Natural boundaries and lacunary series

Consider a power series

Show that the series converges to a holomorphic function on the open unit disk centered at the origin. Prove that the boundary of the disk is the natural boundary of the function.

For $\left|a_n\right|<1$,

Prove that the function

considered in the upper half-plane $\mathrm{Im}\left(z\right) > 0$, is regular there and has the real axis as an essential boundary of continuation.

Prove that the function $\displaystyle \sum_{n=1}^{\infty } \frac{1}{n^{2}}\cdot \frac{1}{z-e^{2\pi ni\sqrt{2}}}$, regular for $\left|z\right| < 1$, cannot be continued beyond the circle $\left|z\right| = 1$.

Prove that the same singularity (problem 5571, the absence of continuation beyond the boundary of the circle of convergence) is exhibited by the series:

if $\sum_{n=0}^{\infty } a_{n}$ diverges.

Let $a$, $b$, $c$ be the numbers represented by the three vertices of a triangle, and $x$, $y$, $z$ proper fractions. The series:

represents, outside the triangle, an analytic function of $\zeta$ that cannot be continued inside the triangle. Prove that, contrary to what is stated in Hermite's book Cours d’analyse, the series also converges inside the triangle on an everywhere-dense set of points.

Let the coefficients of the polynomial

be real. The analytic function defined by the power series

is

1. a rational function, if the $r$ numbers $a_0,a_1,\ldots ,a_{r-1}$ are all rational,

2. not a rational function, if the $r$ numbers $a_0,a_1,\ldots ,a_{r-1}$ are not all rational. (Cf. Problem 168 of Part II.) [Problem 85 of Part I.]

Denote by $Q_n$ the ``cross sum'' of the number $n$, i.e. the sum of its digits. E.g. we have $Q_{137}=1+3+7=11$. The power series

has the unit circle as its natural boundary.

In an infinite table of logarithms let the logarithms of all the positive integers $1,2,3,\ldots$ be written underneath each other in their natural order, in such a fashion that decimal places of the same order appear in the same vertical column. If we interpret the digits in any vertical column as the coefficients of a power series, then we obtain a power series that cannot be analytically continued.

In a different formulation we are required to prove the following theorem: Let $j$ be an integer and in the decimal representation of $\log 1,\log 2,\log 3,\ldots ,\log n,\ldots$ let the $j$th digit after the decimal point be denoted by $d_1,d_2,d_3,\ldots ,d_n,\ldots$ respectively. Then the analytic function defined by the power series

has as its natural boundary the circle $\left|z\right|=1$.

Here is another construction of such a function. Let $\left\{ n_{k}\right\}$ be a sequence of integers such that $n_{1} > 1$ and $n_{k+1} > 2 k n_{k}$. Define

Prove that the series converges if $\left|z\right| < 1$ and prove that there is a constant $c > 0$ such that $\left|h(z)\right| > c \cdot 5^{m}$ for all $z$ with $\left|z\right| = 1-\left(1 / n_{m}\right)$.

Hence $h$ has no finite radial limits. Prove also that $h$ must have infinitely many zeros in $U$. (Compare with Exercise 15, Chap. 12). In fact, prove that to every complex number $\alpha$ there correspond infinitely many $z \in U$ at which $h(z) = \alpha$.

Suppose $f(z) = \Sigma a_{n} z^{n}, a_{n} \geq 0$, and the radius of convergence of the series is 1. Prove that $f$ has a singularity at $z = 1$. Hint: Expand $f$ in powers of $z-\frac{1}{2}$. If 1 were a regular point of $f$, the new series would converge at some $x > 1$. What would this imply about the original series?

Suppose

Find the regions in which the two series converge. Show that this illustrates Theorem 16.5. Find the singular point of $f$ which is nearest to the origin.

Prove that the following statements are correct if $\left\{ n_{k}\right\}$ is a sequence of positive integers which tends to $\infty$ sufficiently rapidly. If

then $\left|f^{\prime }(z)\right| > n_{k} /(10 k)$ for all $z$ such that

Hence

for every $\theta$, although

exists (and is finite) for almost all $\theta$. Interpret this geometrically, in terms of the lengths of the images under $f$ of the radii in $U$.

Prove that the sum of the power series $\sum_{n = 0}^{\infty } z^{2^{n}}$ is a complete analytic function in the disk of convergence $B_{1}(0) = \left\{ z:\left|z\right| < 1\right\}$.

Prove that when the radius of convergence of the power series $\sum_{n = 0}^{+\infty } a_{n} z^{n}$ equals one and all $a_{n} \geqslant 0$, then such a series cannot be continued to the point $z = 1$ (Pringsheim's theorem).

Let $f(z) = \sum_{n = 0}^{\infty } c_{n} z^{n}, \varlimsup_{n \rightarrow \infty }\left|c_{n}\right|^{1 / n} = 1$, and let all coefficients $c_{n}$ be nonnegative. Prove that the point $z = 1$ is a singular point of the function $f(z)$.

Prove that the statement of Problem 43 remains valid also under the weaker assumptions:

Prove that every point of the circle $\left|z\right| = 1$ is a singular point for the functions