This picture, drawn by Elias Wegert, uses colors to show the phase $f/|f|$ of the complex function

$$f(z) = \prod_{k=1}^5 \exp\left(\frac{z+\omega^k}{z-\omega^k}\right)$$

where $\omega$ is a nontrivial fifth root of unity. This is an ‘atomic singular inner function’. To understand what that means, it helps to start with some complex analysis.

The **Hardy space** $H^\infty$ is the space of bounded analytic functions in the complex unit disk $D$. Any function $f$ in the Hardy space $H^\infty$ can be written as a product of an ‘inner function’ $g$ and an ‘outer function’ $h$:

$$ f = gh $$

The **inner function** $g$ has modulus $1$ almost everywhere on $S^1$. The **outer function** $h$ has no zeros in $D$ and is completely determined by the boundary values of $|f|$ on the unit circle $S^1$.

The inner function can be further split into two factors, $g=bs$, where $b$ is a ‘Blaschke product’ and $s$ is a so-called ‘singular’ innner function. The **Blaschke product** is completely determined by the zeros of $f$. The **singular inner function** is an inner function with no zeros in $D$ and modulus 1 almost everywhere on $S^1$.

So, the singular inner function coming from $f$ is a kind of ‘remainder’ which can neither be determined from the zeros of $f$ in the unit disk nor from $|f|$ on the unit circle!

The prototypical example of a singular inner function is

$$ \exp\left(\frac{z+t}{z-t}\right)$$

for any point $t$ on the unit circle. This has an essential singularity at $t$. Starting from this, one gets the most general singular inner function as follows:

$$ s(z)=c\,\exp \int_{S^1}\frac{z+t}{z-t}\,d\mu(t)$$

where $\mu$ is a normalized singular measure on $S^1$ and $|c|=1$. Remember, a measure on the circle is **singular** if it is supported on a set of Lebesgue measure zero. The easiest singular measures to understand are the **atomic** ones, which are (possibly infinite) linear combinations of Dirac delta measures.

In the example at hand we have taken a linear combination of Dirac deltas supported at the fifth roots of unity. So, the picture uses colors to show the phase $f/|f|$ of the function

$$f(z) = \prod_{k=1}^5 \exp\left(\frac{z+\omega^k}{z-\omega^k}\right)$$

where $\omega=\exp(2\pi i/5)$. This function has essential singularities at the fifth roots of unity, no zeros in $D$, and modulus 1 on $S^1$.

You can learn a lot about a function by looking at a color picture of its phase. For more on this, with lots of wonderful pictures, explore Wegert’s website:

• Elias Wegert, Visualizing complex functions using phase portraits.

and read this paper:

• Elias Wegert, Phase plots of complex functions: a journey in illustration.

The factorization of functions in $H^\infty$ into inner and outer functions is due to Beurling, and it has various generalizations:

• Beurling-Lax theorem, *Encyclopedia of Mathematics*, Springer.

*Visual Insight* is a place to share striking images that help explain advanced topics in mathematics. I’m always looking for truly beautiful images, so if you know about one, please drop a comment here and let me know!