## Posts Tagged ‘**complex analysis**’

## Hadamard’s gap theorem

We mentioned briefly at the end of this post that Carlson proved that a function whose power series expansion has integer coefficients and radius of convergence 1 is either rational or has a natural boundary. A function has a natural boundary if every point on its circle of convergence is a singularity. In particular, it cannot be analytically continued beyond its disc of convergence.

An example of a function with natural boundary is . It is enough to show that the singularities of are dense on the unit circle. Let be positive integers and let for some . Then

.

However, is unbounded as tends to 1. Thus, is a singularity of for every positive . This gives a dense set of singular points on the unit circle, completing the proof of the claim.

This phenomemon is captured by a general theorem of Hadamard, which states that if is a real number and is a sequence of positive integers such that for all , , then has a natural boundary.

A much more powerful “gap theorem” due to Fabry, replaces the condition with . This result is capable, for instance, of establishing that has a natural boundary.

## Power series with integer coefficients

In combinatorial enumeration one often works with generating functions. The coefficients of the power series expansion of such a generating function are therefore integers, since they are meant to count objects of a certain size. A curious old result of Fatou states that a function whose power series expansion has integer coefficients and radius of convergence 1 is either rational or transcendental. This result also appears as Problem VIII.167 of Polya and Szego’s famous *Aufgaben *book. We shall attempt to prove this result below.

We begin with the following particular case of Parseval’s identity. Let be a function with radius of convergence . Then for we have

To see this, first let denote the complex conjugate of . Then we have

Since these series converge absolutely, we may multiply them to obtain

.

Writing , we have

.

We now integrate term by term to obtain

Observe that

equals when and otherwise. Thus,

,

which is the claimed identity.

Next, let be a power series with integer coefficients such that infinitely many are non-zero. We claim that if is convergent in the interior of the unit disc, then it is unbounded there.

Suppose to the contrary that is bounded in the interior of the unit disc. In particular, for , is bounded on the circle of radius . Thus, the integral is also bounded. However, we have seen that

and, since the are integers, the series on the right hand side tends to infinity as tends to 1. This contradiction proves the claim.

Now, let us suppose that is a power series with integer coefficients and radius of convergence 1. Suppose contrary to our desired conclusion that is algebraic but not rational. Then satisfies an equation of the form

,

where and the are polynomials with integer coefficients. Since , we can multiply through by to obtain

Setting , we can rewrite this as

Note that cannot be unbounded in the interior of the unit disc, since this term dominates the other terms of lower order in the left hand side, and if were to be unbounded, then the above equation could not hold. It follows then that is bounded, and hence is bounded. However, is a function whose power series has integer coefficients and radius of convergence 1. We have previously seen that such a function cannot be bounded in the interior of the unit disc. This contradiction establishes the desired result: i.e., a function whose power series expansion has integer coefficients and radius of convergence 1 is either rational or transcendental.

Polya conjectured a stronger result, namely that either the function is rational or admits the unit circle as a natural boundary (i.e., has no analytic continuation beyond the unit disc). This was eventually proved by Carlson.