The idea of complex numbers, as points in a number *plane* with rules for adding and multiplying them (etc.), was first formulated clearly early in the 19th century.

Complex numbers had been used “on the side” in maths for hundreds of years before that. For example, they turn out to be necessary in the calculation of roots of cubic equations *even if* all the roots of the equation are real.

But formulating the idea clearly set off a whole new wave of investigation. For example, how does calculus work if we have functions

where z and w range over the whole complex number plane, rather than just along the real number line?

It’s difficult to tell, because if we try to draw a graph, then it has to be in *four dimensions* (two for the real and imaginary components of z, two for the real and imaginary components of w).

It seems it should still be true that if for example

then . And it is.

Calculus using complex numbers (usually called complex analysis) is of course more difficult than calculus using real numbers. Also, it turns out, it is necessary for answering even some questions which seem to be only about calculus with real numbers.

When we do calculus with real numbers, at school we pretty much take it for granted that any function we deal with will have a derivative. The question is how to find it, not whether it exists.

In fact, plenty of functions do not have derivatives. The modulus function

has no derivative (no ) when x=0.

at 0 would have to be the limit (the number approached more and more closely) by the average speed of change in y as x ranges over smaller and smaller neighbourhoods of 0.

But in every neighbourhood of 0, however tiny, there is an average speed of change in y relative to x of +1 as x ranges bigger than 0, and an average speed of change in y relative to x of −1 as x ranges bigger than 0. Choose the neighbourhood (however small) to be mostly x>0, and the average speed will be near +1. Choose the neighbourhood (however small) to be mostly x<0, and the average speed will be −1.

There is no one set value that the average speed approaches, and so no when x=0.

The Weierstrass function, pictured below, is continuous everywhere, but has no *anywhere*.

With complex functions of complex variables, oddly, the separation between differentiable functions and non-differentiable functions is more drastic.

If a function is differentiable everywhere in an open disk around a complex number q in ℂ, then it has *all* derivatives at q for all n, and it is equal to its Taylor series:

The function is then called analytic, or holomorphic (the two terms mean the same) at q. Just as a lot of calculus with real variables is about differentiable functions, a lot of complex analysis is about analytic or holomorphic functions.

It is not easy for a complex function to be analytic or holomorphic. For example the conjugate function

is not differentiable. (The average speed of increase of w relative to z around z=a is +1 if you move parallel to the real axis, but -1 if you move parallel to the imaginary axis).

If f(z) is analytic (holomorphic) at z=0, then the Maclaurin series (i.e. Taylor series based at z=0) for

“works” from z=0 out to the nearest “singularity”, i.e. point when does *not* exist.

A bit more work shows that that the region in which the Maclaurin series “works” is a disk around z=0. (Makes sense, if you think about it: the Maclaurin series not working probably means the z^{n} terms becoming too big for big n).

It follows that the range of convergence for Maclaurin series considered only for *real* variables is always symmetrical around the origin (an interval [-b, b] or, for super well-behaved functions, the whole real line).

The Maclaurin series stops working for both real and complex variables once we get far enough out to hit a singularity. So for example the Maclaurin series for stops working outside the range −1 < x < 1. The function doesn't blow up or misbehave anywhere around x=−1 or x=1. But it does blow up at x=±i, and so its disk of convergence has radius 1.

As Nets Katz says in his Caltech lecture notes: “The complex numbers overshadow the elementary calculus of one variable, silently pulling its strings”. (More here).

The *only* analytic and invertible functions mapping the unit disc in the z-plane to the unit disc in the w-plane are “Möbius transformations”, transformations of the form

for some given a, b, c, d.

Möbius transformations map circles to circles (so long as we interpret a line as a special circle with infinite radius and centre at infinity). We can see what’s happening with Mobius transformations by drawing pictures side by side of circles (or lines) in the w-plane and their images in the z-plane (one example below: for more click here).

For all other complex functions of complex variables, this method of drawing z-shapes and seeing what w-shape they map into does not work well.

The w-shapes may be very complicated. They may well involve overlaps: the function w = z^{2} transforms the circle |z| = 1 into the circle |w| = 2 *travelled twice*, but how do you show the twice-travelling?

There is another problem, and this is where *Riemann surfaces* begin to kick in.

Consider

If z is real, we get round the problem of every z (≠ 0) having *two* square roots by having the convention that √z means the positive square roots. Then we can draw the graph and do calculus on it.

For complex z, though, it’s more complicated. Write z in the form

, choosing θ to be between −π and π, and so defining w as the square root which has argument between −½π and ½π, i.e. the square root with positive real part.

Follow z from the value z=4 (say) anticlockwise as θ increases round to 4e^{iπ}, i.e. − 4: w moves from 2 through 2e^{iπ/4} to 2e^{iπ/2}, i.e. 2i.

Follow z from the value z=4 *clockwise*, and w moves from 2 through 2e^{−iπ/4} to 2e^{−iπ/2}, i.e. −2i.

If we make a consistent choice for which square root w to choose, and then follow it through round a circle, it becomes inconsistent!

This is the best we can do to picture in three dimensions what’s happening. It plots only the real part of w.

But the w-surface does not really cross itself along the half-line where z is real and negative. It just seems to do that because we’re squashing four dimensions down to three.

These pictures (below) could show spiralling (one-dimensional) curves – which don’t cross themselves – in three dimensions. Only when they’re flattened to two dimensions do the curves seem to cross themselves. It’s the same sort of thing when we try to “flatten” the surface formed by all the values of w and z where w^{2}=z – a two-dimensional surface sitting in four-dimensional space – down to a two-dimensional surface which we can see in three dimensions.

This video gives some idea of what’s happening in four dimensions.

The w-surface is a two-dimensional surface, but a two-dimensional surface sitting in four-dimensional space. To get an idea of how this works, think about the (one-dimensional) w-curve, sitting in three dimensions, which is the w-image of the (one-dimensional) curve |z|=1. It is a one-dimensional w-curve which “flies over” itself, as of course it easily can in three dimensions.

To get a two-dimensional picture of this, cut the w-curve at the flyover point and glue the bits back again with a flip to avoid the flyover. We can then get a “flat” version of the w-curve.

which, with a bit of stretching or nudging, is equivalent to a circle. By analogous cutting-and-gluing, the whole w-surface can be flattened to a copy of the complex plane.

This cutting, gluing, stretching etc. obviously changes the detail of the w-surface. But some important properties of the general shape of the surface remain unchanged, and much of the study of Riemann surfaces is to do with those proporties of their general shape.

The pictures below (from Matt Kerr) show what happens with .

This, from Constantin Teleman’s notes, shows how cutting, gluing, and stretching can make a two-dimensional shape *which can be modelled in three dimensions* from the Riemann surface for:

Surfaces which have the same sort of properties as these Riemann surfaces derived from fairly-well-behaved functions w=f(z) are called “abstract” Riemann surfaces; so the basic “moral” definition of a Riemann surface (as Teleman puts it) is that it is a two-dimensional manifold which is similar in structure to ordinary (Euclidean) two-dimensional space in a neighbourhood around every point and “with a ‘good’ notion of complex-analytic functions”.

Those functions are thought of as mapping from the surface to a complex plane, so in the case of the Riemann surface for w=√z we reconceptualise the function as mapping from points on the 2-dimensional surface (sitting in 4-dimensional space) defined by w^{2}=z to points in the w-plane.

An important example of an abstract Riemann surface is the “Riemann sphere”, which means: a sphere conceptualised as a model of the complex plane plus a single point at infinity.

It’s a higher-dimensional equivalent of the idea of a circular (rather than linear) “number line”.

Suppose instead of listing the numbers along a line, we list them round a circle, starting from 0 at the bottom and marking the positions round the circle in the way shown, so that the numbers are closer and closer together as we get nearer the top, and in theory we can get numbers as high as we like marked on the circle.

This “circular number line” will then be not exactly a full circle, but a “punctured” circle, with one point missing, at the top.

The full circle will be a model, not just of the usual number line, but of the number line plus a single infinity. (Not separate +∞ and −∞, but a single infinity).

The Riemann sphere is the equivalent for complex numbers.