$\gdef\CC{\mathbb{C}}$ $\gdef\spaces#1{~ #1 ~}$

The set of complex numbers is

$$ \CC \spaces= \{ a + b \cdot i \mid a, b, \in \R \} $$

For a complex number $z$ and a positive real number $r$, the open disk of radius $r$ centered at $z$ is

$$ D(z,r) \spaces= \{ w \in \CC \mid |w-z| < r \} $$

and the analogous closed disk is

$$ \overline D(z,r) \spaces= \{ w \in \CC \mid |w-z| \le r \} $$

Its boundary is the circle

$$ \partial \overline D(z,r) \spaces= \{ w \in \CC \mid |w-z| = r \} $$

$\gdef\spaces#1{~ #1 ~}$ $\gdef\CC{\mathbb{C}}$

A subset $U \in \CC$ is called open if for every $z \in U$, there exists a positive real number $r$ such that $D(z,r) \subset U$.

It is equivalent to ask that $U$ be a union of open disks.

The reason we want to consider open subsets is that we want to differentiate. To differentiate, you need to know not just the values of your function at a point, but its value at all points sufficiently close to your point. An open subset is exactly a subset which, if it contains a point, contains all sufficiently close points as well.

$\gdef\CC{\mathbb{C}}$

Let $U \in \CC$ be an open subset, and $f: U \to \CC$ a function. We say that $f$ is holomorphic if it is complex differentiable at $z_0$ for every $z_0 \in U$, i.e. for every $z_0 \in U$ the limit (of complex numbers)

$$ \lim\limits_{z \to z_0} \frac{f(z) - f(z_0)}{z - z_0} $$

exists. (The limit value is denoted $f'(z_0)$, as usual.)

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There are two ways of thinking about this definition. One is that it is the complex analog of the usual notion of a differentiable function of one real variable. So insofar as the complex numbers, with its algebraic operations and its notion of absolute value, the notion of a holomorphic function is like the notion of a real differentiable function.

But this perspective hides the magic of holomorphic functions, which in many ways are very different from real differentiable functions. To bring out the magic, it’s helpful to think geometrically, viewing the complex numbers as the plane $\R^2$ and thus thinking of $f$ as a function of two real variables.

Then we can ask, what is it that distinguishes a holomorphic function from an arbitrary real differentiable function of two variables? The answer can be phrased as follows. Suppose give a real differentiable function $f: U \to \R^2$ and a point $p \in U$. The derivative of $f$ at $p$ is by definition a certain linear map

$$ \d f|_p : \R^2 \to \R^2 $$

where we think of the source (resp. target) as the vector emanating from $p$ (resp. $f(p)$). Then an immediate comparison of the definitions shows that $f$ is holomorphic if and only if $\d f|_p$ is given by multiplication by some complex number (which will be $f'(p)$).

Now, it is only certain linear maps which are of that form. Multiplication by a complex number is either the zero map, or it is the composition of a rotation and a scaling. A particularly vivid consequence is that when $f$ is holomorphic, $\d f|_p$ is either zero or it preserves oriented angles. Another consequence is a certain rigidity: the whole linear map $\d f|_p$ is determined by its value on any nonzero vector, since already that fixes the amount of rotation and scaling undergone by the whole map.

To bring this to life, recall the geometric interpretation of $\d f|_p$: if you leave the point $p$ travelling with instantaneous velocity $v$, then your image under $f$ will leave $f(p)$ travelling with instantaneous velocity $\d f|_p(v)$. Thus when $f$ if holomorphic, it will preserve (oriented) angles to first order, since $\d f|_p$ preserves angles. In fact this is almost an equivalent condition: a real-differentiable function with non-vanishing derivative is holomorphic if and only if it is conformal, i.e. preserves angles, or even just if and only if it preserves right angles. That’s because the only linear maps which do that are compositions of rotations and scaling, hence given by multiplication by a complex number. Furthermore, there is the rigidity: the first-order movement under $f$ in any direction determines its first-order movement in every direction. e.g. $\d f|_p(i)$ is the $90$-degree counterclockwise rotation of $\d f|_p(1)$. In coordinates, this is known as the Cauchy–Riemann equation.

Moreover, if you imagine fixing the length of $v$ but sweeping out all possible directions evenly, your image under $f$ will also have fixed length and sweep out all possible directions evenly. Thus, a holomorphic map is balanced (the technical term is harmonic).

This first balanced has a large scale manifestation, the mean value property.

$\gdef\spaces#1{~ #1 ~}$ $\gdef\CC{\mathbb{C}}$ $\gdef\d{\operatorname{d}}$

Let $f: U \to \CC$ be a holomorphic function on an open subset of $\CC$ of, and let $\overline D(z, r)$ be closed disc contained in $U$. Then the average value of $f$ on the boundary $\partial \overline D(z,r)$ is equal to the value of $f$ at the center, i.e.

$$ f(z) \spaces= \frac{1}{2\pi r}\int_{\partial \overline D(z,r)} f(w) \cdot |\d w| $$

Here we should be imagining $w$ tracing around the circle $\partial \overline D(z,r)$, and $\d w$ as picking up the infinitesimal arc length. The $2\pi r$ corresponds to length of the circle, i.e. $\int_{\partial \overline D(z,r)} |\d w|$.

$\gdef\d{\operatorname{d}}$ $\gdef\spaces#1{~ #1 ~}$

There is a way to make this formula nicer. Namely, if we send $w$ around at uniform speed counterclockwise, then we see the following formula

$$ \frac{\d w}{w - z} \spaces= \frac{i \cdot |\d w|}{r} $$

The $i$ comes from the fact that $\d w$ is perpendicular to $w-z$: the tangent to circle is perpendicular to line emanating from the center of circle. Substituting this in to the integral, we find

$$ f(z) \spaces= \frac{1}{2\pi i} \int_{\partial \overline D(z,r)} \frac{f(w)}{w - z} \d w $$

Here it should be implicit that we’re supposed to move around $\partial \overline D(z,r)$ in a counterclockwise direction.

This formula is nicer: one reason is that in fact it is valid with $z$ replaced by any point inside the disk $D(z,r)$. That’s the Cauchy integral formula.

$\gdef\CC{\mathbb{C}}$ $\gdef\spaces#1{~ #1 ~}$ $\gdef\d{\operatorname{d}}$

Let $f: U \to \CC$ be holomorphic, with $U$ open. If $\overline D$ is any closed disk inside $U$ and $z$ is any point in the interior $D$, then

$$ f(z) \spaces= \frac{1}{2\pi i} \int_{\partial D} \frac{f(w)}{w-z} \d w $$

if we go around the boundary once counterclockwise in performing the integration.

This formula is something of a key to the notion of holomorphic function. For example, it explains why holomorphic function are so regular: the dependence on $z$ in the right-hand side of the equation is easy to control; e.g. we can easily differentiate, or even expand into a power series. Here are some corollaries:

Another application of the Cauchy integral formula is the removable singularities theorem:

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Now we have reviewed the basis of holomorphic functions. But there’s one important question we should still discuss: where do holomorphic functions come from? Why do they exist?

The answer is that they come from a variety of sources. We’ll talk about two of the more important ones in the next lecture: they come from solution of polynomial (or more generally holomorphic) equations, and also from solutions of holomorphic differential equations.

But for now we’ll give a different kind of answer, First of all, the identity function $z: \CC \to \CC$ is holomorphic, as is any constant function. Second, the class of holomorphic functions is closed under addition and multiplication. Thus, every polynomial is holomorphic. (The proof of these claims is formally just like the real variable case.) Then, we can access more holomorphic functions by talking limits of polynomials, thanks to the following theorem (also a consequence of Cauchy formula!):

So other examples of holomorphic functions come as uniform limits of polynomial functions. In fact, essentially all examples are of this form, since a power series is uniform limit of its Taylor truncation.

Yet, there are many other ways to construct holomorphic functions than via the above path. One is particularly vivid; it involves going back to the geometric understanding of holomorphic functions, and making it precise will be one of the goals of this course. Before stating a version of it, however, we should define the following important term:

(This last condition in fact follows from the others, but we’ll talk about that later.)

Now we can state Riemann’s mapping theorem, which produces many interesting biholomorphisms: