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).

— 1. Laurent series —

Suppose we are given a holomorphic function ${f: U \rightarrow {\bf C}}$ and a point ${z_0}$ in ${U}$. For a sufficiently small radius ${r > 0}$, the circle ${\gamma_{z_0,r,\circlearrowleft}}$ and its interior both lie in ${U}$, and the Cauchy integral formula tells us that

$\displaystyle f(z) = \frac{1}{2\pi i} \int_{\gamma_{z_0,r,\circlearrowleft}} \frac{f(w)}{w-z}\ dw$

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

$\displaystyle f(z) = \sum_{n=0}^\infty a_n (z-z_0)^n$

in the interior of this circle, where the coefficients ${a_n}$ could be reconstructed from the values of ${f}$ on the circle ${\gamma_{z_0,r,\circlearrowleft}}$ by the formula

$\displaystyle a_n := \frac{1}{2\pi i} \int_{\gamma_{z_0,r,\circlearrowleft}} \frac{f(w)}{(w-z)^{n+1}}\ dw.$

Now suppose that ${f: U \backslash \{z_0\} \rightarrow {\bf C}}$ is only known to be holomorphic outside of ${z_0}$. Then the Cauchy integral formula no longer directly applies, because the interiors of contours such as ${\gamma_{z_0,r,\circlearrowleft}}$ are no longer contained in the region ${U \backslash \{z_0\}}$ where ${f}$ is holomorphic. To deal with this issue, we use the following convenient decomposition.

Lemma 1 (Cauchy integral formula decomposition in annular regions) Let ${f: U \rightarrow {\bf C}}$ be a holomorphic function. Let ${\gamma_1}$, ${\gamma_2}$ be simple closed anticlockwise contours in ${U}$ such that ${\gamma_2}$ is contained in the interior ${\mathrm{int}(\gamma_1)}$ of ${\gamma_1}$ (or equivalently, by Exercise 49 of Notes 3, that ${\gamma_1}$ is contained in the exterior ${\mathrm{ext}(\gamma_2)}$ of ${\gamma_2}$). Suppose also that the “annular region” ${\mathrm{int}(\gamma_1) \cap \mathrm{ext}(\gamma_2)}$ is contained in ${U}$. Then there exists a decomposition

$\displaystyle f = f_1 + f_2$

on ${U}$, where ${f_1: U \cup \mathrm{int}(\gamma_1) \rightarrow {\bf C}}$ is holomorphic on the union of ${U}$ and the interior of ${\gamma_1}$, and ${f_2: U \cup \mathrm{ext}(\gamma_2) \rightarrow {\bf C}}$ is holomorphic on the union of ${U}$ and the exterior of ${\gamma_2}$, with ${f_2(z) \rightarrow 0}$ as ${z \rightarrow \infty}$. Furthermore, this decomposition is unique.

In addition, we have the Cauchy integral type formulae

$\displaystyle f_1(z) = \frac{1}{2\pi i} \int_{\gamma_1} \frac{f(w)}{w-z}\ dw$

for ${z}$ in the interior of ${\gamma_1}$, and

$\displaystyle f_2(z) = - \frac{1}{2\pi i} \int_{\gamma_2} \frac{f(w)}{w-z}\ dw$

for ${z}$ in the exterior of ${\gamma_2}$. In particular, we have

$\displaystyle f(z) = \frac{1}{2\pi i} \int_{\gamma_1} \frac{f(w)}{w-z}\ dw - \frac{1}{2\pi i} \int_{\gamma_2} \frac{f(w)}{w-z}\ dw \ \ \ \ \ (1)$

for ${z}$ in ${\mathrm{int}(\gamma_1) \cap \mathrm{ext}(\gamma_2)}$.

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

$\displaystyle f = f_1 + f_2 = g_1 + g_2$

on ${U}$, where ${f_1,g_1: U \cup \mathrm{int}(\gamma_1) \rightarrow {\bf C}}$ and ${f_2,g_2: U \cup \mathrm{ext}(\gamma_2) \rightarrow {\bf C}}$ holomorphic, and ${f_2, g_2}$ both going to zero at infinity. Then the holomorphic functions ${f_1-g_1: U \cup \mathrm{int}(\gamma_1) \rightarrow {\bf C}}$ and ${g_2 - f_2: U \cup \mathrm{ext}(\gamma_2) \rightarrow {\bf C}}$ agree on the common domain ${U}$ and are hence restrictions of a single entire function ${F: {\bf C} \rightarrow {\bf C}}$. But ${F}$ goes to zero at infinity and is hence bounded; applying Liouville’s theorem (Theorem 28 of Notes 3) we see that ${F}$ vanishes entirely. This gives ${f_1=g_1}$ on ${U \cup \mathrm{int}(\gamma_1)}$ and ${f_2=g_2}$ on ${U \cup \mathrm{ext}(\gamma_1)}$.

Now for existence. Suppose that we can establish the identity (1) for ${z}$ in ${\mathrm{int}(\gamma_1) \cap \mathrm{ext}(\gamma_2)}$. Then we can define ${f_1}$ on ${\mathrm{int}(\gamma_1)}$ by

$\displaystyle f_1(z) := \frac{1}{2\pi i} \int_{\gamma_1} \frac{f(w)}{w-z}\ dw$

and on ${U \cap \mathrm{ext}(\gamma_2)}$ by

$\displaystyle f_1(z) := f(z) + \frac{1}{2\pi i} \int_{\gamma_2} \frac{f(w)}{w-z}\ dw,$

noting from (1) that this consistently defines ${f_1}$ on

$\displaystyle \mathrm{int}(\gamma_1) \cup (U \cap \mathrm{ext}(\gamma_2)) = U \cup \mathrm{int}(\gamma_1).$

From Exercise 36 of Notes 3 we see that ${f_1}$ is holomorphic. Similarly if we define ${f_2}$ on ${\mathrm{ext}(\gamma_2)}$ by

$\displaystyle f_2(z) := - \frac{1}{2\pi i} \int_{\gamma_2} \frac{f(w)}{w-z}\ dw$

and on ${U \cap \mathrm{int}(\gamma_1)}$ by

$\displaystyle f_2(z) := f(z) - \frac{1}{2\pi i} \int_{\gamma_1} \frac{f(w)}{w-z}\ dw.$

One can then verify that ${f_1,f_2}$ obey all the required properties.

Thus it remains to establish (1). This follows from the homology form of the Cauchy integral formula (Exercise 63(v) of Notes 3), but we can also avoid explicit use of homology by the following “keyhole contour” argument. For ${z \in \mathrm{int}(\gamma_1) \cap \mathrm{ext}(\gamma_2)}$, we have

$\displaystyle \frac{1}{2\pi i} \int_{\gamma_1} \frac{1}{w-z}\ dw = 1$

and

$\displaystyle \frac{1}{2\pi i} \int_{\gamma_2} \frac{1}{w-z}\ dw = 0$

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

$\displaystyle \frac{1}{2\pi i} \int_{\gamma_1} \frac{f(w)-f(z)}{w-z}\ dw = \frac{1}{2\pi i} \int_{\gamma_2} \frac{f(w)-f(z)}{w-z}\ dw.$

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

$\displaystyle \int_{\gamma_1} F(w)\ dw = \int_{\gamma_2} F(w)\ dw \ \ \ \ \ (2)$

whenever ${F: U \rightarrow {\bf C}}$ is holomorphic.

By perturbing ${\gamma_1,\gamma_2}$ using Cauchy’s theorem we may assume that these curves are simple closed polygonal paths (if one wishes, one can also restrict the edges to be horizontal and vertical, although this is not strictly necessary for the argument). By connecting a point in ${\gamma_1}$ to a point in ${\gamma_2}$ by a polygonal path in the interior of ${\gamma_1}$, and removing loops, self-intersections, or excursions into the interior (or image) of ${\gamma_2}$, we can find a simple polygonal path ${\gamma_3}$ from a point ${z_1}$ in ${\gamma_1}$ to a point ${z_2}$ in ${\gamma_2}$ that lies entirely in ${\mathrm{int}(\gamma_1) \cap \mathrm{ext}(\gamma_2)}$ except at the endpoints. By rearranging ${\gamma_1}$ and ${\gamma_2}$ we may assume that ${z_1}$ is the initial and terminal point of ${\gamma_1}$, and ${z_2}$ is the initial and terminal point of ${\gamma_2}$. Then the closed polygonal path ${\gamma_1 + \gamma_3 + (-\gamma_2) + (-\gamma_3)}$ has vanishing winding number in the interior of ${\gamma_2}$ or exterior of ${\gamma_1}$, thus ${U}$ contains all the points where the winding number is non-zero. This path is not simple, but we can approximate it to arbitrary accuracy ${\varepsilon}$ by a simple closed polygonal path ${\gamma_\varepsilon}$ by shifting the simple polygonal paths ${\gamma_3}$ and ${-\gamma_3}$ slightly; for ${\varepsilon}$ small enough, the interior of ${\gamma_\varepsilon}$ will then lie in ${U}$. Applying Cauchy’s theorem (Theorem 52 of Notes 3) we conclude that

$\displaystyle \int_{\gamma_\varepsilon} F(w)\ dw = 0;$

taking limits as ${\varepsilon \rightarrow 0}$ we obtain (2) as claimed. $\Box$

Exercise 2 Let ${\gamma_0}$ be a simple closed anticlockwise contour, and let ${\gamma_1,\dots,\gamma_n}$ be simple closed anticlockwise contours in the interior ${\mathrm{int}(\gamma_0)}$ of ${\gamma_0}$ whose images are disjoint, and such that the interiors ${\mathrm{int}(\gamma_1),\dots,\mathrm{int}(\gamma_n)}$ are also disjoint. Let ${U}$ be an open set containing ${\gamma_0,\gamma_1,\dots,\gamma_n}$ and the region

$\displaystyle \Omega := \mathrm{int}(\gamma_0) \cap \mathrm{ext}(\gamma_1) \cap \dots \cap \mathrm{ext}(\gamma_n).$

Let ${f: U \to {\bf C}}$ be holomorphic. Show that for any ${z_0 \in \Omega}$, one has

$\displaystyle f(z_0) = \frac{1}{2\pi i} \int_{\gamma_0} \frac{f(z)}{z-z_0}\ dz - \sum_{j=1}^n \frac{1}{2\pi i} \int_{\gamma_j} \frac{f(z)}{z-z_0}\ dz.$

(Hint: induct on ${n}$ using Lemma 1.)

Exercise 3 (Painlevé’s theorem on removable singularities) Let ${U}$ be an open subset of ${{\bf C}}$. Let ${S}$ be a compact subset of ${U}$ which has zero length in the following sense: for any ${\varepsilon>0}$, one can cover ${S}$ by a countable number of disks ${D(z_n,r_n)}$ such that ${\sum_n r_n < \varepsilon}$. Let ${f: U \backslash S \rightarrow {\bf C}}$ be a bounded holomorphic function. Show that the singularities in ${S}$ are removable in the sense that there is an extension ${\tilde f: U \rightarrow {\bf C}}$ of ${f}$ to ${U}$ which remains holomorphic. (Hint: one can work locally in some disk in ${U}$ that contains a portion of ${S}$. Cover this portion by a finite number of small disks, group them into connected components, use the previous exercise, and take an appropriate limit.) Note that this result generalises Riemann’s theorem on removable singularities, see Exercise 35 from Notes 3. The situation when ${S}$ has positive length is considerably more subtle, and leads to the theory of analytic capacity, which we will not discuss further here.

Now suppose that ${f: U \rightarrow {\bf C}}$ is holomorphic for some open set ${U}$ that contains an annulus of the form

$\displaystyle \{ z \in {\bf C}: r < |z-z_0| < R \} \ \ \ \ \ (3)$

for some ${z_0 \in {\bf C}}$ and ${0 \leq r < R \leq \infty}$. From Lemma 1, we can split ${f = f_1 + f_2}$, where ${f_1}$ is holomorphic in ${D(z_0,R)}$, and ${f_2}$ is holomorphic in the exterior region ${\{ z: |z-z_0| > r \}}$, with ${f_2(z)}$ going to zero as ${z \rightarrow \infty}$. From Corollary 18 of Notes 3, one has a Taylor expansion

$\displaystyle f_1(z) = \sum_{n=0}^\infty a_n (z-z_0)^n$

for some coefficients ${a_0,a_1,\dots \in {\bf C}}$ that is absolutely convergent in the disk ${D(z_0,R)}$. One cannot directly apply this Taylor expansion to ${f_2}$. However, observe that the function ${z \mapsto f_2(z_0 + \frac{1}{z})}$ is holomorphic in the punctured disk ${D(0,1/r) \backslash \{0\}}$, and goes to zero as one approaches zero. By Riemann’s theorem (Exercise 35 from Notes 3), this function may be extended to ${D(0,1/r)}$ to a holomorphic function that vanishes at the origin. Applying Corollary 18 of Notes 3 again, we conclude that there is a Taylor expansion

$\displaystyle f_2( z_0 + \frac{1}{z} ) = \sum_{n=1}^\infty a_{-n} z^n$

for some coefficients ${a_{-1}, a_{-2},\dots \in {\bf C}}$ that is absolutely convergent in the punctured disk ${D(0,1/r) \backslash \{0\}}$. Changing variables, we conclude that

$\displaystyle f_2(z) = \sum_{n=-\infty}^{-1} a_n (z-z_0)^n$

for ${z}$ in ${\{ z: |z-z_0| > r \}}$, and thus

$\displaystyle f(z) = \sum_{n=-\infty}^\infty a_n (z-z_0)^n \ \ \ \ \ (4)$

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

Exercise 4 (Fourier inversion formula) Let ${f: U \rightarrow {\bf C}}$ be holomorphic on some open set ${U}$ that contains an annulus of the form (3), and let ${a_n}$ be the coefficients of the Laurent expansion (4) in this annulus. Show that the coefficients ${a_n}$ are uniquely determined by ${f}$ and ${r,R}$, and are given by the formula

$\displaystyle a_n = \frac{1}{2\pi i} \int_\gamma \frac{f(z)}{(z-z_0)^{n+1}}\ dz$

for all integers ${n}$, whenever ${\gamma}$ is a simple closed curve in the annulus with ${W_\gamma(z_0) = 1}$. Also establish the bounds

$\displaystyle \liminf_{n \rightarrow +\infty} |a_n|^{-1/n} \geq R \ \ \ \ \ (5)$

and

$\displaystyle \liminf_{n \rightarrow -\infty} |a_n|^{-1/n} \geq 1/r. \ \ \ \ \ (6)$

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

Exercise 5 (Fourier inversion formula, again) Let ${0 < r < 1 < R}$.

• (i) Show that if ${f}$ is holomorphic on the annulus ${\{ z: r < |z| < R\}}$, then we have the Fourier expansion

$\displaystyle f(e^{i\theta}) = \sum_{n=-\infty}^\infty a_n e^{in\theta} \ \ \ \ \ (7)$

for all ${\theta \in {\bf R}}$, where the Fourier coefficients ${a_n}$ are given by the formula

$\displaystyle a_n := \frac{1}{2\pi} \int_0^{2\pi} f(e^{i\theta}) e^{-in\theta}\ d\theta. \ \ \ \ \ (8)$

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

• (ii) Conversely, if ${a_n, n \in {\bf Z}}$ are complex numbers obeying the asymptotic bounds (5), (6), show that there exists a function ${f}$ holomorphic on the annulus ${\{ z: r < |z| < R\}}$ obeying the Fourier expansion (7) and the inversion formula (8).

The Laurent series for a given function can vary as one varies the annulus. Consider for instance the function ${f(z) = \frac{1}{1-z}}$. In the annulus ${\{ z: 0 < |z| < 1 \}}$, the Laurent expansion coincides with the Taylor expansion:

$\displaystyle \frac{1}{1-z} = \sum_{n=0}^\infty z^n = 1 + z + z^2 + \dots.$

On the other hand, in the exterior region ${\{ z: |z| > 1\}}$, the Taylor expansion is no longer convergent. Instead, if one writes ${\frac{1}{1-z} = \frac{-1/z}{1-1/z}}$ and uses the geometric series formula, one instead has the Laurent expansion

$\displaystyle \frac{1}{1-z} = \sum_{n=-\infty}^{-1} -z^n = -\frac{1}{z} - \frac{1}{z^2} - \dots$

in this region.

Exercise 6 Find the Laurent expansions for the function ${f(z) := \frac{1}{(z-1)(z-2)}}$ in the regions ${\{ z: 0 < |z| < 1 \}}$, ${\{ z: 1 < |z| 2 \}}$. (Hint: use partial fractions.)

We can use Laurent series to analyse an isolated singularity. Suppose that ${f: D(z_0,r) \backslash \{z_0\} \rightarrow {\bf C}}$ is holomorphic on a punctured disk ${D(z_0,r) \backslash \{z_0\}}$. By the above discussion, we have a Laurent series expansion (4) in this punctured disk. If the singularity is removable, then the Laurent series must coincide with the Taylor series (by the uniqueness component of Exercise 4), so in partcular ${a_n=0}$ for all negative ${n}$; conversely, if ${a_n}$ vanishes for all negative ${n}$, then the Laurent series matches up with a convergent Taylor series and so the singularity is removable. We then adopt the following classification:

• (i) ${f}$ has a removable singularity at ${z_0}$ if one has ${a_n=0}$ for all negative ${n}$. If furthermore there is an ${m \geq 0}$ such that ${a_m \neq 0}$ and ${a_n=0}$ for ${n < m}$, we say that ${f}$ has a zero of order ${m}$ at ${z_0}$ (after removing the singularity). Zeroes of order ${1}$ are known as simple zeroes, zeroes of order ${2}$ are known as double zeroes, and so forth.
• (ii) ${f}$ has a pole of order ${m}$ at ${z_0}$ for some ${m \geq 1}$ if one has ${a_{-m} \neq 0}$, and ${a_n=0}$ for all ${n < -m}$. Poles of order ${1}$ are known as simple poles, poles of order ${2}$ are double poles, and so forth.
• (iii) ${f}$ has an essential singularity if ${a_n \neq 0}$ for infinitely many negative ${n}$.

It is clear that any holomorphic function ${f: D(z_0,r) \backslash \{z_0\} \rightarrow {\bf C}}$ will be of exactly one of the above three categories. Also, from the uniqueness of Laurent series, shrinking ${r}$ does not affect which of the three categories ${f}$ will lie in (or what order of pole ${f}$ will have, in the second category). Thus, we can classify any isolated singularity ${z_0 \in S}$ of a holomorphic function ${f: U \backslash S \rightarrow {\bf C}}$ with singularities as being either removable, a pole of some finite order, or an essential singularity by restricting ${f}$ to a small punctured disk ${D(z_0,r) \backslash \{z_0\}}$ and inspecting the Laurent coefficients ${a_n}$ for negative ${n}$.

Example 7 The function ${z \mapsto e^{1/z}}$ has a Laurent expansion

$\displaystyle e^{1/z} = 1 + \frac{1}{z} + \frac{1}{2! z^2} + \dots$

and thus has an essential singularity at ${z=0}$.

It is clear from the definition (and the holomorphicity of Taylor series) that (as discussed in the introduction), a holomorphic function ${f: U \backslash S \rightarrow {\bf C}}$ has a pole of order ${m}$ at an isolated singularity ${z_0 \in S}$ if and only if it is of the form ${f(z) = \frac{g(z)}{(z-z_0)^m}}$ for some holomorphic ${g: U \backslash S \cup \{z_0\} \rightarrow {\bf C}}$ with ${g(z_0) \neq 0}$. Similarly, a holomorphic function would have a zero of order ${m}$ at ${z_0}$ if and only if ${f(z) = g(z) (z-z_0)^m}$ for some ${g: U \backslash S \cup \{z_0\} \rightarrow {\bf C}}$ with ${g(z_0) \neq 0}$.

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

Definition 8 (Meromorphic functions) Let ${U}$ be an open subset of ${{\bf C}}$. A function ${f:U \backslash S \rightarrow {\bf C}}$ defined on ${U}$ outside of a singular set ${S}$ is said to be meromorphic on ${U}$ if

• (i) ${S}$ is closed and discrete (i.e., all points in ${S}$ are isolated); and
• (ii) Every ${z_0 \in S}$ is either a removable singularity or a pole of finite order.

Two meromorphic functions ${f_1: U \backslash S_1 \rightarrow {\bf C}}$, ${f_2: U \backslash S_2 \rightarrow {\bf C}}$ are said to be equivalent if they agree on their common domain of definition ${U \backslash (S_1 \cup S_2)}$. It is easy to see that this is an equivalence relation. It is common to identify meromorphic functions up to equivalence, similarly to how in measure theory it is common to identify functions which agree almost everywhere.

Exercise 9 (Meromorphic functions form a field) Let ${M(U)}$ denote the space of meromorphic functions on a connected open set ${U \subset {\bf C}}$, up to equivalence. Show that ${M(U)}$ is a field (with the obvious field operations). What happens if ${U}$ is not connected?

Exercise 10 (Order is a valuation) If ${f: U \backslash S \rightarrow {\bf C}}$ is a meromorphic function, and ${z_0 \in U}$, define the order ${\mathrm{ord}_{z_0}(f)}$ of ${f}$ at ${z_0}$ as follows:

• (a) If ${f}$ has a removable singularity at ${z_0}$, and has a zero of order ${m}$ at ${z_0}$ once the singularity is removed, then ${\mathrm{ord}_{z_0}(f) := m}$.
• (b) If ${f}$ is holomorphic at ${z_0}$, and has a zero of order ${m}$ at ${z_0}$, then ${\mathrm{ord}_{z_0}(f) := m}$.
• (c) If ${f}$ has a pole of order ${m}$ at ${z_0}$, then ${\mathrm{ord}_{z_0}(f) := -m}$.
• (d) If ${f}$ is identically zero, then ${\mathrm{ord}_{z_0}(f) = +\infty}$.

Establish the following facts:

• (i) If ${f_1: U \backslash S_1 \rightarrow {\bf C}}$ and ${f_2: U \backslash S_2 \rightarrow {\bf C}}$ are equivalent meromorphic functions, then ${\mathrm{ord}_{z_0}(f_1) = \mathrm{ord}_{z_0}(f_2)}$ for all ${z_0 \in U}$. In particular, one can meaningfully define the order of an element of ${M(U)}$ at any point ${z_0}$ in ${U}$, where ${M(U)}$ is as in the preceding exercise.
• (ii) If ${f,g \in M(U)}$ and ${z_0 \in U}$, show that ${\mathrm{ord}_{z_0}(fg) = \mathrm{ord}_{z_0}(f) + \mathrm{ord}_{z_0}(g)}$. If ${g}$ is not zero, show that ${\mathrm{ord}_{z_0}(f/g) = \mathrm{ord}_{z_0}(f) - \mathrm{ord}_{z_0}(g)}$.
• (iii) If ${f,g \in M(U)}$ and ${z_0 \in U}$, show that ${\mathrm{old}_{z_0}(f+g) \geq \min( \mathrm{ord}_{z_0}(f), \mathrm{ord}_{z_0}(g) )}$. Furthermore, show if ${\mathrm{ord}_{z_0}(f) \neq \mathrm{ord}_{z_0}(g)}$, then the above inequality is in fact an equality.

In the language of abstract algebra, the above facts are asserting that ${\mathrm{ord}_{z_0}}$ is a valuation on the field ${M(U)}$.

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

Theorem 11 Let ${f: U \backslash S \rightarrow {\bf C}}$ be holomorphic on an open set ${U}$ outside of a singular set ${S}$, and let ${z_0}$ be an isolated singularity in ${S}$.

• (i) If ${z_0}$ is a removable singularity of ${f}$, then ${f(z)}$ converges to a finite limit as ${z \rightarrow z_0}$.
• (ii) If ${z_0}$ is a pole of ${f}$, then ${|f(z)| \rightarrow \infty}$ as ${z \rightarrow z_0}$.
• (iii) (Casorati-Weierstrass theorem) If ${z_0}$ is an essential singularity of ${f}$, then every point ${c}$ of ${{\bf C} \cup \{\infty\}}$ is a limit point of ${f(z)}$ as ${z \rightarrow z_0}$, that is to say there exists a sequence ${z_n}$ converging to ${z_0}$ such that ${f(z_n)}$ converges to ${c}$ (where we adopt the convention that ${f(z_n)}$ converges to ${\infty}$ if ${|f(z_n)|}$ converges to ${\infty}$).

Proof: Part (i) is obvious. Part (ii) is immediate from the factorisation ${f(z) = g(z) / (z-z_0)^m}$ and noting that ${g(z)}$ converges to the non-zero value ${g(z_0)}$ as ${z \rightarrow z_0}$. The ${c=\infty}$ case of (iii) follows from Riemann’s theorem on removable singularities (Exercise 35 from Notes 3). Now suppose ${c}$ is finite. If (iii) failed, then there exist ${\varepsilon, r > 0}$ such that ${f}$ avoids the disk ${D(c,\varepsilon)}$ on the domain ${D(z_0,r) \backslash \{z_0\}}$. In particular, the function ${\frac{1}{f-c}}$ is bounded and holomorphic on ${D(z_0,r) \backslash \{z_0\}}$, and thus extends holomorphically to ${D(z_0,r)}$ by Riemann’s theorem. This function cannot vanish identically, so we must have ${\frac{1}{f(z)-c} = (z-z_0)^m g(z)}$ on ${D(z_0,r) \backslash \{z_0\}}$ for some ${m \geq 0}$ and some holomorphic ${g: D(z_0,r) \rightarrow {\bf C}}$ that does not vanish at ${z_0}$. Rearranging this as ${f(z) = c + \frac{1/g(z)}{(z-z_0)^m}}$, we see that ${f}$ has a pole or removable singularity at ${z_0}$, a contradiction. $\Box$

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

Exercise 12 Let ${f: {\bf C} \backslash S \rightarrow {\bf C}}$ be holomorphic in ${{\bf C}}$ outside of a discrete set ${S}$ of singularities. Let ${z_0 \in {\bf C} \backslash S}$. Show that the radius of convergence of the Taylor series of ${f}$ around ${z_0}$ is equal to the distance from ${z_0}$ to the nearest non-removable singularity in ${S}$, or ${+\infty}$ if no such non-removable singularity exists. (This fact provides a neat way to understand the rate of growth of a sequence ${a_n}$: form its generating function ${\sum_{n=0}^\infty a_n z^n}$, locate the singularities of that function, and find out how close they get to the origin. This is a simple example of the methods of analytic combinatorics in action.)

A curious feature of the singularities in complex analysis is that the order of singularity is “quantised”: one can have a pole of order ${1}$, ${2}$, or ${3}$ (for instance), but not a pole of order ${2.5}$ or ${0.7}$. This quantisation can be exploited: if for instance one somehow knows that the order of the pole is less than ${m+1-\varepsilon}$ for some integer ${m}$ and real number ${\varepsilon>0}$, then the singularity must be removable or a pole of order at most ${m}$. The following exercise formalises this assertion:

Exercise 13 Let ${f: D(z_0,r) \backslash \{z_0\} \rightarrow {\bf C}}$ be holomorphic on a disk ${D(z_0,r)}$ except for a singularity at ${z_0}$. Let ${m}$ be an integer, and suppose that there exist ${C>0}$, ${\varepsilon>0}$ such that one has the upper bound

$\displaystyle |f(z)| \leq C |z-z_0|^{-m-1+\varepsilon}$

for all ${z \in D(z_0,r) \backslash \{z_0\}}$. Show that the singularity of ${f}$ at ${z_0}$ is either removable, or a pole of order at most ${m}$ (the latter option is only possible for ${m}$ positive, of course). (Hint: use Lemma 4 and a limiting argument to evaluate the Laurent coefficients ${a_n}$ for ${n \leq -1}$.) In particular, if one has

$\displaystyle |f(z)| \leq C |z-z_0|^{-1+\varepsilon}$

for all ${z \in D(z_0,r) \backslash \{z_0\}}$, then the singularity is removable.

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

Definition 14 (Smooth manifold) Let ${n \geq 1}$, and let ${M}$ be a topological space. An (${n}$-dimensional real) atlas for ${M}$ is an open cover ${(U_\alpha)_{\alpha \in A}}$ of ${M}$ together with a family of homeomorphisms ${\phi_\alpha: U_\alpha \rightarrow V_\alpha}$ (known as coordinate charts) from each ${U_\alpha}$ to an open subset ${V_\alpha}$ of ${{\bf R}^n}$. Furthermore, the atlas is said to be smooth if for any ${\alpha,\beta \in A}$, the transition map ${\phi_\beta \circ \phi_\alpha^{-1}: \phi_\alpha(U_\alpha \cap U_\beta) \rightarrow \phi_\beta(U_\alpha \cap U_\beta)}$, which maps one open subset of ${{\bf R}^n}$ to another, is required to be smooth (i.e., infinitely differentiable). A map ${f: M \rightarrow M'}$ from one topological space ${M}$ (equipped with a smooth atlas of coordinate charts ${\phi_\alpha: U_\alpha \rightarrow V_\alpha}$ for ${\alpha \in A}$) to another ${M'}$ (equipped with a smooth atlas of coordinate charts ${\phi'_\beta: U'_\beta \rightarrow V'_\beta}$ for some ${\beta \in B}$) is said to be smooth if, for any ${\alpha \in A}$ and ${\beta \in B}$, the maps ${\phi'_\beta \circ f \circ \phi_\alpha^{-1}: \phi_\alpha( f^{-1}(U'_\beta) \cap U_\alpha ) \rightarrow V_\beta}$ are smooth; if ${f}$ is invertible and ${f}$ and ${f^{-1}}$ are both smooth, we say that ${f}$ is a diffeomorphism, and that ${M}$ and ${M'}$ are diffeomorphic. Two smooth atlases on ${M}$ are said to be equivalent if the identity map from ${M}$ (equipped with one of the two atlases) to ${M}$ (equipped with the other atlas) is a diffeomorphism; this is easily seen to be an equivalence relation, and an equivalence class of such atlases is called a smooth structure on ${M}$. A smooth ${n}$-dimensional real manifold is a Hausdorff topological space ${M}$ equipped with a smooth structure. (In some texts the mild additional condition of second countability on ${M}$ is also imposed.) A map ${f: M \rightarrow M'}$ between two smooth manifolds is said to be smooth, if the map ${f}$ from ${M}$ (equipped with one of the atlases in the smooth structure on ${M}$) to ${M'}$ (equipped with one of the atlases in the smooth structure on ${M'}$) is smooth; it is easy to see that this definition is independent of the choices of atlas. We may similarly define the notion of a diffeomorphism between two smooth manifolds.

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

Example 15 A simple example of a smooth ${1}$-dimensional manifold is the unit circle ${S^1 = \{ z \in {\bf C}: |z| = 1 \}}$; there are many equivalent atlases one could place on this circle to define the smooth structure, but one example would be the atlas consisting of the two charts ${\phi_1: U_1 \rightarrow V_1}$, ${\phi_2: U_2 \rightarrow V_2}$, defined by setting ${V_1 := (-\pi, \pi)}$, ${V_2 := (0, 2\pi)}$, ${U_1 := \{ e^{i \theta}: \theta \in V_1 \} = S^1 \backslash \{-1\}}$, ${U_2 := \{ e^{i\theta}: \theta \in V_2 \} = S^1 \backslash \{1\}}$, ${\phi_1(e^{i\theta}) := \theta}$ for ${\theta \in U_1}$, and ${\phi_2(e^{i\theta}) := \theta}$ for ${\theta \in U_2}$. Another smooth manifold, which turns out to be diffeomorphic to the unit circle ${S^1}$, is the one-point compactification ${{\bf R} \cup \{\infty\}}$ of the real numbers, with the two charts ${\phi_1: U_1 \rightarrow V_1}$, ${\phi_2: U_2 \rightarrow V_2}$ defined by setting ${V_1=V_2:={\bf R}}$, ${U_1 := {\bf R}}$, ${U_2 := ({\bf R} \backslash \{0\}) \cup \{\infty\}}$, ${\phi_1}$ to be the identity map, and ${\phi_2}$ defined by setting ${\phi_2(x) := 1/x}$ for ${x \in {\bf R} \backslash \{0\}}$ and ${\phi_2(\infty) := 0}$.

Exercise 16 Verify that the unit circle ${S^1}$ is indeed diffeomorphic to the one-point compactification ${{\bf R} \cup \{\infty\}}$.

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

Definition 17 (Riemann surface) Let ${M}$ be a Hausdorff topological space. A holomorphic atlas on ${M}$ is an open cover ${(U_\alpha)_{\alpha \in A}}$ of ${M}$ together with a family of homeomorphisms ${\phi_\alpha: U_\alpha \rightarrow V_\alpha}$ (known as coordinate charts) from each ${U_\alpha}$ to an open subset ${V_\alpha}$ of ${{\bf C}}$, such that, for any ${\alpha,\beta \in A}$, the transition map ${\phi_\beta \circ \phi_\alpha^{-1}: \phi_\alpha(U_\alpha \cap U_\beta) \rightarrow \phi_\beta(U_\alpha \cap U_\beta)}$, which maps one open subset of ${{\bf C}}$ to another, is required to be holomorphic. A map ${f: M \rightarrow M'}$ from one space ${M}$ (equipped with coordinate charts ${\phi_\alpha: U_\alpha \rightarrow V_\alpha}$ for ${\alpha \in A}$) to another ${M'}$ (equipped with coordinate charts ${\phi'_\beta: U'_\beta \rightarrow V'_\beta}$ for some ${\beta \in B}$) is said to be holomorphic if, for any ${\alpha \in A}$ and ${\beta \in B}$, the maps ${\phi'_\beta \circ f \circ \phi_\alpha: \phi_\alpha^{-1}( f^{-1}(U'_\beta) \cap U_\alpha ) \rightarrow V_\beta}$ are holomorphic; if ${f}$ is invertible and ${f}$ and ${f^{-1}}$ are both holomorphic, we say that ${f}$ is a complex diffeomorphism, and that ${M}$ and ${M'}$ are complex diffeomorphic. Two holomorphic atlases on ${M}$ are said to be equivalent if the identity map from ${M}$ (equipped with one of the atlases) to ${M}$ (equipped with the other atlas) is a complex diffeomorphism; this is easily seem to be an equivalence relation, and we refer to such an equivalence class as a (one-dimensional) complex structure on ${M}$. A Riemann surface is a Hausdorff topological space ${M}$, equipped with a one-dimensional complex structure. (Again, in some texts the hypothesis of second countability is imposed. This makes little difference in practice, as most Riemann surfaces one actually encounters will be second countable.)

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

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

Now we come to the Riemann sphere ${{\bf C} \cup \{\infty\}}$, which is to the complex numbers as ${{\bf R} \cup \{\infty\}}$ is to the real numbers. As a set, this is the complex numbers ${{\bf C}}$ with one additional point (the point at infinity) ${\infty}$ attached. Topologically, this is the one-point compactification of the complex numbers ${{\bf C}}$: the open sets of ${{\bf C} \cup \{\infty\}}$ are either subsets of ${{\bf C}}$ that were already open, or complements ${({\bf C} \cup \{\infty\}) \backslash K}$ of compact subsets ${K}$ of ${{\bf C}}$. As a Riemann surface, the complex structure can be described by the atlas of coordinate charts ${\phi_1: U_1 \rightarrow V_1}$, ${\phi_2: U_2 \rightarrow V_2}$, where ${V_1 = V_2 := {\bf C}}$, ${U_1 := {\bf C}}$, ${U_2 := ({\bf C} \backslash \{0\}) \cup \{\infty\}}$, ${\phi_1}$ is the identity map, and ${\phi_2(z)}$ equals ${1/z}$ for ${z \in {\bf C} \backslash \{0\}}$ with ${\phi_2(\infty) = 0}$. It is not difficult to verify that this is indeed a Riemann surface (basically because the map ${z \mapsto \frac{1}{z}}$ is holomorphic on ${{\bf C} \backslash \{0\}}$). One can identify the Riemann sphere with a geometric sphere, and specifically the sphere ${S^2 := \{ (z,t) \in {\bf C} \times {\bf R}: |z|^2 + (t-\frac{1}{2})^2 = \frac{1}{4} \}}$, through the device of stereographic projection through the north pole ${N := (0,1) \in {\bf C} \times {\bf R}}$, identifying a point ${z}$ in ${{\bf C} \subset {\bf C} \cup \{\infty\}}$ with the point ${(\frac{z}{1+|z|^2}, \frac{|z|^2}{1+|z|^2})}$ on ${S^2 \backslash \{N\}}$ collinear with that point, and the point at infinity ${\infty}$ identified with the north pole ${N}$. This geometric perspective is especially helpful when thinking about Möbius transformations, as is for instance exemplified by this excellent video. (We may cover Möbius transformations in a subsequent set of notes.)

By unpacking the definitions, we can now work out what it means for a function to be holomorphic to or from the Riemann sphere. For instance, if ${f: U \rightarrow {\bf C} \cup \{\infty\}}$ is a map from an open subset ${U}$ of ${{\bf C}}$ to the Riemann sphere ${{\bf C} \cup \{\infty\} }$, then ${f}$ is holomorphic if and only if

• (i) ${f}$ is continuous;
• (ii) ${f}$ is holomorphic on the set ${\{ z \in U: f(z) \neq \infty\}}$ (which is open thanks to (i)); and
• (iii) ${1/f}$ is holomorphic on the set ${\{ z \in U: f(z) \neq 0 \}}$ (which is open thanks to (i)), where we adopt the convention ${1/\infty = 0}$.

Similarly, if a function ${f: U \rightarrow {\bf C} \cup \{ \infty\}}$ is a map from an open subset ${U}$ of the Riemann sphere ${{\bf C} \cup \{\infty\}}$ to the Riemann sphere, then ${f}$ is holomorphic if and only if

• (i) ${z \mapsto f(z)}$ is holomorphic on ${U \cap {\bf C}}$; and
• (ii) ${z \mapsto f(1/z)}$ is holomorphic on ${\{ z \in {\bf C} \cup \{\infty\}: 1/z \in U \}}$, where we again adopt the convention ${1/\infty=0}$.

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

Exercise 18 Let ${U \subset {\bf C}}$ be open connected, let ${S}$ be a discrete subset of ${U}$, and let ${f: U \backslash S \rightarrow {\bf C}}$ be a function. Show that the following are equivalent:

• (i) ${f}$ is meromorphic on ${U}$.
• (ii) ${f}$ is the restriction of a holomorphic function ${\tilde f: U \rightarrow {\bf C} \cup \{\infty\}}$ to the Riemann sphere, which is not identically equal to ${\infty}$.

Furthermore, if (ii) holds, show that ${\tilde f}$ is uniquely determined by ${f}$, and is unaffected if one replaces ${f}$ with an equivalent meromorphic function.

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

Exercise 19 Let ${f: {\bf C} \cup \{\infty\} \rightarrow {\bf C} \cup \{\infty\}}$ be a holomorphic map from the Riemann sphere to itself. Show that either ${f}$ is identically equal to $\{\infty\}$, or ${f}$ is a rational function in the sense that there exist polynomials ${P(z), Q(z)}$ of one complex variable, with ${Q}$ not identically zero, such that ${f(z) = P(z) / Q(z)}$ for all ${z \in {\bf C}}$ with ${Q(z) \neq 0}$. (Hint: show that ${f}$ has finitely many poles, and eliminate them by multiplying ${f}$ by appropriate linear factors. Then use Exercise 29 from Notes 3.)

Exercise 20 (Partial fractions) Let ${P(z)}$ be a polynomial of one complex variable, which by the fundamental theorem of algebra we may write as

$\displaystyle P(z) = a (z-z_1)^{d_1} \dots (z-z_k)^{d_k}$

for some distinct roots ${z_1,\dots,z_k \in {\bf C}}$, some non-zero ${a}$, and some positive integers ${d_1,\dots,d_k}$. Let ${Q(z)}$ be another polynomial of one complex variable. Show that there exist unique polynomials ${R_1(z),\dots,R_k(z),S(z)}$, with each ${R_i}$ having degree less than ${d_i}$ for ${i=1,\dots,k}$, such that one has the partial fraction decomposition

$\displaystyle \frac{Q(z)}{P(z)} = \sum_{j=1}^k \frac{R_j(z)}{(z-z_j)^{d_j}} + S(z)$

for all ${z \in {\bf C} \backslash \{z_1,\dots,z_k\}}$. Furthermore, show that ${S}$ vanishes if the degree ${\mathrm{deg}(Q)}$ of ${Q}$ is less than the degree ${\mathrm{deg}(P)}$ of ${P}$, and has degree ${\mathrm{deg}(Q)-\mathrm{deg}(P)}$ otherwise.

— 2. The residue theorem —

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

Suppose one has a function ${f: U \backslash S \rightarrow {\bf C}}$ holomorphic on an open set ${U \subset {\bf C}}$ outside of a singular set ${S}$. If ${z_0 \in S}$ is an isolated singularity of ${f}$, then we have a Laurent expansion

$\displaystyle f(z) = \sum_{n=-\infty}^\infty a_n (z-z_0)^n$

which is convergent in some punctured disk ${D(z_0,r) \backslash \{z_0\}}$. The coefficient ${a_{-1}}$ plays a privileged role and is known as the residue of ${f}$ at ${z_0}$; we denote it by ${\mathrm{Res}(f; z_0)}$. Clearly this is quantity is local in the sense that it only depends on the behaviour of ${f}$ in a neighbourhood of ${z_0}$; in particular, it does not depend on the domain ${U}$ so long as ${z_0}$ remains inside of that domain. By convention, we also set ${\mathrm{Res}(f;z_0)=0}$ if ${f}$ is holomorphic at ${z_0}$ (i.e., if ${z_0 \in U \backslash S}$).

We then have

Theorem 21 (Residue theorem) Let ${U \subset {\bf C}}$ be a simply connected open set, and let ${f: U \backslash S \rightarrow {\bf C}}$ be holomorphic outside of a closed discrete singular set ${S}$ (thus all singularities in ${S}$ are isolated singularities). Let ${\gamma}$ be a closed curve in ${U \backslash S}$. Then

$\displaystyle \frac{1}{2\pi i} \int_\gamma f(z)\ dz = \sum_{z_0 \in S} W_\gamma(z_0) \mathrm{Res}(f; z_0),$

where only finitely many of the terms on the right-hand side are non-zero.

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

Next, we reduce to the case where all the residues ${\mathrm{Res}(f;z_0)}$ vanish. We introduce the rational function ${g: {\bf C} \backslash S \rightarrow {\bf C}}$ defined by

$\displaystyle g(z) := \sum_{z_0 \in S} \frac{\mathrm{Res}(f;z_0)}{z-z_0}.$

From Laurent expansion around each singularity ${z_0}$ we see that ${\mathrm{Res}(g;z_0) = \mathrm{Res}(f;z_0)}$ for all ${z_0 \in S}$, thus ${\mathrm{Res}(f-g;z_0) = 0}$. Also, from the definition of winding number (see Definition 38 of Notes 3) we have

$\displaystyle \frac{1}{2\pi i} \int_\gamma g(z)\ dz = \sum_{z_0 \in S} W_\gamma(z_0) \mathrm{Res}(f; z_0).$

Setting ${F := f-g}$, it thus suffices to show that

$\displaystyle \int_\gamma F(z)\ dz = 0. \ \ \ \ \ (9)$

As ${U}$ is simply connected, ${\gamma: [a,b] \rightarrow U}$ is homotopic in ${U}$ (as closed contours) to a point. Let ${\tilde \gamma: [0,1] \times [a,b] \rightarrow U}$ denote the homotopy. We would like to mimic the proof of Cauchy’s theorem (Theorem 4 of Notes 3) to conclude (9). The difficulty is that the homotopy ${\tilde \gamma}$ may pass through points ${z_0}$ in ${S}$. However, note from the vanishing of the residue ${\mathrm{Res}(F;z_0)}$ that one has a Laurent expansion of the form

$\displaystyle F(z) = \sum_{n=0}^\infty a_n (z-z_0)^n + \sum_{n=-\infty}^{-2} a_n (z-z_0)^n$

for some coefficients ${a_n, n \neq -1}$, in some punctured disk ${D(z_0,r) \backslash \{z_0\}}$, with both series being absolutely convergent in this punctured disk. From term by term differentiation (see Theorem 15 of Notes 1) we see that ${F}$ has an antiderivative in this punctured disk, namely

$\displaystyle G(z) := \sum_{n=0}^\infty a_n \frac{(z-z_0)^n}{n+1} + \sum_{n=-\infty}^{-2} a_n \frac{(z-z_0)^n}{n+1}$

(note how crucial it is that the ${n=-1}$ term is absent in order to form this antiderivative). The absolute convergence of the series on the right-hand side in ${D(z_0,r)}$ can be seen from the comparison test. From the fundamental theorem of calculus, we thus conclude that ${F}$ is conservative on ${D(z_0,r) \backslash \{z_0\}}$. Also, for any ${z_0 \in U}$ that is not in ${S}$, we see from Cauchy’s theorem that ${F}$ is conservative on ${D(z_0,r)}$ for some radius ${r>0}$. Putting this together using a compactness argument, we conclude that there exists a radius ${r>0}$, such that for all ${z_0}$ in the image of the homotopy ${\tilde \gamma}$, the function ${F}$ is conservative in ${D(z_0,r) \backslash S}$.

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

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

Corollary 22 (Residue theorem for simple closed contours) Let ${\gamma}$ be a simple closed anticlockwise contour in ${{\bf C}}$. Suppose that ${f: U \backslash S \rightarrow {\bf C}}$ is holomorphic on an open set ${U}$ containing the image and interior of ${\gamma}$, outside of a closed discrete ${S}$ that does not intersect the image of ${\gamma}$. Then we have

$\displaystyle \frac{1}{2\pi i} \int_\gamma f(z)\ dz = \sum_{z_0 \in S \cap \mathrm{int}(\gamma)} \mathrm{Res}(f;z_0).$

If ${\gamma}$ is oriented clockwise instead of anticlockwise, then we instead have

$\displaystyle \frac{1}{2\pi i} \int_\gamma f(z)\ dz = -\sum_{z_0 \in S \cap \mathrm{int}(\gamma)} \mathrm{Res}(f;z_0).$

Exercise 23 (Homology version of residue theorem) Show that the residue theorem continues to hold when the closed curve ${\gamma}$ is replaced by a ${1}$-cycle (as in Exercise 63 of Notes 3) that avoids all the singularities in ${S}$, and the requirement that ${U}$ be simply connected is replaced by the requirement that ${U}$ contains all the points ${z_0}$ outside of the image of ${\gamma}$ where ${W_\gamma(z_0) \neq 0}$.

Exercise 24 (Exterior version of residue theorem) Let ${\gamma}$ be a simple closed anticlockwise contour in ${{\bf C}}$. Suppose that ${f: U \backslash S \rightarrow {\bf C}}$ is holomorphic on an open set ${U}$ containing the image and exterior of ${\gamma}$, outside of a finite ${S}$ that does not intersect the image of ${\gamma}$. Suppose also that ${z f(z)}$ converges to a finite limit ${c}$ in the limit ${|z| \rightarrow \infty}$. Show that

$\displaystyle \frac{1}{2\pi i} \int_\gamma f(z)\ dz = c-\sum_{z_0 \in S \cap \mathrm{ext}(\gamma)} \mathrm{Res}(f;z_0).$

If ${\gamma}$ is oriented clockwise instead of anticlockwise, show instead that

$\displaystyle \frac{1}{2\pi i} \int_\gamma f(z)\ dz = -c+\sum_{z_0 \in S \cap \mathrm{ext}(\gamma)} \mathrm{Res}(f;z_0).$

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

Exercise 25 Let ${f: U \backslash S \rightarrow {\bf C}}$ be holomorphic on an open set ${U}$ outside of a singular set ${S}$, and let ${z_0}$ be an isolated point of ${S}$.

• (i) If ${f}$ has a removable singularity at ${z_0}$, show that ${\mathrm{Res}(f;z_0)=0}$.
• (ii) If ${f}$ has a simple pole at ${z_0}$, show that ${\mathrm{Res}(f;z_0)=\lim_{z \rightarrow z_0; z \in U \backslash S} f(z) (z-z_0)}$.
• (iii) If ${f}$ has a pole of order at most ${m}$ at ${z_0}$ for some ${m \geq 1}$, show that

$\displaystyle \mathrm{Res}(f;z_0)= \frac{1}{(m-1)!} \lim_{z \rightarrow z_0; z \in U \backslash S} \frac{d^{m-1}}{dz^{m-1}} (f(z) (z-z_0)^m).$

In particular, if ${f(z) = g(z)/(z-z_0)^m}$ near ${z_0}$ for some ${g}$ that is holomorphic at ${z_0}$, then

$\displaystyle \mathrm{Res}(f;z_0)= \frac{1}{m!} \frac{d^{m-1}}{dz^{m-1}} g(z_0).$

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

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

Exercise 26 Use the residue theorem to give an alternate proof of the fundamental theorem of algebra, by considering the integral ${\int_{\gamma_{0,R,\circlearrowleft}} \frac{z^{n-1}}{P(z)}\ dz}$ for a polynomial ${P}$ of degree ${n}$ and some large radius ${R}$.

Exercise 27 Let ${f: {\bf C} \rightarrow {\bf C}}$ be a Dirichlet polynomial of the form

$\displaystyle f(s) := \sum_{n=1}^\infty \frac{a_n}{n^s}$

for some sequence ${a_1,a_2,\dots}$ of complex numbers, with only finitely many of the ${a_n}$ non-zero. Establish Perron’s formula

$\displaystyle \sum_{n \leq x} a_n = \lim_{T \rightarrow +\infty} \frac{1}{2\pi i} \int_{\gamma_{c-iT \rightarrow c+iT}} f(s) \frac{x^s}{s}\ ds$

for any real numbers ${x,c>0}$ with ${x}$ not an integer. What happens if ${x}$ is an integer? Generalisations and variants of this formula, particularly with the Dirichlet polynomial replaced by more general Dirichlet series in which infinitely many of the ${a_n}$ are allowed to be non-zero, are of particular use in analytic number theory; see for instance this previous blog post.

Exercise 28 (Spectral theorem for matrices) This exercise presumes some familiarity with linear algebra. Let ${n}$ be a positive integer, and let ${M_n({\bf C})}$ denote the ring of ${n \times n}$ complex matrices. Let ${A}$ be a matrix in ${M_n({\bf C})}$. The characteristic polynomial ${\Delta(z) := \mathrm{det}(A - zI)}$, where ${I}$ is the ${n \times n}$ identity matrix, is a polynomial of degree ${n}$ in ${z}$ with leading coefficient ${(-1)^n}$; we let ${z_1,\dots,z_k}$ be the distinct zeroes of this polynomial, and let ${d_1,\dots,d_k}$ be the multiplicities; thus by the fundamental theorem of algebra we have

$\displaystyle \mathrm{det}(A-zI) = (-1)^n (z-z_1)^{d_1} \dots (z-z_k)^{d_k}.$

We refer to the set ${\{z_1,\dots,z_k\}}$ as the spectrum of ${A}$. Let ${\gamma}$ be any closed anticlockwise curve that contains the spectrum of ${A}$ in its interior, and let ${U}$ be an open subset of ${{\bf C}}$ that contains ${\gamma}$ and its interior.

• (i) Show that the resolvent ${(A-zI)^{-1}}$ is a meromorphic function on ${{\bf C}}$ with poles at the spectrum of ${A}$, where we call a matrix-valued function meromorphic if each of its ${n^2}$ components are meromorphic. (Hint: use the adjugate matrix.)
• (ii) For any holomorphic ${f: U \rightarrow {\bf C}}$, we define the matrix ${f(A) := M_n({\bf C})}$ by the formula

$\displaystyle f(A) := - \frac{1}{2\pi i} \int_\gamma f(z) (A-zI)^{-1}\ dz$

(cf. the Cauchy integral formula). We refer to ${f(A)}$ as the holomorphic functional calculus for ${A}$ applied to ${f}$. Show that the matrix ${f(A)}$ does not depend on the choice of ${\gamma}$, depends linearly on ${f}$, and equals the identity matrix when ${f}$ is the constant function ${1}$. Furthermore, if ${g: U \rightarrow {\bf C}}$ is the function ${g(z) := z f(z)}$, show that

$\displaystyle g(A) = A f(A) = f(A) A.$

Conclude in particular that if ${P}$ is a polynomial

$\displaystyle P(z) = a_m z^m + \dots + a_1 z + a_0$

with complex coefficients ${a_0,\dots,a_m \in {\bf C}}$, then the function ${P(A)}$ (as defined by the holomorphic functional calculus) matches how one would define ${P(A)}$ algebraically, in the sense that

$\displaystyle P(A) = a_m A^m + \dots + a_1 A + a_0 I.$

• (iii) Prove the Cayley-Hamilton theorem ${\Delta(A)=0}$. (Note from (ii) that it does not matter whether one interprets ${\Delta(A)}$ algebraically, or via the holomorphic functional calculus.)
• (iv) If ${f: U \rightarrow {\bf C}}$ is holomorphic, show that the matrix-valued function ${z \mapsto (f(A)-f(z)) (A-zI)^{-1}}$ has only removable singularities in ${U}$.
• (v) If ${f,g: U \rightarrow {\bf C}}$ are holomorphic, establish the identity

$\displaystyle (fg)(A) = f(A) g(A).$

• (vi) Show that there exist matrices ${P_1,\dots,P_k \in M_n({\bf C})}$ that are idempotent (thus ${P_j^2=P_j}$ for all ${j=1,\dots,k}$), commute with each other and with ${A}$, sum to the identity (thus ${P_1+\dots+P_k = I}$), annihilate each other (thus ${P_j P_{j'} = 0}$ for all distinct ${j,j' = 1,\dots,k}$) and are such that for each ${j=1,\dots,k}$, one has the nilpotency property

$\displaystyle (P_j (A - z_j I) P_j)^{d_j} = 0.$

In particular, we have the spectral decomposition

$\displaystyle A = \sum_{j=1}^k P_j (z_j I + N_j) P_j$

where each ${N_j}$ is a nilpotent matrix with ${N_j^{d_j} = 0}$. Finally, show that the range of ${P_j}$ (viewed as a linear operator from ${{\bf C}^n}$ to itself) has dimension ${d_j}$. Find a way to interpret each ${P_j}$ as the (negative of the) “residue” of the resolvent operator ${(A-zI)^{-1}}$ at ${z_j}$.

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

— 3. The argument principle —

We have not yet defined the complex logarithm ${\log z}$ of a complex number ${z}$, but one of the properties we would expect of this logarithm is that its derivative should be the reciprocal function: ${\frac{d}{dz} \log z = \frac{1}{z}}$. In particular, by the chain rule we would expect the formula

$\displaystyle \frac{d}{dz} \log f(z) = \frac{f'(z)}{f(z)} \ \ \ \ \ (10)$

for a holomorphic function ${f}$, at least away from the zeroes of ${f}$. Inspired by this formal calculation, we refer to the function ${\frac{f'(z)}{f(z)}}$ as the log-derivative of ${f}$. Observe the product rule and quotient rule, when applied to complex differentiable functions ${f,g}$ that are non-zero at some point ${z}$, gives the formulae

$\displaystyle \frac{(fg)'(z)}{(fg)(z)} = \frac{f'(z)}{f(z)} + \frac{g'(z)}{g(z)} \ \ \ \ \ (11)$

and

$\displaystyle \frac{(f/g)'(z)}{(f/g)(z)} = \frac{f'(z)}{f(z)} - \frac{g'(z)}{g(z)} \ \ \ \ \ (12)$

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

$\displaystyle P(z) = a (z-z_1)^{d_1} \dots (z-z_k)^{d_k}$

and

$\displaystyle Q(z) = b (z-w_1)^{e_1} \dots (z-w_l)^{e_l}$

for some non-zero complex numbers ${a,b}$, distinct complex numbers ${z_1,\dots,z_k,w_1,\dots,w_l}$, and positive integers ${d_1,\dots,d_k,e_1,\dots,e_l}$, then the log-derivative of the rational function ${P/Q}$ is given by

$\displaystyle \frac{(P/Q)'(z)}{(P/Q)(z)} = \frac{d_1}{z-z_1} + \dots + \frac{d_k}{z-z_k} - \frac{e_1}{z-w_1} - \dots - \frac{e_l}{z-w_l}.$

In particular, the log-derivative of ${P/Q}$ is meromorphic with poles at ${z_1,\dots,z_k, w_1,\dots,w_l}$, with a residue of ${+d_j}$ at each zero ${z_j}$ of ${P/Q}$, and a residue of ${-e_j}$ at each pole ${w_j}$ of ${P/Q}$.

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

Lemma 29 Let ${f: U \backslash S \rightarrow {\bf C}}$ be a holomorphic function on an open set ${U \subset {\bf C}}$ outside of a singular set ${S}$, and let ${z_0}$ be either an element of ${U \backslash S}$ or an isolated point of ${S}$.

• (i) If ${f}$ is holomorphic and non-zero at ${z_0}$, then the log-derivative ${\frac{f'(z)}{f(z)}}$ is also holomorphic at ${z_0}$.
• (ii) If ${f}$ is holomorphic at ${z_0}$ with a zero of order ${m \geq 1}$, then the log-derivative ${\frac{f'(z)}{f(z)}}$ has a simple pole at ${z_0}$ with residue ${m}$.
• (iii) If ${f}$ has a removable singularity at ${z_0}$, and is non-zero once the singularity is removed, then the log-derivative ${\frac{f'(z)}{f(z)}}$ has a removable singularity at ${z_0}$.
• (iv) If ${f}$ has a removable singularity at ${z_0}$, and has a zero of order ${m \geq 1}$ once the singularity is removed, then the log-derivative ${\frac{f'(z)}{f(z)}}$ has a simple pole at ${z_0}$ with residue ${m}$.
• (v) If ${f}$ has a pole of order ${m\geq 1}$ at ${z_0}$, then the log-derivative ${\frac{f'(z)}{f(z)}}$ has a simple pole at ${z_0}$ with residue ${-m}$.

Proof: The claim (i) is obvious. For (ii), we use Taylor expansion to factor ${f(z) = (z-z_0)^m g(z)}$ for some ${g}$ holomorphic and non-zero near ${z_0}$, and then from (11) we have

$\displaystyle \frac{f'(z)}{f(z)} = \frac{g'(z)}{g(z)} + \frac{m}{z-z_0}.$

Since ${\frac{g'}{g}}$ is holomorphic at ${z_0}$, the claim (ii) follows. The claim (v) is proven similarly using a factorisation ${f(z) = g(z) / (z-z_0)^m}$, and using (12) in place of (11). The claims (iii), (iv) then follow from (i), (ii) respectively after removing the singularity. $\Box$

Remark 30 Note that the lemma does not cover all possible singularity and zero scenarios. For instance, ${f}$ could be identically zero, in which case the log-derivative is nowhere defined. If ${f}$ has an essential singularity then the log-derivative can be a pole (as seen for instance by the example ${f(z) = \exp( 1 / z^m )}$ for some ${m \geq 1}$) or another essential singularity (as can be seen for instance by the example ${f(z) = \exp(\exp(1/z))}$). Finally, if ${f}$ has a non-isolated singularity, then the log-derivative could exhibit a wide range of behaviour (but probably will be quite wild as one approaches the singular set).

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

Theorem 31 (Argument principle) Let ${\gamma: [a,b] \rightarrow {\bf C}}$ be a simple closed anticlockwise contour. Let ${U}$ be an open set containing ${\gamma}$ and its interior. Let ${f}$ be a meromorphic function on ${U}$ that is holomorphic and non-zero on the image of ${\gamma}$. Suppose that after removing all the removable singularities of ${f}$, ${f}$ has zeroes ${z_1,\dots,z_k}$ in the interior of ${\gamma}$ (of orders ${d_1,\dots,d_k}$ respectively), and poles ${w_1,\dots,w_l}$ in the interior of ${\gamma}$ (of orders ${e_1,\dots,e_l}$ respectively). (${f}$ is also allowed to have zeroes and poles in the exterior of ${\gamma}$.) Then we have

$\displaystyle \sum_{j=1}^k d_j - \sum_{j=1}^l e_j = \frac{1}{2\pi i} \int_\gamma \frac{f'(z)}{f(z)}\ dz \ \ \ \ \ (13)$

$\displaystyle = W_{f \circ \gamma}(0)$

where ${f \circ \gamma: [a,b] \rightarrow {\bf C}}$ is the closed contour ${t \mapsto f(\gamma(t))}$.

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

$\displaystyle \frac{1}{2\pi i} \int_\gamma \frac{f'(z)}{f(z)}\ dz = \frac{1}{2\pi i} \int_{f \circ \gamma} \frac{1}{z}\ dz$

and the second identity also follows. $\Box$

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

Corollary 32 (Special case of argument principle) Let ${\gamma}$ be a simple closed anticlockwise contour, let ${U}$ be an open set containing the image of ${\gamma}$ and its interior, and let ${f: U \rightarrow {\bf C}}$ be holomorphic. Suppose that ${f}$ has no zeroes on the image of ${\gamma}$. Then the number of zeroes of ${f}$ (counting multiplicity) in the interior of ${\gamma}$ is equal to the winding number ${W_{f \circ \gamma}(0)}$ of ${f \circ \gamma}$ around the origin.

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

Corollary 33 (Stability of number of zeroes) Let ${U}$ be an open set. Let ${\gamma_0: [a,b] \rightarrow U}$, ${\gamma_1: [a,b] \rightarrow U}$ be simple closed anticlockwise contours that are homotopic as closed curves via some homotopy ${\gamma: [0,1] \times [a,b] \rightarrow U}$; suppose also that ${U}$ contains the interiors of ${\gamma_0}$ and ${\gamma_1}$. Let ${f_0, f_1: U \rightarrow {\bf C}}$ be holomorphic, and let ${f: [0,1] \times U \rightarrow {\bf C}}$ be a continuous function such that ${f(0,z) = f_0(z)}$ and ${f(1,z) = f_1(z)}$ for all ${z \in U}$. Suppose that ${f(s,\gamma(s,t)) \neq 0}$ for all ${s \in [0,1]}$ and ${t \in [a,b]}$ (i.e., at time ${s}$, the curve ${\gamma(s,\cdot)}$ never encounters any zeroes of ${f(s,\cdot)}$). Then the number of zeroes (counting multiplicity) of ${f_0}$ in the interior of ${\gamma_0}$ equals the number of zeroes of ${f_1}$ in the interior of ${\gamma_1}$ (counting multiplicity).

Proof: By Corollary 32, it suffices to show that

$\displaystyle W_{f_0 \circ \gamma_0}(0) = W_{f_1 \circ \gamma_1}(0).$

But the curves ${f_0 \circ \gamma_0: [a,b] \rightarrow {\bf C} \backslash \{0\}}$ and ${f_1 \circ \gamma_1: [a,b] \rightarrow {\bf C} \backslash \{0\}}$ are homotopic as closed curves in ${{\bf C} \backslash \{0\}}$, using the homotopy ${F: [0,1] \times [a,b] \rightarrow {\bf C} \backslash \{0\}}$ defined by

$\displaystyle F( s, t ):= f(s,\gamma(s,t))$

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

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

Example 34 Let ${\gamma}$ be the unit circle ${\gamma = \gamma_{0,1,\circlearrowleft}}$. The polynomial ${f_0(z) := z^2}$ has a double zero at ${0}$, so (counting multiplicity) has two zeroes in the interior of ${\gamma}$. If we consider instead the perturbation ${f_\varepsilon(z) = z^2 + \varepsilon}$ for some ${\varepsilon>0}$, this has simple zeroes at ${+i\sqrt{\varepsilon}}$ and ${-i\sqrt{\varepsilon}}$ respectively, so as long as ${\varepsilon<1}$, the holomorphic function ${f_\varepsilon}$ also has two zeroes in the interior of ${\gamma}$; but as ${\varepsilon}$ crosses ${1}$, the zeroes of ${f_\varepsilon}$ pass through ${\gamma}$, and one no longer has any zeroes of ${f_\varepsilon}$ in the interior of ${\gamma}$. The situation can be contrasted with the real case: the function ${x \mapsto x^2+\varepsilon}$ has a double zero at the origin when ${\varepsilon=0}$, but as soon as ${\varepsilon}$ becomes positive, the zeroes immediately disappear from the real line. Note that the stability of zeroes fails if we do not count zeroes with multiplicity; thus, as a general rule of thumb, one should always try to count zeroes with multiplicity when doing complex analysis. (Heuristically, one can think of a zero of order ${m}$ as ${m}$ simple zeroes that are “infinitesimally close together".)

Example 35 When one considers meromorphic functions instead of holomorphic ones, then the number of zeroes inside a region need not be stable any more, but the number of zeroes minus the number of poles will be stable. Consider for instance the meromorphic function ${f_0(z) = \frac{z^2}{z^2}}$, which has a removable singularity at ${0}$ but no zeroes or poles. If we perturb it to ${f_\varepsilon(z) := \frac{z^2+\varepsilon}{z^2}}$ for some ${\varepsilon>0}$, then we suddenly have a double pole at ${0}$, but this is balanced by two simple zeroes at ${+i\sqrt{\varepsilon}}$ and ${-i\sqrt{\varepsilon}}$; in the limit as ${\varepsilon \rightarrow 0}$ we see that the two zeroes “collide” with the double pole, annihilating both the zeroes and the poles.

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

Theorem 36 (Rouche’s theorem) Let ${\gamma}$ be a simple closed contour, and let ${U}$ be an open set containing the image of ${\gamma}$ and its interior. Let ${f, g: U \rightarrow {\bf C}}$ be holomorphic. If one has ${|g(z)| < |f(z)|}$ for all ${z}$ in the image of ${\gamma}$, then ${f}$ and ${f+g}$ have the same number of zeroes (counting multiplicity) in the interior of ${\gamma}$.

Proof: We may assume without loss of generality that ${\gamma}$ is anticlockwise. By hypothesis, ${f}$ and ${f+g}$ cannot have zeroes on the image of ${\gamma}$. The claim then follows from Corollary 33 with ${\gamma_0=\gamma_1=\gamma}$, ${\gamma(s,t) := \gamma(t)}$, ${f_0(z) := f(z)}$, ${f_1(z) := f(z) + g(z)}$, and ${f(s,z) := f(z) + s g(z)}$. $\Box$

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

Theorem 37 (Open mapping theorem) Let ${U}$ be an open connected non-empty subset of ${{\bf C}}$, and let ${f: U \rightarrow {\bf C}}$ be holomorphic and not constant. Then ${f(U)}$ is also open.

Proof: Let ${z_0 \in U}$. As ${f}$ is not constant, the zeroes of ${f(z)-f(z_0)}$ are isolated (Corollary 24 of Notes 3). Thus, for ${r}$ sufficiently small, ${f(z)-f(z_0)}$ is nonvanishing on the image of the circle ${\gamma_{z_0,r,\circlearrowleft}}$. Clearly ${f(z)-f(z_0)}$ has at least one zero in the interior of this circle. Thus, by Rouche’s theorem, if ${w}$ is sufficiently close to ${f(z_0)}$, then ${f(z)-w}$ will also have at least one zero in the interior of this circle. In particular, ${f(U)}$ contains a neighbourhood of ${f(z_0)}$, and the claim follows. $\Box$

Exercise 38 Use Rouche’s theorem to obtain another proof of the fundamental theorem of algebra, by showing that a polynomial ${P(z) = a_n z^n + \dots + a_0}$ with ${a_n \neq 0}$ and ${n \geq 1}$ has exactly ${n}$ zeroes (counting multiplicity) in the complex plane. (Hint: compare ${P(z)}$ with ${a_n z^n}$ inside some large circle ${\gamma_{0,R,\circlearrowleft}}$.)

Exercise 39 (Inverse function theorem) Let ${U}$ be an open subset of ${{\bf C}}$, let ${z_0 \in U}$, and let ${f: U \rightarrow {\bf C}}$ be a holomorphic function such that ${f'(z_0) \neq 0}$. Show that there exists a neighbourhood ${V}$ of ${z_0}$ in ${U}$ such that the map ${f: V \rightarrow f(V)}$ is a complex diffeomorphism; that is to say, it is holomorphic, invertible, and the inverse is also holomorphic. Finally, show that

$\displaystyle (f^{-1})'(w) = \frac{1}{f'(f^{-1}(w))}$

for all ${w \in f(V)}$. (Hint: one can either mimic the real-variable proof of the inverse function theorem using the contraction mapping theorem, or one can use Rouche’s theorem and the open mapping theorem to construct the inverse.)

Exercise 40 Let ${U}$ be an open subset of ${{\bf C}}$, and ${f: U \rightarrow {\bf C}}$ be a map. Show that the following are equivalent:

• (i) ${f}$ is a local complex diffeomorphism. That is to say, for every ${z_0 \in U}$ there is a neighbourhood ${V}$ of ${z_0}$ in ${U}$ such that ${f(V)}$ is open and ${f: V \rightarrow f(V)}$ is a complex diffeomorphism (as defined in the preceding exercise).
• (ii) ${f}$ is holomorphic on ${U}$ and is a local homeomorphism. That is to say, for every ${z_0 \in U}$ there is a neighbourhood ${V}$ of ${z_0}$ in ${U}$ such that ${f(V)}$ is open and ${f: V \rightarrow f(V)}$ is a homeomorphism.
• (iii) ${f}$ is holomorphic on ${U}$ and is locally injective. That is to say, for every ${z_0 \in U}$ there is a neighbourhood ${V}$ of ${z_0}$ in ${U}$ such that ${f: V \rightarrow f(V)}$ is injective.
• (iv) ${f}$ is holomorphic on ${U}$, and the derivative ${f'}$ is nowhere vanishing.

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

• (i) If none of the ${f_n}$ have any zeroes in ${U}$, show that either ${f}$ also has no zeroes in ${U}$, or is identically zero.
• (ii) If all of the ${f_n}$ are univalent (that is to say, they are injective holomorphic functions), show that either ${f}$ is also univalent, or is constant.

Exercise 42 (Bloch’s theorem) The purpose of this exercise is to establish a more quantitative variant of the open mapping theorem, due to Bloch; this will be useful later in this notes for proving the Picard and Montel theorems. Let ${f: D(z_0,R) \rightarrow {\bf C}}$ be a holomorphic function on a disk ${D(z_0,R)}$, and suppose that ${f'(z_0)}$ is non-zero

• (i) Suppose that ${|f'(z)| \leq 2 |f'(z_0)|}$ for all ${z \in D(z_0,R)}$. Show that there is an absolute constant ${c>0}$ such that ${f(D(z_0,R))}$ contains the disk ${D(f(z_0), c|f'(z_0)| R)}$. (Hint: one can normalise ${z_0=0}$, ${R=1}$, ${f'(z_0)=1}$. Use the higher order Cauchy integral formula to get some bound on ${f''(z)}$ for ${z}$ near the origin, and use this to approximate ${f}$ by ${z}$ near the origin. Then apply Rouche’s theorem.)
• (ii) Without the hypothesis in (i), show that there is an absolute constant ${c'>0}$ such that ${f(D(z_0,R))}$ contains a disk of radius ${c' |f'(z_0)| R}$. (Hint: if one has ${|f'(z)| \leq 2 |f'(z_0)|}$ for all ${z \in D(z_0,R/4)}$, then we can apply (i) with ${R}$ replaced by ${R/4}$. If not, pick ${z_1 \in D(z_0,R/4)}$ with ${|f'(z_1)| > 2 |f'(z_0)|}$, and start over with ${z_0}$ replaced by ${z_1}$ and ${R}$ replaced by ${R/2}$. One cannot iterate this process indefinitely as it will create a singularity of ${f}$ in ${D(z_0,R)}$.)

— 4. Branches of the complex logarithm —

We have refrained until now from discussing one of the most basic transcendental functions in complex analysis, the complex logarithm. In real analysis, the real logarithm ${\ln: (0,+\infty) \rightarrow {\bf R}}$ can be defined as the inverse of the exponential function ${\exp: {\bf R} \rightarrow (0,+\infty)}$; it can also be equivalently defined as the antiderivative of the function ${x \mapsto \frac{1}{x}}$, with the initial condition ${\ln(1) = 0}$. (We use ${\ln}$ here for the real logarithm in order to distinguish it from the complex logarithm below.)

Let’s see what happens when one tries to extend these definitions to the complex domain. We begin with the inversion of the complex exponential. From Euler’s formula we have that ${e^{2\pi i} = 1}$; more generally, we have ${e^z = e^w}$ whenever ${z = w + 2 k \pi i}$ for some integer ${k}$. In particular, the exponential function ${\exp: {\bf C} \rightarrow {\bf C}}$ is not injective. Indeed, for any non-zero ${z \in {\bf C}}$, we have a multi-valued logarithm

$\displaystyle \log(z) := \{ w: e^w = z \}$

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

$\displaystyle \log(z) = \ln |z| + i \mathrm{arg}(z)$

where

$\displaystyle \mathrm{arg}(z) := \{ \theta \in {\bf R}: \cos \theta + i \sin \theta = \frac{z}{|z|} \}$

denotes all the possible arguments of ${z}$ in polar form. These arguments are a coset of the group ${2\pi {\bf Z} := \{ 2\pi k: k \in {\bf Z} \}}$, and so the complex logarithm ${\log(z)}$ is a coset of the group ${2\pi i {\bf Z} := \{ 2\pi i k: k \in {\bf Z}\}}$. For instance, if ${z = 2i}$, then

$\displaystyle \mathrm{arg}(2i) = \{ \frac{\pi}{2} + 2 k \pi: k \in {\bf Z} \}$

and

$\displaystyle \log(2i) = \{ \ln 2 + \frac{\pi i}{2} + 2 k \pi i: k \in {\bf Z} \}.$

The complex exponential never vanishes, so by our definitions we see that ${\log(0)}$ is the empty set. As such, we will usually omit the origin from the domain ${{\bf C}}$ when discussing the complex exponential.

Of course, one also encounters multi-valued functions in real analysis, starting when one tries to invert the squaring function ${x \mapsto x^2}$, as any given positive number ${y}$ has two square roots. In the real case, one can eliminate this multi-valuedness by picking a branch of the square root function – a function which selects one of the multiple choices for that function at each point in the domain. In particular, we have the positive branch ${x \mapsto \sqrt{x}}$ of the square root function on ${[0,+\infty)}$, as well as the negative branch ${x \mapsto - \sqrt{x}}$. One could also create more discontinuous branches of the square root function, for instance the function that sends ${x}$ to ${\sqrt{x}}$ for ${0 \leq x \leq 5}$, and ${x}$ to ${-\sqrt{x}}$ for ${x > 5}$.

Suppose now that we have a branch ${f: {\bf C} \backslash \{0\} \rightarrow {\bf C}}$ of the logarithm function, thus

$\displaystyle \exp(f(z))=z \ \ \ \ \ (14)$

for any ${z \in {\bf C} \backslash \{0\}}$. If ${f}$ is complex differentiable at some point ${z_0 \in {\bf C} \backslash \{0\}}$, then by differentiating (14) at ${z_0}$ using the chain rule, we see that

$\displaystyle f'(z_0) \exp( f(z_0) ) = 1$

and hence by (14) again we have

$\displaystyle f'(z_0) = \frac{1}{z_0}$

(which is of course consistent with the real-variable formula ${\frac{d}{dx} \log x = \frac{1}{x}}$). If now ${\gamma}$ is a closed contour in ${{\bf C} \backslash \{0\}}$, and ${f}$ is differentiable on the entire image of ${\gamma}$, then the fundamental theorem of calculus then tells us that

$\displaystyle \int_\gamma \frac{dz}{z} = \int_\gamma f'(z)\ dz = 0.$

On the other hand, ${\int_\gamma \frac{dz}{z}}$ is equal to ${2\pi i W_\gamma(0)}$. We thus conclude that for any branch ${f}$ of the complex logarithm, the set on which ${f}$ is complex differentiable cannot contain any closed curve that winds non-trivially around the origin. Thus for instance one cannot find a branch of ${\log}$ that is holomorphic on all of ${{\bf C} \backslash \{0\}}$, or even on a neighbourhood of the unit circle (or any other curve going around the origin).

On the other hand, if ${U}$ is a simply connected open subset of ${{\bf C} \backslash \{0\}}$, then from Cauchy’s theorem the function ${\frac{1}{z}}$ is conservative on ${U}$. If we pick a point ${z_0}$ in ${U}$ and arbitrarily select a logarithm ${w_0 \in \log(z_0)}$ of ${z_0}$, we can then use the fundamental theorem of calculus to find an antiderivative ${f: U \rightarrow {\bf C}}$ of ${\frac{1}{z}}$ on ${U}$ with ${f(z_0)=w_0}$. By definition, ${f}$ is holomorphic, and from the chain rule we have for all ${z \in U}$ that

$\displaystyle \frac{d}{dz} \exp(f(z)) = f'(z) \exp(f(z))$

$\displaystyle = \frac{1}{z} \exp(f(z))$

and hence by the quotient rule

$\displaystyle \frac{d}{dz} \frac{\exp(f(z))}{z} = 0.$

As ${U}$ is connected, ${\frac{\exp(f(z))}{z}}$ must therefore be constant; by construction we have ${\frac{\exp(f(z_0))}{z_0}=1}$, and thus

$\displaystyle \frac{\exp(f(z))}{z} = 1$

for all ${z \in U}$. In other words, ${f}$ is a branch of the complex logarithm.

Thus, for instance, the region ${{\bf C} \backslash \{ -x: x \geq 0\}}$ formed by excluding the negative real axis from the complex plane is simply connected (it is star-shaped around ${1}$), and so must admit a holomorphic branch of the complex logarithm. One such branch is the standard branch ${\mathrm{Log}: {\bf C} \backslash \{0\} \rightarrow {\bf C}}$ of the complex logarithm, defined as

$\displaystyle \mathrm{Log}(z) := \ln |z| + i \mathrm{Arg}(z)$

where ${\mathrm{Arg}(z)}$ is the standard branch of the argument, defined as the unique argument in ${\mathrm{arg}(z)}$ in the interval ${(-\pi,\pi]}$. This branch of the logarithm is continuous on ${{\bf C} \backslash \{ -x: x \geq 0\}}$, and hence (by the exercise below) is holomorphic on this region, and is thus an antiderivative of ${1/z}$ here. Similarly if one replaces the negative real axis by other rays emenating from the origin (or indeed from arbitrary simple curves from zero to infinity, see Exercise 44 below.)

Exercise 43 Let ${U}$ be a connected non-empty open subset of ${{\bf C} \backslash \{0\}}$.

• (i) If ${f: U \rightarrow {\bf C}}$ and ${g: U \rightarrow {\bf C}}$ are continuous branches of the complex logarithm, show that there exists a natural number ${k}$ such that ${f(z) = g(z) + 2\pi i k}$ for all ${z \in U}$.
• (ii) Show that any continuous branch ${f: U \rightarrow {\bf C}}$ of the complex logarithm is holomorphic.
• (iii) Show that there is a continuous branch ${f: U \rightarrow {\bf C}}$ of the logarithm if and only if ${0}$ and ${\infty}$ lie in the same connected component of ${({\bf C} \cup \{\infty\}) \backslash U}$. (Hint: for the “if” direction, use a continuity argument to show that the winding number of any closed curve in ${U}$ around ${0}$ vanishes. For the “only if”, you may use without proof the fact that if two points ${x,y}$ in a compact Hausdorff space ${X}$ do not lie in the same connected component, then there exists a clopen subset ${K}$ of ${X}$ that contains ${x}$ but not ${y}$ (sketch of proof: show that the intersection of all the clopen neighbourhoods of ${x}$ is connected and also contains all the connected sets containing ${x}$). Use this fact to encircle ${0}$ by a simple polygonal path in ${U}$.)

Exercise 44 Let ${\gamma: [0,+\infty) \rightarrow {\bf C}}$ be a continuous injective map with ${\gamma(0)=0}$ and ${\gamma(t) \rightarrow \infty}$ as ${t \rightarrow \infty}$.

• (i) Show that ${\gamma([0,+\infty))}$ is not all of ${{\bf C}}$. (Hint: modify the construction in Section 4 of Notes 3 that showed that a simple closed curve admitted at least one point with non-zero winding number.)
• (ii) Show that the complement ${{\bf C} \backslash \gamma([0,+\infty))}$ is simply connected. (Hint: modify the remaining arguments in Section 4 of Notes 3). In particular, by the preceding discussion, there is a branch of the complex logarithm that is holomorphic outside of ${\gamma([0,+\infty))}$.

It is instructive to view the identity

$\displaystyle \int_{\gamma_{0,1,\circlearrowleft}} \frac{dz}{z} = 2\pi i \ \ \ \ \ (15)$

, through the lens of branches of the complex logarithm such as the standard branch ${\mathrm{Log}}$. From the fundamental theorem of calculus, one has

$\displaystyle \int_\gamma \frac{dz}{z} = \mathrm{Log}(\gamma(b)) - \mathrm{Log}(\gamma(a))$

for any curve ${\gamma}$ that avoids the negative real axis. Of course, the contour ${\gamma_{0,1,\circlearrowleft}}$ does not avoid this negative axis, but it can be approximated by (non-closed) contours that do. More precisely, one has

$\displaystyle \int_{\gamma_{0,1,\circlearrowleft}} \frac{dz}{z} = \lim_{\varepsilon \rightarrow 0^+} \int_{\gamma_{[-\pi+\varepsilon,\pi-\varepsilon]}} \frac{dz}{z}$

where ${\gamma_{[-\pi+\varepsilon,\pi-\varepsilon]}: [-\pi+\varepsilon,\pi-\varepsilon] \rightarrow {\bf C}}$ is the map ${t \mapsto e^{it}}$. As each ${\gamma_{[-\pi+\varepsilon,\pi-\varepsilon]}}$ avoids the negative real axis, we thus have

$\displaystyle \int_{\gamma_{0,1,\circlearrowleft}} \frac{dz}{z} = \lim_{\varepsilon \rightarrow 0^+} \mathrm{Log}( e^{i (\pi-\varepsilon)}) - \mathrm{Log}( e^{i (-\pi+\varepsilon)}).$

We observe that ${\mathrm{Log}}$ has a jump discontinuity of ${2\pi i}$ on the negative real axis, and specifically

$\displaystyle \lim_{\varepsilon \rightarrow 0^+} \mathrm{Log}(e^{i(\pi-\varepsilon)}) = i \pi$

and

$\displaystyle \lim_{\varepsilon \rightarrow 0^+} \mathrm{Log}(e^{i(-\pi+\varepsilon)}) = -i \pi,$

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

$\displaystyle \int_{\gamma} \frac{dz}{z} = 2\pi i W_\gamma(0)$

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

One can use branches of the complex logarithm to create branches of the ${n^{\mathrm{th}}}$ root functions ${z \mapsto z^{1/n}}$ for natural numbers ${n>1}$. As with the complex exponential, the function ${z \mapsto z^n}$ is not injective, and so ${z^{1/n}}$ is multivalued (see Exercise 15 of Notes 0). One cannot form a continuous branch of this function on ${{\bf C} \backslash \{0\}}$ for any ${n \geq 2}$, as the corresponding branch of ${1/z^{1/n}}$ would then contradict the quantisation of order of singularities (Exercise 13). However, on any domain ${U}$ where there is a holomorphic branch ${f: U \rightarrow {\bf C}}$ of the complex logarithm, one can define a holomorphic branch ${g: U \rightarrow {\bf C}}$ of the ${n^{\mathrm{th}}}$ function by the formula

$\displaystyle g(z) := \exp( f(z) / n ).$

It is easy to see that ${g}$ is indeed holomorphic with ${g(z)^n = z}$ for all ${z \in U}$. Thus for instance we have the standard branch ${z \mapsto \exp( (\mathrm{Log} z) / n )}$ of the ${n^{\mathrm{th}}}$ root function, which is holomorphic away from the negative real axis. More generally, one can define a “standard branch of ${z \mapsto z^\alpha}$” for any complex ${\alpha}$ by the formula ${z \mapsto \exp( \alpha \mathrm{Log} z )}$, for instance the standard branch of ${i^i}$ can be computed to be ${e^{-\pi/2}}$.

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

Exercise 45 Compute the improper integral

$\displaystyle \int_0^\infty \frac{dx}{\sqrt{x} (x+1)} := \lim_{\varepsilon \rightarrow 0, R \rightarrow \infty} \int_\varepsilon^R \frac{dx}{\sqrt{x} (x+1)}$

by applying the residue theorem to the function ${\frac{f(z)}{z+1}}$ for some branch ${f(z)}$ of ${z \mapsto z^{-1/2}}$ with branch cut on the positive real axis, and using a “keyhole” contour that is a perturbation of

$\displaystyle \gamma_{0,\varepsilon,\circlearrowright} + \gamma_{\varepsilon \rightarrow R} + \gamma_{0,R,\circlearrowleft} + \gamma_{R \rightarrow \varepsilon};$

the key point is that the branch cut makes the contribution of (the perturbations) of ${\gamma_{\varepsilon \rightarrow R}}$ and ${\gamma_{R \rightarrow \varepsilon}}$ fail to cancel each other.

The construction of holomorphic branches of ${\log z}$ can be extended to other logarithms:

Exercise 46 Let ${U}$ be a simply connected subset of ${{\bf C}}$, and let ${f: U \rightarrow {\bf C}}$ be a holomorphic function with no zeroes on ${U}$.

• (i) Show that there exists a holomorphic branch ${g: U \rightarrow {\bf C}}$ of the complex logarithm ${\log f}$, thus ${\exp(g) = f}$.
• (ii) Show that for any natural number ${n > 1}$, there exists a holomorphic branch ${h: U \rightarrow {\bf C}}$ of the root function ${f^{1/n}}$, thus ${h^n = f}$.

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

Definition 47 (Covering map) Let ${f: M \rightarrow N}$ be a continuous map between two connected topological spaces ${M, N}$. We say that ${f}$ is a covering map if, for each ${z_0 \in N}$, there exists an open neighbourhood ${U}$ of ${z_0}$ in ${N}$ such that the preimage ${f^{-1}(U)}$ is the disjoint union of open subsets ${V_\alpha, \alpha \in A}$ of ${M}$, such that for each ${\alpha \in A}$, the map ${f: V_\alpha \rightarrow U}$ is a homeomorphism. In this situation, we call ${M}$ a covering space of ${N}$.

In complex analysis, one specialises to the situation in which ${M,N}$ are Riemann surfaces (e.g. they could be open subsets of ${{\bf C}}$), and ${f}$ is a holomorphic map. In that case, the homeomorphisms ${f: V_\alpha \rightarrow U}$ are in fact complex diffeomorphisms, thanks to Exercise 40.

Example 48 The exponential map ${\exp: {\bf C} \rightarrow {\bf C} \backslash \{0\}}$ is a covering map, because for any element of ${{\bf C} \backslash \{0\}}$ written in polar form as ${r e^{i\theta}}$, one can pick (say) the neighbourhood

$\displaystyle U := \{ s e^{i\alpha}: r/2 < s < 2r; \theta - \frac{\pi}{2} < \alpha < \theta + \frac{\pi}{2} \}$

of ${re^{i\theta}}$, and observe that the preimage ${\exp^{-1}(U) = \log U}$ of ${U}$ is the disjoint union of the open sets

$\displaystyle V_k := \{ x + i \alpha: \ln(r/2) < x < \ln(r); \theta + 2 k \pi - \frac{\pi}{2} < \alpha < \theta + \frac{\pi}{2} \}$

for ${k \in {\bf Z}}$, and that the exponential map ${\exp: V_k \rightarrow U}$ is a diffeomorphism. A similar calculation shows that for any natural number ${n > 1}$, the map ${z \mapsto z^n}$ is a covering map from ${{\bf C} \backslash \{0\}}$ to ${{\bf C} \backslash \{0\}}$. However, the map ${z \mapsto z^n}$ is not a covering map from ${{\bf C}}$ to ${{\bf C}}$, because it fails to be a local diffeomorphism at zero due to the vanishing derivative (here we use Exercise 40). One final (non-)example: the map ${z \mapsto z^3}$ is not a covering map from the upper half-plane ${\{ z \in {\bf C}: \mathrm{Im}(z)>0\}}$ to ${{\bf C} \backslash \{0\}}$, because the preimage of any small disk ${D(1,r)}$ around ${1}$ splits into two disconnected regions, and only one of them is homeomorphic to ${D(1,r)}$ via the map ${z \mapsto z^3}$.

From topology we have the following lifting property:

Lemma 49 (Lifting lemma) Let ${f: M \rightarrow N}$ be a continuous covering map between two path-connected and locally path-connected topological spaces ${M,N}$. Let ${U}$ be a simply connected and path connected topological space, and let ${g: U \rightarrow N}$ be continuous. Let ${z_0 \in U}$, and let ${p \in M}$ be such that ${f(p) = g(z_0)}$. Then there exists a unique continuous map ${h: U \rightarrow M}$ such that ${g = f \circ h}$ and ${h(z_0)=p}$, which we call a lift of ${g}$ by ${f}$.

Proof: We first verify uniqueness. If we have two continuous functions ${h_1, h_2: U \rightarrow N}$ with ${h_1(z_0) = h_2(z_0) = p}$ and ${g = f \circ h_1 = f \circ h_2}$, then the set ${\Omega := \{ z \in U: h_1(z) = h_2(z) \}}$ is clearly closed in ${N}$ and contains ${z_0}$. From the covering map property we also see that ${\Omega}$ is open, and hence by connectedness we have ${h_1=h_2}$ on all of ${U}$, giving the claim.

To verify existence of the lift, we first prove the existence of monodromy. More precisely, given any curve ${\gamma:[a,b] \rightarrow U}$ with ${\gamma(a) = z_0}$ we show that there exists a unique curve ${\tilde \gamma: [a,b] \rightarrow M}$ such that ${\tilde \gamma(a) = p}$ and ${f \circ \tilde \gamma = g \circ \gamma}$ (the reader is encouraged to draw a picture to describe this situation). Uniqueness follows from the connectedness argument used to prove uniqueness of the lift ${h}$, so we turn to existence. As in previous notes, we rely on a continuity argument. Let ${\Omega}$ be the set of all ${a \leq t \leq b}$ for which there exists a curve ${\tilde \gamma_{[a,t]}: [a,t] \rightarrow M}$ such that ${f \circ \tilde \gamma_{[a,t]} = g \circ \gamma_{[a,t]}}$, where ${\gamma_{[a,t]}}$ is the restriction of ${\gamma}$ to ${[a,t]}$. Clearly ${\Omega}$ is closed in ${[a,b]}$ and contains ${a}$; using the covering map property it is not difficult to show that ${\Omega}$ is also open in ${[a,b]}$. Thus ${\Omega}$ is all of ${[a,b]}$, giving the claim.

Now let ${\gamma_0: [a,b] \rightarrow U}$, ${\gamma_1: [a,b] \rightarrow U}$ be homotopic curves with fixed endpoints, with initial point ${\gamma_0(a)=\gamma_1(a)=z_0}$ and some terminal point ${z_1}$, and let ${\gamma: [0,1] \times [a,b] \rightarrow U}$ be a homotopy. For each ${s \in [0,1]}$, we have a curve ${\gamma_s: [a,b] \rightarrow U}$ given by ${\gamma_s(t) := \gamma(s,t)}$, and by the preceding paragraph we can associate a curve ${\tilde \gamma_s: [a,b] \rightarrow M}$ such that ${\tilde \gamma_s(a) = p}$ and ${f \circ \tilde \gamma_s = g \circ \gamma_s}$. Another application of the continuity method shows that for all ${t \in [a,b]}$, the map ${s \mapsto \tilde \gamma_s(t)}$ is continuous; in particular, the map ${b \rightarrow \tilde \gamma_s(b)}$. On the other hand, ${\tilde \gamma_s(b)}$ lies in ${f^{-1}(g(z_1))}$, which is a discrete set thanks to the covering map property. We conclude that ${\tilde \gamma_s(b)}$ is constant in ${s}$, and in particular that ${\tilde \gamma_0(b) = \tilde \gamma_1(b)}$.

Since ${U}$ is simply connected, any two curves ${\gamma_0, \gamma_1:[a,b] \rightarrow U}$ with fixed endpoints are homotopic. We can thus define a function ${h: U \rightarrow M}$ by declaring ${h(z_1)}$ for any ${z_1 \in U}$ to be the point ${\tilde \gamma(1)}$, where ${\gamma: [0,1] \rightarrow U}$ is any curve from ${z_0}$ to ${z_1}$, and ${\tilde \gamma}$ is constructed as before. By construction we have ${g = f \circ h}$, and from the local path connectedness of ${N}$ and the covering map property of ${f}$ we can check that ${h}$ is continuous. The claim follows. $\Box$

We can specialise this to the complex case and obtain

Corollary 50 (Holomorphic lifting lemma) Let ${f: M \rightarrow N}$ be a holomorphic covering map between two connected Riemann surfaces ${M,N}$. Let ${U}$ be a simply connected and path connected Riemann surface, and let ${g: U \rightarrow N}$ be holomorphic. Let ${z_0 \in U}$, and let ${p \in M}$ be such that ${f(p) = g(z_0)}$. Then there exists a unique holomorphic map ${h: U \rightarrow M}$ such that ${g = f \circ h}$ and ${h(z_0)=p}$, which we call a lift of ${g}$ by ${f}$.

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

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

Exercise 52 Establish Exercise 46 using Corollary 50.

Exercise 53 Let ${U}$ be simply connected, and let ${f: U \rightarrow {\bf C} \backslash \{-1,+1\}}$ be holomorphic and avoid taking the values ${+1,-1}$. Show that there exists a holomorphic function ${g: U \rightarrow {\bf C}}$ such that ${f = \cos g}$. (This can be proven either through Corollary 50, or by using the quadratic formula to solve for ${e^{ig}}$ and then applying Exercise 46.)

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

Exercise 54 Show that there exists a holomorphic function ${f: {\bf C} \backslash [0,1] \rightarrow {\bf C}}$ such that ${\exp(f(z)) = \frac{z-1}{z}}$ for all ${z \in {\bf C} \backslash [0,1]}$. (Hint: use the Schwartz reflection principle, see Exercise 37 of Notes 3.)

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

Theorem 55 (Little Picard theorem) Let ${f: {\bf C} \rightarrow {\bf C}}$ be entire and non-constant. Then ${f({\bf C})}$ omits at most one point of ${{\bf C}}$.

The example of the exponential function ${\exp: {\bf C} \rightarrow {\bf C}}$, whose range omits the origin, shows that one cannot make any stronger conclusion about ${f({\bf C})}$.

Proof: Suppose for contradiction that we have an entire non-constant function ${f: {\bf C} \rightarrow {\bf C}}$ such that ${f({\bf C})}$ omits at least two points. After applying a linear transformation, we may assume that ${f({\bf C})}$ avoids ${0}$ and ${1}$, thus ${f}$ takes values in ${{\bf C} \backslash \{0,1\}}$.

At this point, the most natural thing to do from a Riemann surface point of view would be to cover ${{\bf C} \backslash \{0,1\}}$ by a bounded region, so that Liouville’s theorem may be applied. This can be done easily once one has the machinery of elliptic functions; but as we do not have this machinery yet, we will instead use a more ad hoc covering of ${{\bf C} \backslash \{0,1\}}$ using the exponential and trigonometric functions to achieve a passable substitute for this strategy.

We turn to the details. Since ${f({\bf C})}$ avoids ${0}$, we may apply Exercise 46 to write ${f = \exp(2\pi i g)}$ for some entire ${g: {\bf C} \rightarrow {\bf C}}$. As ${f({\bf C})}$ avoids ${1}$, ${g({\bf C})}$ must avoid the integers ${{\bf Z}}$.

Next, we apply Exercise 53 to write ${g = \cos(h)}$ for some entire ${h: {\bf C} \rightarrow {\bf C}}$. The set ${h({\bf C})}$ must now avoid all complex numbers of the form ${\pm i \cosh^{-1}(j) + 2 \pi k}$ for natural numbers ${j}$ and integers ${k}$. In particular, if ${C}$ is large enough, we see that ${h({\bf C})}$ does not contain any disk of the form ${D(w,C)}$. Applying Bloch’s theorem (Exercise 42(ii)) in the contrapositive, we conclude that for any disk ${D(z_0,R)}$ in ${{\bf C}}$, one has ${|h'(z_0)|\leq C'/R}$ for some absolute constant ${C'}$. Sending ${R}$ to infinity and using the fundamental theorem of calculus, we conclude that ${h}$ is constant, hence ${g}$ and ${f}$ are also constant, a contradiction. $\Box$

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

Theorem 56 (Great Picard theorem) Let ${f: D(z_0,r) \backslash \{z_0\} \rightarrow {\bf C}}$ be holomorphic on a disk ${D(z_0,r)}$ outside of a singularity at ${z_0}$. If this singularity is essential, then ${f( D(z_0,r) \backslash \{z_0\} )}$ omits at most one point of ${{\bf C}}$.

Note that if one only has a pole at ${z_0}$, e.g. if ${f(z) = (z-z_0)^{-m}}$ for some natural number ${m}$, then the conclusion of the great Picard theorem fails. This result easily implies both the little Picard theorem (because if ${f: {\bf C} \rightarrow {\bf C}}$ is entire and non-polynomial, then ${f(1/z)}$ has an essential singularity at the origin) and the Casorati-Weierstrass theorem (Theorem 11(iii)). By repeatedly passing to smaller neighbourhoods, one in fact sees that with at most one exception, every complex number ${c \in {\bf C}}$ is attained infinitely often by a function holomorphic in a punctured disk around an essential singularity.

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

Assume for contradiction that ${f: D(z_0,r) \backslash \{z_0\} \rightarrow {\bf C}}$ has an essential singularity at ${z_0}$ and avoids at least two points in ${{\bf C}}$. Applying linear transformations to both the domain and range of ${f}$, we may normalise ${z_0=0}$, ${r=1}$, and assume that ${f}$ avoids ${0}$ and ${1}$, thus we have a holomorphic map ${f: D(0,1) \backslash \{0\} \rightarrow {\bf C} \backslash \{0,1\}}$ with an essential singularity at ${0}$.

The domain ${D(0,1) \backslash \{0\}}$ is not simply connected, so we work instead with the function

$\displaystyle F: \{ z \in {\bf C}: \mathrm{Re}(z) > 0 \} \rightarrow {\bf C} \backslash \{0,1\}$

defined by

$\displaystyle F(z) := f( \exp(-z) ).$

Clearly ${F}$ is holomorphic on the right-half plane ${\{ z \in {\bf C}: \mathrm{Re}(z) > 0 \}}$ and avoids ${0,1}$. We also observe that ${F}$ obeys the periodicity property

$\displaystyle F(z + 2\pi i) = F(z). \ \ \ \ \ (16)$

As the right-half plane is simply connected, we may (as before) express ${F = \exp(2\pi i G)}$ for some holomorphic function ${G: \{ z \in {\bf C}: \mathrm{Re}(z) > 0 \} \rightarrow {\bf C} \backslash {\bf Z}}$, and then write ${G = \cos(H)}$ for some holomorphic function ${H: \{z \in {\bf C}: \mathrm{Re}(z) > 0 \} \rightarrow {\bf C}}$ that avoids all numbers of the form ${\pm i \cosh^{-1}(j) + 2 \pi k}$ for natural numbers ${j}$ and integers ${k}$. Using Bloch’s theorem as before, we see that for any disk ${D(z_0,R)}$ in the right-half plane ${\{z \in {\bf C}: \mathrm{Re}(z) > 0 \}}$, we have ${|H'(z_0)| \leq C / R}$ for some absolute constant ${C}$. We cannot set ${R}$ to infinity any more, but we can make ${R}$ as large as the real part of ${z_0}$, giving the bound

$\displaystyle |H'(z_0)| \leq \frac{C}{\mathrm{Re}(z_0)}.$

In particular, on integrating along a line segment from ${2+iy}$ to ${x+iy}$, and using the boundedness of ${H}$ on the compact set ${\{ 2+iy: 0 \leq y \leq 2\pi\}}$, we obtain a bound of the form

$\displaystyle |H(x+iy)| \leq A \log x$

for some ${A > 0}$, and all ${x \geq 2}$ and ${0 \leq y \leq 2\pi}$. Taking cosines using the formula ${\cos(z) = (e^z+e^{-z})/2}$, we obtain a polynomial type bound

$\displaystyle |G(x+iy)| \leq x^{A} \ \ \ \ \ (17)$

for all ${x \geq 2}$ and ${0 \leq y \leq 2\pi}$.

On the other hand, from (16) one has

$\displaystyle \exp( G( z + 2\pi i ) ) = \exp( G(z) )$

and hence

$\displaystyle G(z+2\pi i) - G(z) \in 2 \pi i {\bf Z}$

for all ${z}$ in the right half-plane. The set ${2\pi i {\bf Z}}$ is discrete, the function ${z \mapsto G(z+2\pi i)-G(z)}$ is continuous, and the right half-plane is connected, so this function must in fact be constant. That is to say, there exists an integer ${k}$ such that

$\displaystyle G(z+2\pi i) - G(z) = 2\pi i k$

for all ${z}$ in the upper half plane. Equivalently, the function ${G(z) - kz}$ is periodic with period ${2\pi i}$. From (17) and the triangle inequality we conclude that

$\displaystyle |G(x+iy) - k(x+iy)| \leq x^A + 2 \pi |k| |y| \ \ \ \ \ (18)$

for all ${x \geq 2}$ and ${y \in {\bf R}}$.

We now upgrade this bound on (18) by exploiting the quantisation of pole orders (Exercise 13). As the function ${G(z)-kz}$ is periodic with period ${2\pi i}$ on the right half-plane, we may write

$\displaystyle G(z) - k z = g( \exp( -z ) ) \ \ \ \ \ (19)$

for some function ${g: D(0,1) \backslash \{0\} \rightarrow {\bf C}}$, which is holomorphic thanks to the chain rule. From (18) we have

$\displaystyle |g(z)| \leq \log^A \frac{1}{|z|} + 2\pi |k| \log \frac{1}{|z|}$

when ${0 < |z| \leq e^{-2}}$. Applying Exercise 13 (with, say, ${m=0}$ and ${\varepsilon=1/2}$), we conclude that ${g}$ has a removable singularity and is thus in particular bounded on (say) the disk ${D(0,e^{-2})}$. From (19) we conclude that ${G(z)-kz}$ is bounded on the region ${\{ z: \mathrm{Re}(z) \geq 2 \}}$; taking exponentials, we conclude that ${F(z) e^{-kz}}$ is also bounded on this region. Since ${F(z) = f(\exp(-z))}$, we conclude that ${f(z) z^k}$ is bounded on ${D(0,e^{-2}) \backslash \{0\}}$, and thus by Riemann’s theorem (Exercise 35 from Notes 3) has a removable singularity at the origin. But by taking Laurent series, this implies that ${f}$ has a pole of order at most ${k}$ at the origin, contradicting the hypothesis that the singularity of ${f}$ at the origin was essential. $\Box$

Exercise 57 (Montel’s theorem) Let ${U}$ be an open subset of the complex plane. Define a holomorphic normal family on ${U}$ to be a collection ${{\mathcal F}}$ of holomorphic functions ${f: U \rightarrow {\bf C}}$ with the following property: given any sequence ${f_n}$ in ${{\mathcal F}}$, there exists a subsequence ${f_{n_j}}$ which is uniformly convergent on compact sets (i.e., for every compact subset ${K}$ of ${U}$, the sequence ${f_{n_j}}$ converges uniformly on ${K}$ to some limit). Similarly, define a meromorphic normal family to be a collection ${{\mathcal F}}$ of meromorphic functions ${f: U \rightarrow {\bf C} \cup \{\infty\}}$ such that for any sequence ${f_n}$ in ${{\mathcal F}}$, there exists a subsequence ${f_{n_j}: U \rightarrow {\bf C} \cup \{\infty\}}$ that are uniformly convergent on compact sets, using the metric on the Riemann sphere induced by the identification with the geometric sphere ${\{ (z,t) \in {\bf C} \times {\bf R}: |z|^2 + (t-1/2)^2 = 1/4\}}$. (More succinctly, normal families are those families of holomorphic or meromorphic functions that are precompact in the locally uniform topology.)

• (i) (Little Montel theorem) Suppose that ${{\mathcal F}}$ is a collection of holomorphic functions ${f: U \rightarrow {\bf C}}$ that are uniformly bounded on compact sets (i.e., for each compact ${K \subset U}$ there exists a constant ${C_K}$ such that ${|f(z)| \leq C_K}$ for all ${f \in {\mathcal F}}$ and ${z \in K}$). Show that ${{\mathcal F}}$ is a holomorphic normal family. (Hint: use the higher order Cauchy integral formula to establish some equicontinuity on this family on compact sets, then use the Arzelá-Ascoli theorem.
• (ii) (Great Montel theorem) Let ${z_0,z_1,z_2 \in {\bf C} \cup \{\infty\}}$ be three distinct elements of the Riemann sphere, and suppose that ${{\mathcal F}}$ is a family of meromorphic functions ${f: U \rightarrow ({\bf C} \cup \{\infty\}) \backslash \{z_0,z_1,z_2\}}$ which avoid the three points ${z_0,z_1,z_2}$. Show that ${{\mathcal F}}$ is a meromorphic normal family. (Hint: use some elementary transformations to reduce to the case ${z_0=0, z_1=1, z_2=\infty}$. Then, as in the proof of the Picard theorems, express each element ${f}$ of ${{\mathcal F}}$ locally in the form ${f = \exp(2\pi i \cos(h))}$ and use Bloch’s theorem to get some uniform bounds on ${h'}$.)

Exercise 58 (Harnack principle) Let ${U}$ be an open connected subset of ${{\bf C}}$, and let ${u_n: U \rightarrow {\bf R}}$ be a sequence of harmonic functions which is pointwise nondecreasing (thus ${u_{n+1}(z) \geq u_n(z)}$ for all ${z \in U}$ and ${n \geq 1}$). Show that ${\sup_n u_n}$ is either infinite everywhere on ${U}$, or is harmonic. (Hint: work locally in a disk. Write each ${u_n}$ on this disk as the real part of a holomorphic function ${f_n}$, and apply Montel’s theorem followed by the Hurwitz theorem to ${e^{-f_n}}$.) This result is known as Harnack’s principle.

Exercise 59

• (i) Show that the function ${z \mapsto \log|z|}$ is harmonic on ${{\bf C} \backslash \{0\}}$ but has no harmonic conjugate.
• (ii) Let ${r>0}$, and let ${u: D(0,r) \backslash \{0\} \rightarrow {\bf R}}$ be a harmonic function obeying the bounds

$\displaystyle |u(z)| \leq C_1 \log \frac{1}{|z|} + C_2$

for all ${z \in D(0,r) \backslash \{0\}}$ and some constants ${C_1,C_2}$. Show that there exists a real number ${c}$ and a harmonic function ${w: D(0,r) \rightarrow {\bf R}}$ such that

$\displaystyle u(z) = c \log |z| + w(z)$

for all ${z \in D(0,r)}$. (Hint: one can find a conjugate of ${u}$ outside of some branch cut, say the negative real axis restricted to ${B(0,r)}$. Adjust ${u}$ by a multiple of ${\log |z|}$ until the conjugate becomes continuous on this branch cut.)

Exercise 60 (Local description of holomorphic maps) Let ${f: U \rightarrow {\bf C}}$ be a holomorphic function on an open subset ${U}$ of ${{\bf C}}$, let ${z_0}$ be a point in ${U}$, and suppose that ${f}$ has a zero of order ${n}$ at ${z_0}$ for some ${n \geq 1}$. Show that there exists a neighbourhood ${V}$ of ${z_0}$ in ${U}$ on which one has the factorisation ${f(z) = g(z)^n}$, where ${g: V \rightarrow {\bf C}}$ is holomorphic with a simple zero at ${z_0}$ (and hence a complex diffeomorphism from a sufficiently small neighbourhood of ${z_0}$ to a neighbourhood of ${0}$). Use this to give an alternate proof of the open mapping theorem (Theorem 37).

Exercise 61 (Winding number and lifting) Let ${z_0 \in {\bf C}}$, let ${\gamma: [a,b] \rightarrow {\bf C} \backslash \{z_0\}}$ be a closed curve avoiding ${z_0}$, and let ${k}$ be an integer. Show that the following are equivalent:

• (i) ${W_{\gamma}(z_0) = k}$.
• (ii) There exists a complex number ${w}$ and a curve ${\eta: [a,b] \rightarrow {\bf C}}$ from ${w}$ to ${w+2k\pi i}$ such that ${\gamma(t) = z_0 + \exp(\eta(t))}$ for all ${t \in [a,b]}$.
• (iii) ${\gamma}$ is homotopic up to reparameterisation as closed curves in ${{\bf C} \backslash \{z_0\}}$ to the curve ${\gamma_{z_0,r,m \circlearrowleft}: [0,2\pi] \rightarrow {\bf C} \backslash \{z_0\}}$ that maps ${t}$ to ${z_0 + r e^{imt}}$ for some ${r>0}$.