natural boundary

It is not always possible to analytically continue afunction given in a certain region. It might turnout that, as one approaches the boundary of the region(or a portion of the boundary), the function always blowsup, so there is no way of extending it past that portionof the boundary to a larger region. When this happens,we say that our function has a natural boundary.More formally, we may make a definition as follows:

Definition 1

Let $\mathrm{D}$ be an open subset of the complex planeand let $f\mathrm{:}\mathrm{D}\mathrm{\to }\mathrm{C}$ be analytic.Then the natural boundary of $f$ is that subset$B$ of $\mathrm{\partial }\mathit{}\mathrm{D}$ such that, if $z\mathrm{\in }B$,then there exists no open neighborhood $\mathrm{N}$ of$z$ and no analytic function $g\mathrm{:}\mathrm{N}\mathrm{\to }\mathrm{C}$ such that $f\mathit{}\mathrm{(}w\mathrm{)}\mathrm{=}g\mathit{}\mathrm{(}w\mathrm{)}$ for all $w\mathrm{\in }\mathrm{D}\mathrm{\cup }\mathrm{N}$.

As an example of this phenomenon, consider the power series

$$\sum _{k=0}^{\mathrm{\infty }}{z}^{k!}.$$

By comparison with the geometric series, it is seen that thisseries converges absolutely when :

However, when we try to take the limit $|z|\to 1$, we findthat the series diverges. Namely, let $p/q$ be a rationalnumber and let $r$ be a positive real variable. Then, if weset $z=r\exp (2i\pi p/q)$, then, when $k\ge q$, wehave that $q$ divides $k!$, so $z^{k!}=r^{k!}$. However,

$$\underset{r\to 1}{lim}\sum _{k=q}^{\mathrm{\infty }}{r}^{k!}$$

diverges, so our powers series diverges when we try to takethe limit $z\to \exp (2i\pi p/q)$. Since numbers of theform $\exp (2i\pi p/q)$ are dense amongst complex numberswith norm $1$, it follows that the limit $z\to z_0$ divergeswhenver $|z_0|=1$. Hence, the unit circle $|z|=1$ formsa natural boundary for the function defined by our power series.

Natural boundaries are not so familiar to beginners becausethe functions which one encounters in the more elementarypart of the subject, such as algebraic functions, exponentialfunctions, and functions defined by linear differentialequations, do not have natural boundaries. To be sure, onecould technically call a singular point a natural boundary,but this is usually not done, the term “natural boundary”being reserved for cases where the set on which thefunction misbehaves consists of more than just isolated points,as in the example above.

However, when one gains some more experience and studies moreadvanced material, then natural boundaries arise ratherfrequently. For instance, theta functions, elliptic modularfunctions, and functions defined by non-linear differentialequations have natural boundaries. Natural boundaries alsoplay an important role in applications — for instance, instatistical mechanics, phase transitions (such as freezingand boiling) are associated with natural boundaries of thepartition function.