My Path to Complex Analysis

In some ways my first ‘proper’ course in mathematics, 1340 Funktionentheorie has been a pleasure to work through. As my first two oral exams at the University of Hagen are approaching, I wanted to wrap up the content or rather the gist of the course in a short post that should serve the following purposes.

  • Be a reminder to myself of what things I have learnt in the course.
  • Be an exercise in putting math on a webpage. (Which may be comfortably done using MathJax.)
  • Be an exercise in using the English language equivalents of some German mathematical terms.

To begin with the last point in above list: Funktionentheorie is the German name of a discipline called complex analysis, a venerable and, by provenance, strongly German and due to Cauchy equally French discipline, with different approaches to its main subject of study tied to its founding fathers. The Cauchy-Riemannian path, along which the course at hand first moved before taking a junction into power series, builds from complex differentiability and, most importantly, Cauchy’s integral theorem and formula. Weierstraß's proofs and concept formations on the other hand are based on said power series.

Before anything else, it would probably be useful to clarify the subject around which complex analysis revolves. Complex analysis is most interested in studying holomorphic functions. A holomorphic function is a complex-valued function that is complex differentiable.

Let $U$ be an open set in the complex plane. A function $f:U\rightarrow\mathbb{C}$ is holomorphic at $z_0 \in U$ if there exists a neighbourhood of $z_0$ where $f$ is complex differentiable. If $f$ is complex differentiable at every point in $U$, $f$ is said to be holomorphic on $U$.

$f$ is complex differentiable at $z_0$ if there exists a function $f_1:U\rightarrow\mathbb{C}$ which is continuous at $z_0$ and with which: $$\quad \forall z \in U\ f(z)=f(z_0) + f_1(z)\cdot (z-z_0)$$ $f_1$ then is unique and is called the (complex) derivative of $f$ at $z_0$.

When first studying complex analysis, it may seem as though much of complex analysis consisted of lifting concepts from real analysis into the complex plane. And, to be sure, that is the case for things such as continuity and the rules of differentiation. One salient feature of the complex differentiable function however that sets it apart from its real counterpart is that, once it is known that the complex derivative exists at a point $z_0$, we know that the function is infinitely many times differentiable there.

Complex analysis also departs fundamentally from real analysis where, due to Cauchy’s integral formula, a holomorphic function's value at any point in an open disk may be calculated by integrating over the disk's bounding circle. This stunning result is said to compress much of complex analysis, since many other results can be derived from it. Hence it is regarded as somewhat of a centerpiece of the discipline, and the pattern of making a statement about a disk by integrating over its bounding circle (or, in the annulus version, about an annulus by integrating over its bounding circles) is found throughout. Cauchy’s integral formula for disks:

Let $f:U\rightarrow\mathbb{C}$ be a holomorphic function on the open set $U \subset \mathbb{C}$. Let $\Delta$ be a disk for which $\bar{\Delta} \subset U$. (I.e. $U$ contains the disk as well as its bounding circle.) Then: $$\forall\ z \in \Delta \quad f(z) = \frac{1}{2\pi i}\int_{\delta\Delta} \frac{f(\zeta)}{\zeta-z}d\zeta$$

In fact, the aforementioned property of holomorphic functions—that they are infinitely differentiable where they are holomorphic—can be derived from Cauchy's integral formula by induction. Another central result that ensues, and a bridge of sorts to the Weierstraßian side of things, is the expandability of holomorphic functions in power series—a property called analyticity by some. The theorem has it that for a function $f:U\rightarrow\C$ for which the integral formula holds within an open disk $\Delta$ of center $a$, the power series

$$\sum_{n=0}^{\infty} c_n(z-a)^n \\\text{where}\quad c_n=\frac{1}{2\pi i} \int_{\delta\Delta}\frac{f(\zeta)}{(\zeta-a)^{n+1}}d\zeta$$

converges (normally) in $\Delta$ to $f$, and conversely, that for any power series $\sum_{n=0}^{\infty} b_n(z-a)^n$ that converges to $f$ in a neighbourhood of $a$, $\forall n \in \N \ \ b_n=c_n$ (i.e. the uniqueness around $a$ of the power series representing $f$).

One theorem that lends itself particularly well to proof by complex analysis is none less than the Fundamental Theorem of Algebra. In its complex version, the theorem states that any non-constant polynomial $p \in \C[z]$ has at least one root in $\C$. It may be proven very succinctly using another consequence of Cauchy's integral formula—Liouville's theorem:

Let $f: \C \rightarrow \C$ be an entire function, i.e. a function that is holomorphic on the entire complex plane. If $f$ is also bounded, then $f$ is constant on $\C$.

Now, suppose $p \in \C[z]$ has no roots. The function defined by $f := \frac{1}{p}$ then is an entire function, and bounded, because any non-constant complex polynomial is unbounded. Due to Liouville, $p$ and thereby also $f$ must be constant. (The other proof, which shall not be given here, uses the open mapping theorem, according to which $f(V)$ is an open subset of $\C$ for any open subset $V \subset U$ where $U$ is a domain in the complex plane, and any function $f:U\rightarrow \C$ holomorphic on that domain.)

No introduction to complex analysis would be complete without featuring the residue theorem, using which an integral along a closed curve $\gamma$ in a simply connected open subset of $\C$ may be calculated for a function $f$ holomorphic on the subset with the exception of a finite number of points $a_i \ (i=1..n)$. More specifically,

$$\int_\gamma f(\zeta) d\zeta = 2\pi i \sum_{\nu = 1}^n I(\gamma, a_\nu)res(f, a_\nu)$$

Where $I(\gamma, a_\nu)$ is the winding number of $\gamma$ around $a_\nu$—visually speaking, the number of times the curve travels counterclockwise around the point—and $res(f, a_\nu)$ is the residue of $f$ in $a_\nu$—a number equal to the coefficient at index $-1$ of the Laurent series representing $f$ around the point.

Much more can be said, and will be said, when things get serious and I take the oral exam in October. Until then, I'll try not to forget too much while preparing for the other exams, and looking forward to going on an autumn vacation in the beautiful towns of Ghent and Bruges with my wife—to celebrate, I hope.