|}$ and the area integrals ${\int_\Omega f(x+iy)\ dx dy}$, but these play only an auxiliary role in the subject. Finally, we still have the notion of an antiderivative ${F(z)}$ (also known as a primitive) of a complex function ${f(z)}$.

As it turns out, the fundamental theorem of calculus continues to hold in the complex plane: under suitable regularity assumptions on a complex function ${f}$ and a primitive ${F}$ of that function, one has

$\displaystyle \int_\gamma f(z)\ dz = F(z_1) - F(z_0)$

whenever ${\gamma}$ is a contour from ${z_0}$ to ${z_1}$ that lies in the domain of ${f}$. In particular, functions ${f}$ that possess a primitive must be conservative in the sense that ${\int_\gamma f(z)\ dz = 0}$ for any closed contour. This property of being conservative is not typical, in that “most” functions ${f}$ will not be conservative. However, there is a remarkable and far-reaching theorem, the Cauchy integral theorem (also known as the Cauchy-Goursat theorem), which asserts that any holomorphic function is conservative, so long as the domain is simply connected (or if one restricts attention to contractible closed contours). We will explore this theorem and several of its consequences in the next set of notes.
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Previous set of notes: Notes 0. Next set of notes: Notes 2.

At the core of almost any undergraduate real analysis course are the concepts of differentiation and integration, with these two basic operations being tied together by the fundamental theorem of calculus (and its higher dimensional generalisations, such as Stokes’ theorem). Similarly, the notion of the complex derivative and the complex line integral (that is to say, the contour integral) lie at the core of any introductory complex analysis course. Once again, they are tied to each other by the fundamental theorem of calculus; but in the complex case there is a further variant of the fundamental theorem, namely Cauchy’s theorem, which endows complex differentiable functions with many important and surprising properties that are often not shared by their real differentiable counterparts. We will give complex differentiable functions another name to emphasise this extra structure, by referring to such functions as holomorphic functions. (This term is also useful to distinguish these functions from the slightly less well-behaved meromorphic functions, which we will discuss in later notes.)

In this set of notes we will focus solely on the concept of complex differentiation, deferring the discussion of contour integration to the next set of notes. To begin with, the theory of complex differentiation will greatly resemble the theory of real differentiation; the definitions look almost identical, and well known laws of differential calculus such as the product rule, quotient rule, and chain rule carry over verbatim to the complex setting, and the theory of complex power series is similarly almost identical to the theory of real power series. However, when one compares the “one-dimensional” differentiation theory of the complex numbers with the “two-dimensional” differentiation theory of two real variables, we find that the dimensional discrepancy forces complex differentiable functions to obey a real-variable constraint, namely the Cauchy-Riemann equations. These equations make complex differentiable functions substantially more “rigid” than their real-variable counterparts; they imply for instance that the imaginary part of a complex differentiable function is essentially determined (up to constants) by the real part, and vice versa. Furthermore, even when considered separately, the real and imaginary components of complex differentiable functions are forced to obey the strong constraint of being harmonic. In later notes we will see these constraints manifest themselves in integral form, particularly through Cauchy’s theorem and the closely related Cauchy integral formula.

Despite all the constraints that holomorphic functions have to obey, a surprisingly large number of the functions of a complex variable that one actually encounters in applications turn out to be holomorphic. For instance, any polynomial ${z \mapsto P(z)}$ with complex coefficients will be holomorphic, as will the complex exponential ${z \mapsto \exp(z)}$. From this and the laws of differential calculus one can then generate many further holomorphic functions. Also, as we will show presently, complex power series will automatically be holomorphic inside their disk of convergence. On the other hand, there are certainly basic complex functions of interest that are not holomorphic, such as the complex conjugation function ${z \mapsto \overline{z}}$, the absolute value function ${z \mapsto |z|}$, or the real and imaginary part functions ${z \mapsto \mathrm{Re}(z), z \mapsto \mathrm{Im}(z)}$. We will also encounter functions that are only holomorphic at some portions of the complex plane, but not on others; for instance, rational functions will be holomorphic except at those few points where the denominator vanishes, and are prime examples of the meromorphic functions mentioned previously. Later on we will also consider functions such as branches of the logarithm or square root, which will be holomorphic outside of a branch cut corresponding to the choice of branch. It is a basic but important skill in complex analysis to be able to quickly recognise which functions are holomorphic and which ones are not, as many of useful theorems available to the former (such as Cauchy’s theorem) break down spectacularly for the latter. Indeed, in my experience, one of the most common “rookie errors” that beginning complex analysis students make is the error of attempting to apply a theorem about holomorphic functions to a function that is not at all holomorphic. This stands in contrast to the situation in real analysis, in which one can often obtain correct conclusions by formally applying the laws of differential or integral calculus to functions that might not actually be differentiable or integrable in a classical sense. (This latter phenomenon, by the way, can be largely explained using the theory of distributions, as covered for instance in this previous post, but this is beyond the scope of the current course.)

Remark 1 In this set of notes it will be convenient to impose some unnecessarily generous regularity hypotheses (e.g. continuous second differentiability) on the holomorphic functions one is studying in order to make the proofs simpler. In later notes, we will discover that these hypotheses are in fact redundant, due to the phenomenon of elliptic regularity that ensures that holomorphic functions are automatically smooth.

Next set of notes: Notes 1.

Kronecker is famously reported to have said, “God created the natural numbers; all else is the work of man”. The truth of this statement (literal or otherwise) is debatable; but one can certainly view the other standard number systems ${{\bf Z}, {\bf Q}, {\bf R}, {\bf C}}$ as (iterated) completions of the natural numbers ${{\bf N}}$ in various senses. For instance:

• The integers ${{\bf Z}}$ are the additive completion of the natural numbers ${{\bf N}}$ (the minimal additive group that contains a copy of ${{\bf N}}$).
• The rationals ${{\bf Q}}$ are the multiplicative completion of the integers ${{\bf Z}}$ (the minimal field that contains a copy of ${{\bf Z}}$).
• The reals ${{\bf R}}$ are the metric completion of the rationals ${{\bf Q}}$ (the minimal complete metric space that contains a copy of ${{\bf Q}}$).
• The complex numbers ${{\bf C}}$ are the algebraic completion of the reals ${{\bf R}}$ (the minimal algebraically closed field that contains a copy of ${{\bf R}}$).

These descriptions of the standard number systems are elegant and conceptual, but not entirely suitable for constructing the number systems in a non-circular manner from more primitive foundations. For instance, one cannot quite define the reals ${{\bf R}}$ from scratch as the metric completion of the rationals ${{\bf Q}}$, because the definition of a metric space itself requires the notion of the reals! (One can of course construct ${{\bf R}}$ by other means, for instance by using Dedekind cuts or by using uniform spaces in place of metric spaces.) The definition of the complex numbers as the algebraic completion of the reals does not suffer from such a non-circularity issue, but a certain amount of field theory is required to work with this definition initially. For the purposes of quickly constructing the complex numbers, it is thus more traditional to first define ${{\bf C}}$ as a quadratic extension of the reals ${{\bf R}}$, and more precisely as the extension ${{\bf C} = {\bf R}(i)}$ formed by adjoining a square root ${i}$ of ${-1}$ to the reals, that is to say a solution to the equation ${i^2+1=0}$. It is not immediately obvious that this extension is in fact algebraically closed; this is the content of the famous fundamental theorem of algebra, which we will prove later in this course.
The two equivalent definitions of ${{\bf C}}$ – as the algebraic closure, and as a quadratic extension, of the reals respectively – each reveal important features of the complex numbers in applications. Because ${{\bf C}}$ is algebraically closed, all polynomials over the complex numbers split completely, which leads to a good spectral theory for both finite-dimensional matrices and infinite-dimensional operators; in particular, one expects to be able to diagonalise most matrices and operators. Applying this theory to constant coefficient ordinary differential equations leads to a unified theory of such solutions, in which real-variable ODE behaviour such as exponential growth or decay, polynomial growth, and sinusoidal oscillation all become aspects of a single object, the complex exponential ${z \mapsto e^z}$ (or more generally, the matrix exponential ${A \mapsto \exp(A)}$). Applying this theory more generally to diagonalise arbitrary translation-invariant operators over some locally compact abelian group, one arrives at Fourier analysis, which is thus most naturally phrased in terms of complex-valued functions rather than real-valued ones. If one drops the assumption that the underlying group is abelian, one instead discovers the representation theory of unitary representations, which is simpler to study than the real-valued counterpart of orthogonal representations. For closely related reasons, the theory of complex Lie groups is simpler than that of real Lie groups.

Meanwhile, the fact that the complex numbers are a quadratic extension of the reals lets one view the complex numbers geometrically as a two-dimensional plane over the reals (the Argand plane). Whereas a point singularity in the real line disconnects that line, a point singularity in the Argand plane leaves the rest of the plane connected (although, importantly, the punctured plane is no longer simply connected). As we shall see, this fact causes singularities in complex analytic functions to be better behaved than singularities of real analytic functions, ultimately leading to the powerful residue calculus for computing complex integrals. Remarkably, this calculus, when combined with the quintessentially complex-variable technique of contour shifting, can also be used to compute some (though certainly not all) definite integrals of real-valued functions that would be much more difficult to compute by purely real-variable methods; this is a prime example of Hadamard’s famous dictum that “the shortest path between two truths in the real domain passes through the complex domain”.

Another important geometric feature of the Argand plane is the angle between two tangent vectors to a point in the plane. As it turns out, the operation of multiplication by a complex scalar preserves the magnitude and orientation of such angles; the same fact is true for any non-degenerate complex analytic mapping, as can be seen by performing a Taylor expansion to first order. This fact ties the study of complex mappings closely to that of the conformal geometry of the plane (and more generally, of two-dimensional surfaces and domains). In particular, one can use complex analytic maps to conformally transform one two-dimensional domain to another, leading among other things to the famous Riemann mapping theorem, and to the classification of Riemann surfaces.

If one Taylor expands complex analytic maps to second order rather than first order, one discovers a further important property of these maps, namely that they are harmonic. This fact makes the class of complex analytic maps extremely rigid and well behaved analytically; indeed, the entire theory of elliptic PDE now comes into play, giving useful properties such as elliptic regularity and the maximum principle. In fact, due to the magic of residue calculus and contour shifting, we already obtain these properties for maps that are merely complex differentiable rather than complex analytic, which leads to the striking fact that complex differentiable functions are automatically analytic (in contrast to the real-variable case, in which real differentiable functions can be very far from being analytic).

The geometric structure of the complex numbers (and more generally of complex manifolds and complex varieties), when combined with the algebraic closure of the complex numbers, leads to the beautiful subject of complex algebraic geometry, which motivates the much more general theory developed in modern algebraic geometry. However, we will not develop the algebraic geometry aspects of complex analysis here.

Last, but not least, because of the good behaviour of Taylor series in the complex plane, complex analysis is an excellent setting in which to manipulate various generating functions, particularly Fourier series ${\sum_n a_n e^{2\pi i n \theta}}$ (which can be viewed as boundary values of power (or Laurent) series ${\sum_n a_n z^n}$), as well as Dirichlet series ${\sum_n \frac{a_n}{n^s}}$. The theory of contour integration provides a very useful dictionary between the asymptotic behaviour of the sequence ${a_n}$, and the complex analytic behaviour of the Dirichlet or Fourier series, particularly with regard to its poles and other singularities. This turns out to be a particularly handy dictionary in analytic number theory, for instance relating the distribution of the primes to the Riemann zeta function. Nowadays, many of the analytic number theory results first obtained through complex analysis (such as the prime number theorem) can also be obtained by more “real-variable” methods; however the complex-analytic viewpoint is still extremely valuable and illuminating.

We will frequently touch upon many of these connections to other fields of mathematics in these lecture notes. However, these are mostly side remarks intended to provide context, and it is certainly possible to skip most of these tangents and focus purely on the complex analysis material in these notes if desired.

Note: complex analysis is a very visual subject, and one should draw plenty of pictures while learning it. I am however not planning to put too many pictures in these notes, partly as it is somewhat inconvenient to do so on this blog from a technical perspective, but also because pictures that one draws on one’s own are likely to be far more useful to you than pictures that were supplied by someone else.