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

The integers are the additive completion of the natural numbers (the minimal additive group that contains a copy of ).

are the additive completion of the natural numbers (the minimal additive group that contains a copy of ). The rationals are the multiplicative completion of the integers (the minimal field that contains a copy of ).

are the multiplicative completion of the integers (the minimal field that contains a copy of ). The reals are the metric completion of the rationals (the minimal complete metric space that contains a copy of ).

are the metric completion of the rationals (the minimal complete metric space that contains a copy of ). The complex numbers are the algebraic completion of the reals (the minimal algebraically closed field that contains a copy of ).

These descriptions of the standard number systems are elegant and conceptual, but not entirely suitable for constructing the number systems in a non-circular manner from more primitive foundations. For instance, one cannot quite define the reals from scratch as the metric completion of the rationals , because the definition of a metric space itself requires the notion of the reals! (One can of course construct by other means, for instance by using Dedekind cuts or by using uniform spaces in place of metric spaces.) The definition of the complex numbers as the algebraic completion of the reals does not suffer from such a non-circularity issue, but a certain amount of field theory is required to work with this definition initially. For the purposes of quickly constructing the complex numbers, it is thus more traditional to first define as a quadratic extension of the reals , and more precisely as the extension formed by adjoining a square root of to the reals, that is to say a solution to the equation . It is not immediately obvious that this extension is in fact algebraically closed; this is the content of the famous fundamental theorem of algebra, which we will prove later in this course.

The two equivalent definitions of – as the algebraic closure, and as a quadratic extension, of the reals respectively – each reveal important features of the complex numbers in applications. Because is algebraically closed, all polynomials over the complex numbers split completely, which leads to a good spectral theory for both finite-dimensional matrices and infinite-dimensional operators; in particular, one expects to be able to diagonalise most matrices and operators. Applying this theory to constant coefficient ordinary differential equations leads to a unified theory of such solutions, in which real-variable ODE behaviour such as exponential growth or decay, polynomial growth, and sinusoidal oscillation all become aspects of a single object, the complex exponential (or more generally, the matrix exponential ). Applying this theory more generally to diagonalise arbitrary translation-invariant operators over some locally compact abelian group, one arrives at Fourier analysis, which is thus most naturally phrased in terms of complex-valued functions rather than real-valued ones. If one drops the assumption that the underlying group is abelian, one instead discovers the representation theory of unitary representations, which is simpler to study than the real-valued counterpart of orthogonal representations. For closely related reasons, the theory of complex Lie groups is simpler than that of real Lie groups.

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

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

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

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

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

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

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

— 1. The construction and algebra of the complex numbers —

Note: this section will be far more algebraic in nature than the rest of the course; we are concentrating almost all of the algebraic preliminaries in this section in order to get them out of the way and focus subsequently on the analytic aspects of the complex numbers.

Thanks to the laws of high-school algebra, we know that the real numbers are a field: it is endowed with the arithmetic operations of addition, subtraction, multiplication, and division, as well as the additive identity and multiplicative identity , that obey the usual laws of algebra (i.e. the field axioms).

The algebraic structure of the reals does have one drawback though – not all (non-trivial) polynomials have roots! Most famously, the polynomial equation has no solutions over the reals, because is always non-negative, and hence is always strictly positive, whenever is a real number.

As mentioned in the introduction, one traditional way to define the complex numbers is as the smallest possible extension of the reals that fixes this one specific problem:

Definition 1 (The complex numbers) A field of complex numbers is a field that contains the real numbers as a subfield, as well as a root of the equation . (Thus, strictly speaking, a field of complex numbers is a pair , but we will almost always abuse notation and use as a metonym for the pair .) Furthermore, is generated by and , in the sense that there is no subfield of , other than itself, that contains both and ; thus, in the language of field extensions, we have .

(We will take the existence of the real numbers as a given in this course; constructions of the real number system can of course be found in many real analysis texts, including my own.)

Definition 1 is short, but proposing it as a definition of the complex numbers raises some immediate questions:

(Existence) Does such a field even exist?

even exist? (Uniqueness) Is such a field unique (up to isomorphism)?

unique (up to isomorphism)? (Non-arbitrariness) Why the square root of ? Why not adjoin instead, say, a fourth root of , or the solution to some other algebraic equation? Also, could one iterate the process, extending further by adding more roots of equations?

The third set of questions can be answered satisfactorily once we possess the fundamental theorem of algebra. For now, we focus on the first two questions.

We begin with existence. One can construct the complex numbers quite explicitly and quickly using the Argand plane construction; see Remark 7 below. However, from the perspective of higher mathematics, it is more natural to view the construction of the complex numbers as a special case of the more general algebraic construction that can extend any field by the root of an irreducible nonlinear polynomial over that field; this produces a field of complex numbers when specialising to the case where and . We will just describe this construction in that special case, leaving the general case as an exercise.

Starting with the real numbers , we can form the space of (formal) polynomials

with real co-efficients and arbitrary non-negative integer in one indeterminate variable . (A small technical point: we do not view this indeterminate as belonging to any particular domain such as , so we do not view these polynomials as functions but merely as formal expressions involving a placeholder symbol (which we have rendered in Roman type to indicate its formal character). In this particular characteristic zero setting of working over the reals, it turns out to be harmless to identify each polynomial with the corresponding function formed by interpreting the indeterminate as a real variable; but if one were to generalise this construction to positive characteristic fields, and particularly finite fields, then one can run into difficulties if polynomials are not treated formally, due to the fact that two distinct formal polynomials might agree on all inputs in a given finite field (e.g. the polynomials and agree for all in the finite field ). However, this subtlety can be ignored for the purposes of this course.) This space of polynomials has a pretty good algebraic structure, in particular the usual operations of addition, subtraction, and multiplication on polynomials, together with the zero polynomial and the unit polynomial , give the structure of a (unital) commutative ring. This commutative ring also contains as a subring (identifying each real number with the degree zero polynomial ). The ring is however not a field, because many non-zero elements of do not have multiplicative inverses. (In fact, no non-constant polynomial in has an inverse in , because the product of two non-constant polynomials has a degree that is the sum of the degrees of the factors.)

If a unital commutative ring fails to be a field, then it will instead possess a number of non-trivial ideals. The only ideal we will need to consider here is the principal ideal

This is clearly an ideal of – it is closed under addition and subtraction, and the product of any element of the ideal with an element of the full ring remains in the ideal .

We now define to be the quotient space

of the commutative ring by the ideal ; this is the space of cosets of in . Because is an ideal, there is an obvious way to define addition, subtraction, and multiplication in , namely by setting

and

for all ; these operations, together with the additive identity and the multiplicative identity , can be easily verified to give the structure of a commutative ring. Also, the real line embeds into by identifying each real number with the coset ; note that this identification is injective, as no real number is a multiple of the polynomial .

If we define to be the coset

then it is clear from construction that . Thus contains both and a solution of the equation . Also, since every element of is of the form for some polynomial , we see that every element of is a polynomial combination of with real coefficients; in particular, any subring of that contains and will necessarily have to contain every element of . Thus is generated by and .

The only remaining thing to verify is that is a field and not just a commutative ring. In other words, we need to show that every non-zero element of has a multiplicative inverse. This stems from a particular property of the polynomial , namely that it is irreducible in . That is to say, we cannot factor into non-constant polynomials

with . Indeed, as has degree two, the only possible way such a factorisation could occur is if both have degree one, which would imply that the polynomial has a root in the reals , which of course it does not.

Because the polynomial is irreducible, it is also prime: if divides a product of two polynomials in , then it must also divide at least one of the factors , . Indeed, if does not divide , then by irreducibility the greatest common divisor of and is . Applying the Euclidean algorithm for polynomials, we then obtain a representation of as

for some polynomials ; multiplying both sides by , we conclude that is a multiple of .

Since is prime, the quotient space is an integral domain: there are no zero-divisors in other than zero. This brings us closer to the task of showing that is a field, but we are not quite there yet; note for instance that is an integral domain, but not a field. But one can finish up by using finite dimensionality. As is a ring containing the field , it is certainly a vector space over ; as is generated by and , and , we see that it is in fact a two-dimensional vector space over , spanned by and (which are linearly independent, as clearly cannot be real). In particular, it is finite dimensional. For any non-zero , the multiplication map is an -linear map from this finite-dimensional vector space to itself. As is an integral domain, this map is injective; by finite-dimensionality, it is therefore surjective (by the rank-nullity theorem). In particular, there exists such that , and hence is invertible and is a field. This concludes the construction of a complex field .

Remark 2 One can think of the action of passing from a ring to a quotient by some ideal as the action of forcing some relations to hold between the various elements of , by requiring all the elements of the ideal (or equivalently, all the generators of ) to vanish. Thus one can think of as the ring formed by adjoining a new element to the existing ring and then demanding the constraint . With this perspective, the main issues to check in order to obtain a complex field are firstly that these relations do not collapse the ring so much that two previously distinct elements of become equal, and secondly that all the non-zero elements become invertible once the relations are imposed, so that we obtain a field rather than merely a ring or integral domain.

Remark 3 It is instructive to compare the complex field , formed by adjoining the square root of to the reals, with other commutative rings such as the dual numbers (which adjoins an additional square root of to the reals) or the split complex numbers (which adjoins a new root of to the reals). The latter two objects are perfectly good rings, but are not fields (they contain zero divisors, and the first ring even contains a nilpotent). This is ultimately due to the reducible nature of the polynomials and in .

Uniqueness of up to isomorphism is a straightforward exercise:

Exercise 4 (Uniqueness up to isomorphism) Suppose that one has two complex fields and . Show that there is a unique field isomorphism that maps to and is the identity on .

Now that we have existence and uniqueness up to isomorphism, it is safe to designate one of the complex fields as the complex field; the other complex fields out there will no longer be of much importance in this course (or indeed, in most of mathematics), with one small exception that we will get to later in this section. One can, if one wishes, use the above abstract algebraic construction as the choice for “the” complex field , but one can certainly pick other choices if desired (e.g. the Argand plane construction in Remark 7 below). But in view of Exercise 4, the precise construction of is not terribly relevant for the purposes of actually doing complex analysis, much as the question of whether to construct the real numbers using Dedekind cuts, equivalence classes of Cauchy sequences, or some other construction is not terribly relevant for the purposes of actually doing real analysis. So, from here on out, we will no longer refer to the precise construction of used; the reader may certainly substitute his or her own favourite construction of in place of if desired, with essentially no change to the rest of the lecture notes.

Exercise 5 Let be an arbitrary field, let be the ring of polynomials with coefficients in , and let be an irreducible polynomial in of degree at least two. Show that is a field containing an embedded copy of , as well as a root of the equation , and that this field is generated by and . Also show that all such fields are unique up to isomorphism. (This field is an example of a field extension of , the further study of which can be found in any advanced undergraduate or early graduate text on algebra, and is the starting point in particular for the beautiful topic of Galois theory, which we will not discuss here.)

Exercise 6 Let be an arbitrary field. Show that every non-constant polynomial in can be factored as the product of irreducible non-constant polynomials. Furthermore show that this factorisation is unique up to permutation of the factors , and multiplication of each of the factors by a constant (with the product of all such constants being one). In other words: the polynomiail ring is a unique factorisation domain.

Remark 7 (Real and imaginary coordinates) As a complex field is spanned (over ) by the linearly independent elements and , we can write with each element of having a unique representation of the form , thus for real . The addition, subtraction, and multiplication operations can then be written down explicitly in these coordinates as and with a bit more work one can compute the division operation as if . One could take these coordinate representations as the definition of the complex field and its basic arithmetic operations, and this is indeed done in many texts introducing the complex numbers. In particular, one could take the Argand plane as the choice of complex field, where we identify each point in with (so for instance becomes endowed with the multiplication operation ). This is a very concrete and direct way to construct the complex numbers; the main drawback is that it is not immediately obvious that the field axioms are all satisfied. For instance, the associativity of multiplication is rather tedious to verify in the coordinates of the Argand plane. In contrast, the more abstract algebraic construction of the complex numbers given above makes it more evident what the source of the field structure on is, namely the irreducibility of the polynomial .

Remark 8 Because of the Argand plane construction, we will sometimes refer to the space of complex numbers as the complex plane. We should warn, though, that in some areas of mathematics, particularly in algebraic geometry, is viewed as a one-dimensional complex vector space (or a one-dimensional complex manifold or complex variety), and so is sometimes referred to in those cases as a complex line. (Similarly, Riemann surfaces, which from a real point of view are two-dimensional surfaces, can sometimes be referred to as complex curves in the literature; the modular curve is a famous instance of this.) In this current course, though, the topological notion of dimension turns out to be more important than the algebraic notions of dimension, and as such we shall generally refer to as a plane rather than a line.

Elements of of the form for real are known as purely imaginary numbers; the terminology is colourful, but despite the name, imaginary numbers have precisely the same first-class mathematical object status as real numbers. If is a complex number, the real components of are known as the real part and imaginary part of respectively. Complex numbers that are not real are occasionally referred to as strictly complex numbers. In the complex plane, the set of real numbers forms the real axis, and the set of imaginary numbers forms the imaginary axis. Traditionally, elements of are denoted with symbols such as , , or , while symbols such as are typically intended to represent real numbers instead.

Remark 9 We noted earlier that the equation had no solutions in the reals because was always positive. In other words, the properties of the order relation on prevented the existence of a root for the equation . As does have a root for , this means that the complex numbers cannot be ordered in the same way that the reals are ordered (that is to say, being totally ordered, with the positive numbers closed under both addition and multiplication). Indeed, one usually refrains from putting any order structure on the complex numbers, so that statements such as for complex numbers are left undefined (unless are real, in which case one can of course use the real ordering). In particular, the complex number is considered to be neither positive nor negative, and an assertion such as is understood to implicitly carry with it the claim that are real numbers and not just complex numbers. (Of course, if one really wants to, one can find some total orderings to place on , e.g. lexicographical ordering on the real and imaginary parts. However, such orderings do not interact too well with the algebraic structure of and are rarely useful in practice.)

As with any other field, we can raise a complex number to a non-negative integer by declaring inductively and for ; in particular we adopt the usual convention that (when thinking of the base as a complex number, and the exponent as a non-negative integer). For negative integers , we define for non-zero ; we leave undefined when is zero and is negative. At the present time we do not attempt to define for any exponent other than an integer; we will return to such exponentiation operations later in the course, though we will at least define the complex exponential for any complex later in this set of notes.

By definition, a complex field is a field together with a root of the equation . But if is a root of the equation , then so is (indeed, from the factorisation we see that these are the only two roots of this quadratic equation. Thus we have another complex field which differs from only in the choice of root . By Exercise 4, there is a unique field isomorphism from to that maps to (i.e. a complex field isomorphism from to ); this operation is known as complex conjugation and is denoted . In coordinates, we have

Being a field isomorphism, we have in particular that

and

for all complex numbers . It is also clear that complex conjugation fixes the real numbers, and only the real numbers: if and only if is real. Geometrically, complex conjugation is the operation of reflection in the complex plane across the real axis. It is clearly an involution in the sense that it is its own inverse:

One can also relate the real and imaginary parts to complex conjugation via the identities

Remark 10 Any field automorphism of has to map to a root of , and so the only field automorphisms of that preserve the real line are the identity map and the conjugation map; conversely, the real line is the subfield of fixed by both of these automorphisms. In the language of Galois theory, this means that is a Galois extension of , with Galois group consisting of two elements. There is a certain sense in which one can think of the complex numbers (or more precisely, the scheme of complex numbers) as a double cover of the real numbers (or more precisely, the scheme of real numbers), analogous to how the boundary of a Möbius strip can be viewed as a double cover of the unit circle formed by shrinking the width of the strip to zero. (In this analogy, points on the unit circle correspond to specific models of the real number system , and lying above each such point are two specific models , of the complex number system; this analogy can be made precise using Grothendieck’s “functor of points” interpretation of schemes.) The operation of complex conjugation is then analogous to the operation of monodromy caused by looping once around the base unit circle, causing the two complex fields sitting above a real field to swap places with each other. (This analogy is not quite perfect, by the way, because the boundary of a Möbius strip is not simply connected and can in turn be finitely covered by other curves, whereas the complex numbers are algebraically complete and admit no further finite extensions; one should really replace the unit circle here by something with a two-element fundamental group, such as the projective plane that is double covered by the sphere , but this is harder to visualize.) The analogy between (absolute) Galois groups and fundamental groups suggested by this picture can be made precise in scheme theory by introducing the concept of an étale fundamental group, which unifies the two concepts, but developing this further is well beyond the scope of this course; see this book of Szamuely for further discussion.

Observe that if we multiply a complex number by its complex conjugate , we obtain a quantity which is invariant with respect to conjugation (i.e. ) and is therefore real. The map produced this way is known in field theory as the norm form of over ; it is clearly multiplicative in the sense that , and is only zero when is zero. It can be used to link multiplicative inversion with complex conjugation, in that we clearly have

for any non-zero complex number . In coordinates, we have

(thus recovering, by the way, the inversion formula implicit in Remark 7). In coordinates, the multiplicativity takes the form of Lagrange’s identity

— 2. The geometry of the complex numbers —

The norm form of the complex numbers has the feature of being positive definite: is always non-negative (and strictly positive when is non-zero). This is a feature that is somewhat special to the complex numbers; for instance, the quadratic extension of the rationals by has the norm form , which is indefinite. One can view this positive definiteness of the norm form as the one remaining vestige in of the order structure on the reals, which as remarked previously is no longer present directly in the complex numbers. (One can also view the positive definiteness of the norm form as a consequence of the topological connectedness of the punctured complex plane : the norm form is positive at , and cannot change sign anywhere in , so is forced to be positive on the rest of this connected region.)

One consequence of positive definiteness is that the bilinear form

becomes a positive definite inner product on (viewed as a vector space over ). In particular, this turns the complex numbers into an inner product space over the reals. From the usual theory of inner product spaces, we can then construct a norm

(thus, the norm is the square root of the norm form) which obeys the triangle inequality

(which implies the usual permutations of this inequality, such as ), and from the multiplicativity of the norm form we also have

(and hence also if is non-zero) and from the involutive nature of complex conjugation we have

The norm clearly extends the absolute value operation on the real numbers, and so we also refer to the norm of a complex number as its absolute value or magnitude. In coordinates, we have

thus for instance , and from (6) we also immediately have the useful inequalities

As with any other normed vector space, the norm defines a metric on the complex numbers via the definition

Note that using the Argand plane representation of as that this metric coincides with the usual Euclidean metric on . This metric in turn defines a topology on (generated in the usual manner by the open disks ), which in turn generates all the usual topological notions such as the concept of an open set, closed set, compact set, connected set, and boundary of a set; the notion of a limit of a sequence ; the notion of a continuous map, and so forth. For instance, a sequence of complex numbers converges to a limit if as , and a map is continuous if one has whenever , or equivalently if the inverse image of any open set is open. Again, using the Argand plane representation, these notions coincide with their counterparts on the Euclidean plane .

As usual, if a sequence of complex numbers converges to a limit , we write . From the triangle inequality (3) and the multiplicativity property (4) we see that the addition operation , subtraction operation , and multiplication operation , thus we have the familiar limit laws

and

whenever the limits on the right-hand side exist. Similarly, from (5) we see that complex conjugation is an isometry of the complex numbers, thus

when the limit on the right-hand side exists. As a consequence, the norm form and the absolute value are also continuous, thus

whenever the limit on the right-hand side exists. Using the formula (2) for the reciprocal of a complex number, we also see that division is a continuous operation as long as the denominator is non-zero, thus

as long as the limits on the right-hand side exist, and the limit in the denominator is non-zero.

From (7) we see that

in particular

and

whenever the limit on the right-hand side exists. One consequence of this is that is complete: every sequence of complex numbers that is a Cauchy sequence (thus as ) converges to a unique complex limit . (As such, one can view the complex numbers as a (very small) example of a Hilbert space.)

As with the reals, we have the fundamental fact that any formal series of complex numbers which is absolutely convergent, in the sense that the non-negative series is finite, is necessarily convergent to some complex number , in the sense that the partial sums converge to as . This is because the triangle inequality ensures that the partial sums are a Cauchy sequence. As usual we write to denote the assertion that is the limit of the partial sums . We will occasionally have need to deal with series that are only conditionally convergent rather than absolutely convergent, but in most of our applications the only series we will actually evaluate are the absolutely convergent ones. Many of the limit laws imply analogues for series, thus for instance

whenever the series on the right-hand side is absolutely convergent (or even just convergent). We will not write down an exhaustive list of such series laws here.

An important role in complex analysis is played by the unit circle

In coordinates, this is the set of points for which , and so this indeed has the geometric structure of a unit circle. Elements of the unit circle will be referred to in these notes as phases. Every non-zero complex number has a unique polar decomposition as where is a positive real and lies on the unit circle . Indeed, it is easy to see that this decomposition is given by and , and that this is the only polar decopmosition of . We refer to the polar components and of a non-zero complex number as the magnitude and phase of respectively.

From (4) we see that the unit circle is a multiplicative group; it contains the multiplicative identity , and if lie in , then so do and . From (2) we see that reciprocation and complex conjugation agree on the unit circle, thus

for . It is worth emphasising that this useful identity does not hold as soon as one leaves the unit circle, in which case one must use the more general formula (2) instead! If are non-zero complex numbers with polar decompositions and respectively, then clearly the polar decompositions of and are given by and respectively. Thus polar coordinates are very convenient for performing complex multiplication, although they turn out to be atrocious for performing complex addition. (This can be contrasted with the usual Cartesian coordinates , which are very convenient for performing complex addition and mildly inconvenient for performing complex multiplication.) In the language of group theory, the polar decomposition splits the multiplicative complex group as the direct product of the positive reals and the unit circle : .

If is an element of the unit circle , then from (4) we see that the operation of multiplication by is an isometry of , in the sense that

for all complex numbers . This isometry also preserves the origin . As such, it is geometrically obvious (see Exercise 11 below) that the map must either be a rotation around the origin, or a reflection around a line. The former operation is orientation preserving, and the latter is orientation reversing. Since the map is clearly orientation preserving when , and the unit circle is connected, a continuity argument shows that must be orientation preserving for all , and so must be a rotation around the origin by some angle. Of course, by trigonometry, we may write

for some real number . The rotation clearly maps the number to the number , and so the rotation must be a counter-clockwise rotation by (adopting the usual convention of placing to the right of the origin and above it). In particular, when applying this rotation to another point on the unit circle, this point must get rotated to . We have thus given a geometric proof of the multiplication formula

taking real and imaginary parts, we recover the familiar trigonometric addition formulae

We can also iterate the multiplication formula to give de Moivre’s formula

for any natural number (or indeed for any integer ), which can in turn be used to recover familiar identities such as the double angle formulae

or triple angle formulae

after expanding out de Moivre’s formula for or and taking real and imaginary parts.

Exercise 11 (i) Let be an isometry of the Euclidean plane that fixes the origin . Show that is either a rotation around the origin by some angle , or the reflection around some line through the origin. (Hint: try to compose with rotations or reflections to achieve some normalisation of , e.g. that fixes . Then consider what must do to other points in the plane, such as . Alternatively, one can use various formulae relating distances to angle, such as the sine rule or cosine rule, or the formula for the inner product.) For this question, you may use any result you already know from Euclidean geometry or trigonometry.

be an isometry of the Euclidean plane that fixes the origin . Show that is either a rotation around the origin by some angle , or the reflection around some line through the origin. (Hint: try to compose with rotations or reflections to achieve some normalisation of , e.g. that fixes . Then consider what must do to other points in the plane, such as . Alternatively, one can use various formulae relating distances to angle, such as the sine rule or cosine rule, or the formula for the inner product.) For this question, you may use any result you already know from Euclidean geometry or trigonometry. (ii) Show that all isometries of the complex numbers take the form or for some complex number and phase .

Every non-zero complex number can now be written in polar form as

with and ; we refer to as an argument of , and can be interpreted as an angle of counterclockwise rotation needed to rotate the positive real axis to a position that contains . The argument is not quite unique, due to the periodicity of sine and cosine: if is an argument of , then so is for any integer , and conversely these are all the possible arguments that can have. The set of all such arguments will be denoted ; it is a coset of the discrete group , and can thus be viewed as an element of the -torus .

The operation of multiplying a complex number by a given non-zero complex number now has a very appealing geometric interpretation when expressing in polar coordinates (9): this operation is the composition of the operation of dilation by around the origin, and counterclockwise rotation by around the origin. For instance, multiplication by performs a counter-clockwise rotation by around the origin, while multiplication by performs instead a clockwise rotation by . As complex multiplication is commutative and associative, it does not matter in which order one performs the dilation and rotation operations. Similarly, using Cartesian coordinates, we see that the operation of adding a complex number by a given complex number is simply a spatial translation by a displacement of . The multiplication operation need not be isometric (due to the presence of the dilation ), but observe that both the addition and multiplication operations are conformal (angle-preserving) and also orientation-preserving (a counterclockwise loop will transform to another counterclockwise loop, and similarly for clockwise loops). As we shall see later, these conformal and orientation-preserving properties of the addition and multiplication maps will extend to the larger class of complex differentiable maps (at least outside of critical points of the map), and are an important aspect of the geometry of such maps.

Remark 12 One can also interpret the operations of complex arithmetic geometrically on the Argand plane as follows. As the addition law on coincides with the vector addition law on , addition and subtraction of complex numbers is given by the usual parallelogram rule for vector addition; thus, to add a complex number to another , we can translate the complex plane until the origin gets mapped to , and then gets mapped to ; conversely, subtraction by corresponds to translating back to . Similarly, to multiply a complex number with another , we can dilate and rotate the complex plane around the origin until gets mapped to , and then will be mapped to ; conversely, division by corresponds to dilating and rotating back to .

When performing computations, it is convenient to restrict the argument of a non-zero complex number to lie in a fundamental domain of the -torus , such as the half-open interval or , in order to recover a unique parameterisation (at the cost of creating a branch cut at one point of the unit circle). Traditionally, the fundamental domain that is most often used is the half-open interval . The unique argument of that lies in this interval is called the standard argument of and is denoted , and is called the standard branch of the argument function. Thus for instance , , , and . Observe that the standard branch of the argument has a discontinuity on the negative real axis , which is the branch cut of this branch. Changing the fundamental domain used to define a branch of the argument can move the branch cut around, but cannot eliminate it completely, due to non-trivial monodromy (if one continuously loops once counterclockwise around the origin, and varies the argument continuously as one does so, the argument will increment by , and so no branch of the argument function can be continuous at every point on the loop).

The multiplication formula (8) resembles the multiplication formula

for the real exponential function . The two formulae can be unified through the famous Euler formula involving the complex exponential . There are many ways to define the complex exponential. Perhaps the most natural is through the ordinary differential equation with boundary condition . However, as we have not yet set up a theory of complex differentiation, we will proceed (at least temporarily) through the device of Taylor series. Recalling that the real exponential function has the Taylor expansion

which is absolutely convergent for any real , one is led to define the complex exponential function by the analogous expansion

noting from (4) that the absolute convergence of the real exponential for any implies the absolute convergence of the complex exponential for any . We also frequently write for . The multiplication formula (10) for the real exponential extends to the complex exponential:

Exercise 13 Use the binomial theorem and Fubini’s theorem for (complex) doubly infinite series to conclude that for any complex numbers .

If one compares the Taylor series for with the familiar Taylor expansions

and

for the (real) sine and cosine functions, one obtains Euler formula

for any real number ; in particular we have the famous identities

and

We now see that the multiplication formula (8) can be written as a special form

of (12); similarly, de Moivre’s formula takes the simple and intuitive form

From (12) and (13) we also see that the exponential function basically transforms Cartesian coordinates to polar coordinates:

Later on in the course we will study (the various branches of) the logarithm function that inverts the complex exponential, thus converting polar coordinates back to Cartesian ones.

From (13) and (1), together with the easily verified identity

we see that we can recover the trigonometric functions from the complex exponential by the formulae

(Indeed, if one wished, one could take these identities as the definition of the sine and cosine functions, giving a purely analytic way to construct these trigonometric functions.) From these identities one can derive all the usual trigonometric identities from the basic properties of the exponential (and in particular (12)). For instance, using a little bit of high school algebra we can prove the familiar identity

from (16):

Thus, in principle at least, one no longer has a need to memorize all the different trigonometric identities out there, since they can now all be unified as consequences of just a handful of basic identities for the complex exponential, such as (12), (14), and (15).

In view of (16), it is now natural to introduce the complex sine and cosine functions and by the formula

These complex trigonometric functions no longer have a direct trigonometric interpretation (as one cannot easily develop a theory of complex angles), but they still inherit almost all of the algebraic properties of their real-variable counterparts. For instance, one can repeat the above high school algebra computations verbatim to conclude that

for all . (We caution however that this does not imply that and are bounded in magnitude by – note carefully the lack of absolute value signs outside of and in the above formula! See also Exercise 16 below.) Similarly for all of the other trigonometric identities. (Later on in this series of lecture notes, we will develop the concept of analytic continuation, which can explain why so many real-variable algebraic identities naturally extend to their complex counterparts.) From (11) we see that the complex sine and cosine functions have the same Taylor series expansion as their real-variable counterparts, namely

and

The formulae (17) for the complex sine and cosine functions greatly resemble those of the hyperbolic trigonometric functions , defined by the formulae

Indeed, if we extend these functions to the complex domain by defining to be the functions

then on comparison with (17) we obtain the complex identities

or equivalently

for all complex . Thus we see that once we adopt the perspective of working over the complex numbers, the hyperbolic trigonometric functions are “rotations by 90 degrees” of the ordinary trigonometric functions; this is a simple example of what physicists call a Wick rotation. In particular, we see from these identities that any trigonometric identity will have a hyperbolic counterpart, though due to the presence of various factors of , the signs may change as one passes from trigonometric to hyperbolic functions or vice versa (a fact quantified by Osborne’s rule). For instance, by substituting (19) or (20) into (18) (and replacing by or as appropriate), we end up with the analogous identity

for the hyperbolic trigonometric functions. Similarly for all other trigonometric identities. Thus we see that the complex exponential single-handedly unites the trigonometry, hyperbolic trigonometry, and the real exponential function into a single coherent theory!

Exercise 14 (i) If is a positive integer, show that the only complex number solutions to the equation are given by the complex numbers for ; these numbers are thus known as the roots of unity. Conclude the identity for any complex number .

is a positive integer, show that the only complex number solutions to the equation are given by the complex numbers for ; these numbers are thus known as the roots of unity. Conclude the identity for any complex number . (ii) Show that the only compact subgroups of the multiplicative complex numbers are the unit circle and the roots of unity for . (Hint: there are two cases, depending on whether is a limit point of or not.)

of the multiplicative complex numbers are the unit circle and the roots of unity (iii) Give an example of a non-compact subgroup of .

. (iv) (Warning: this one is tricky.) Show that the only connected closed subgroups of are the whole group , the trivial group , and the one-parameter groups of the form for some non-zero complex number .

The next exercise gives a special case of the fundamental theorem of algebra, when considering the roots of polynomials of the specific form .

Exercise 15 Show that if is a non-zero complex number and is a positive integer, then there are exactly distinct solutions to the equation , and any two such solutions differ (multiplicatively) by an root of unity. In particular, a non-zero complex number has two square roots, each of which is the negative of the other. What happens when ?

Exercise 16 Let be a sequence of complex numbers. Show that is bounded if and only if the imaginary part of is bounded, and similarly with replaced by .

Exercise 17 (This question was drawn from a previous version of this course taught by Rowan Killip.) Let be distinct complex numbers, and let be a positive real that is not equal to . (i) Show that the set defines a circle in the complex plane. (Ideally, you should be able to do this without breaking everything up into real and imaginary parts.)

defines a circle in the complex plane. (Ideally, you should be able to do this without breaking everything up into real and imaginary parts.) (ii) Conversely, show that every circle in the complex plane arises in such a fashion (for suitable choices of , of course).

, of course). (iii) What happens if ?

? (iv) Let be a circle that does not pass through the origin. Show that the image of under the inversion map is a circle. What happens if is a line? What happens if the passes through the origin (and one then deletes the origin from before applying the inversion map)?

Exercise 18 If is a complex number, show that .