Exegesis. Why do holomorphic functions exist? [rmsf-1404][edit]

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

    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!):

    Theorem 1. Uniform limits of polynomial [rmsf-1405][edit]

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

      Suppose $f_1, f_2, \cdots$ is a sequence of holomorphic functions $U \to \CC$ with pointwise limit $f: U \to \CC$. Suppose also the following stronger property: for every closed ball $\overline D \subset U$, the value

      $$ \sup_{z \in \overline D} |f(z) - f_i(z)| $$

      tends to zero as $i$ tend to $\infty$ (uniform convergence on closed balls.) Then $f$ is holomorphic.

      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:

      Definition 2. Biholomorphism [rmsf-1406][edit]

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

        Let $U$ and $V$ be open subsets of $\CC$. A biholomorphism between $U$ and $V$ is a holomorphic map $f: U \to C$ which is bijective and has holomorphic inverse.

        (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:

        Theorem 3. Riemann’s mapping theorem [rmsf-1407][edit]

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

          Let $U \in \CC$ be an open subset which is simply-connected and not equal to $\CC$. Then there exists a biholomorphism between $U$ and the unit disk $D = D(0,1)$.

          We will explain the term “simply-connected” later.

          There is something of a physical “proof” of this theorem, which also give some intuition for the notion of a biholomorphism. The idea is this: we can decorate the unit disk by imagining the radial lines joining the center $0$ to the boundary, and also imagining all the circles of radius $< 1$ centered at $0$. We want to find the corresponding decorations of $U$; then the biholomorphism will be constructed to match these up.

          For this, we start by fixing a point $p \in U$ which will correspond to $0 \in D$. Then we place an electric charge at $p$, and consider the magnetic field it generates. But we do this subject to the boundary condition that the potential should be zero on the boundary of $U$. Then we can decorate $U$ by the field lines on the one hand, and by the equipotential lines on the other. These will correspond to the radial lines and the circles in $D$.