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.

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One Response to Fixed points of entire functions

  1. Pingback: The equation z=exp(z) via Newton’s method | Negro's notes

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