EC-4.1 Complex Numbers, and Functions of a Complex Variable

What this is and why it exists

Engineers use complex numbers long before they study them. A phasor is a complex number, and using one to handle phase is a perfectly good habit. But that use is bookkeeping, and it is not the subject.

The subject begins with a question that sounds harmless. What does it mean for a function of a complex variable to be differentiable?

On the real line a limit can be approached from two directions. In the plane it can be approached from infinitely many, and every one of them must give the same answer. That is a far stronger demand, and everything remarkable in complex analysis follows from it. Functions that meet it turn out to be extraordinarily well behaved. The residue theorem in the next topic is then a consequence rather than a rule to memorise.

The vocabulary

• Complex plane — the plane in which a complex number is a point. One axis carries the real part and the other the imaginary part.
• Modulus — the distance of a point from the origin.
• Argument — the angle a point makes with the positive real axis.
• Polar form — a complex number written as a modulus multiplied by a complex exponential of the argument.
• Analytic — differentiable at every point of some open region.
• Cauchy-Riemann conditions — two partial derivative equations that hold exactly when a function is differentiable.
• Singularity — a point where a function fails to be analytic.
• Principal value — the choice made to give a single answer where a function would otherwise have many.

The mental model

Start with geometry. A complex number is a point in a plane. Adding two of them is adding arrows. Multiplying two of them adds their angles and multiplies their lengths. That last fact is why the polar form is the one to reach for. Multiplication in rectangular form is four real multiplications and a sign to remember; in polar form it is one addition and one multiplication.

Euler's relation is the bridge: e raised to i theta equals cosine theta plus i sine theta. Read as geometry it says that raising e to an imaginary power gives a point on the unit circle at angle theta. That single identity is standing underneath every phasor, every Fourier coefficient and every transfer function you will meet.

Now differentiability. The derivative is defined by the same limit as on the real line. It is the change in the function divided by the change in the variable, as the change shrinks to nothing. On the real line the change can only come from the left or the right. In the plane it can come from any direction at all.

Requiring every direction to agree is the whole of the subject in one sentence.

Write the function in terms of its real and imaginary parts, each a function of the two coordinates. Approach along the real direction and you get one expression for the derivative. Approach along the imaginary direction and you get another. Setting them equal gives two equations relating the partial derivatives, and those are the Cauchy-Riemann conditions. Checking them is the practical way to decide whether a function is differentiable at a point.

A function analytic throughout a region is not merely differentiable once. It is infinitely differentiable, and it equals its own power series about any point of the region. Nothing like that is true on the real line, where a function can have a first derivative and no second. This is the payoff of the strong requirement.

The isolated points where a function stops being analytic are its singularities, and they turn out to carry all the useful information. The next topic is entirely about them.

One last piece, because it trips people. The complex exponential is periodic in the imaginary direction: adding two pi i to the exponent changes nothing. So the logarithm cannot have a single answer, and neither can a root. The convention is to pick a principal value and state that you have. Roots follow the same pattern, spacing themselves evenly around a circle, which is a picture worth drawing once.

What you should now be able to explain or do

• Move between rectangular and polar form and multiply two numbers the quick way.
• State Euler's relation and say what it means geometrically.
• Explain why complex differentiability is a stronger demand than real differentiability.
• Apply the Cauchy-Riemann conditions to test whether a given function is analytic.
• Say why the complex logarithm is many-valued and what a principal value is.

Check yourself