Fixed points of entire functions

This story has a long personal history. In my first attempted complex analysis exam I was asked if the entire function $e^z$ has any fixed points. I had no idea how to approach this question and I failed the exam. This first failure was soon overcome but that particular question was —much to my advantage— never addressed during my study again. Much later I found several ways to solve this question.

The image of $e^z-z$ is invariant under a shift $z \mapsto z + 2 \pi \textrm{i}$. The “Little Picard” theorem then asserts that this function maps onto $\mathbb{C}$ since it cannot omit just a single value and so it must have a root. Another way uses “Great Picard” and the fact that $z e^{-z}$ only has a single root so it must attain the value $1$ infinitely often. Both approaches seem nice enough but depend on non-trivial theorems (“Great Picard” more so than “Little Picard”) and there is no indication of the location of the fixed points. A more pedestrian approach shows that there must be infinitely many fixed points along the curve $|e^{2z}| = |z|^2$ but this is hardly in the spirit of complex analysis. This was the state of affairs until this week. Then I found a simple and much more satisfying answer.

The Banach fixed point theorem asserts that every contraction of a complete metric space has a single fixed point. I will use that theorem in the following setting.

Let $E$ be a non-empty closed convex subset of an open set $U$ and let $f{:U}\to E$ be holomorphic. If there exists a constant $k < 1$ such that $|f'| \leq k$ on $E$ then $f$ restricted to $E$ is a contraction and therefore $f$ has a single fixed point in $E$.

For $m \in \mathbb{Z}$ let $E_m$ denote the horizontal strip

$\displaystyle E_m = \{ z \mid |\textrm{Im}(z) - 2m \pi| \leq \pi \}$

and let $\log_m$ be the branch of the logarithm that maps the slit complex plane $\mathbb{C} \setminus (-\infty, 0]$ onto the interior of $E_m$. If $m \neq 0$ then

$\displaystyle |\log_m'| \leq \frac{1}{(2|m|-1)\pi} < 1$

on $E_m$ so the conditions of the theorem above apply and therefore $\log_m$ has a single fixed point $z_m \in E_m$ (which must lie in the interior.) Each $z_m$ is a fixed point of $e^z$. Unfortunately the same argument does not work for $E_0$ where $e^z$ has two more fixed points.

The fixed point method works equally well for a number of other functions. Here are two more examples:

Example 1. Let $m \in \mathbb{Z}$ and $m \neq 0$. The branch of $\arcsin$ that maps the (closed) upper half plane onto the half strip

$\displaystyle E_m = \{ z \mid \textrm{Im}(z) \geq 0 \textrm { and } |\textrm{Re}(z) - 2m\pi| \leq \frac{\pi}{2} \}$

is a contraction on $E_m$. So $\sin(z)$ has exactly one fixed point in each such strip (and therefore also exactly one in its complex conjugate).

Example 2. Let $m \in \mathbb{Z}$ and $m$ odd. The branch of $\arctan$ that maps the complement of the open unit disc into the strip

$\displaystyle E_m = \{ z \mid |\textrm{Re}(z) - \tfrac{1}{2}m \pi| \leq \frac{\pi}{4} \}$

is a contraction on $E_m$. So $\tan(z)$ has exactly one fixed point in each such strip.

These example for $e^z$, $\sin$ and $\tan$ work so well because their inverses have a nice derivative that is less than one except in a small bounded region. Put in another way these functions all satisfy a differential equation of the form $(f')^d = p(f)$ for some $d \in \{1,2\}$ and some polynomial $p$.