Analytic Continuation

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 $(f, D)$ and $(g, D)$ are function elements, $P$ is a polynomial in two variables, and $P(f, g) = 0$ in $D$. Suppose $f$ and $g$ can be analytically continued along a curve $\gamma$, to $f_{1}$ and $g_{1}$. Prove that $P\left(f_{1}, g_{1}\right) = 0$. Extend this to more than two functions. Is there such a theorem for some class of functions $P$ which is larger than the polynomials?

Suppose $\Omega$ is a simply connected region, and $u$ is a real harmonic function in $\Omega$. Prove that there exists an $f \in H(\Omega )$ such that $u = \mathrm{Re}\left(f\right)$. Show that this fails in every region which is not simply connected.

Suppose $X$ is the closed unit square in the plane, $f$ is a continuous complex function on $X$, and $f$ has no zero in $X$. Prove that there is a continuous function $g$ on $X$ such that $f = e^{g}$. For what class of spaces $X$ (other than the above square) is this also true?

Prove that the transformations $z \rightarrow z+1$ and $z \rightarrow -1 / z$ generate the full modular group $G$. Let $R$ consist of all $z = x+i y$ such that $\left|x\right| < \frac{1}{2}, y > 0$, and $\left|z\right| > 1$, plus those limit points which have $x \leq 0$. Prove that $R$ is a fundamental domain of $G$.

Prove that $G$ is also generated by the transformations $\varphi$ and $\psi$, where

Show that $\varphi$ has period $2, \psi$ has period 3.

Find the relation between composition of linear fractional transformations and matrix multiplication. Try to use this to construct an algebraic proof of Theorem 16.19(c) or of the first part of Exercise 5.

Let $E$ be a compact set on the real axis, of positive Lebesgue measure, let $\Omega$ be the complement of $E$, relative to the plane, and define

Answer the following questions:

Check your answers in Exercise 8 against the special case

Call a compact set $E$ in the plane removable if there are no nonconstant bounded holomorphic functions in the complement of $E$.

For each positive number $\alpha$, let $\Gamma_{\alpha }$ be the path with parameter interval $(-\infty , \infty )$ defined by

Let $\Omega_{\alpha }$ be the component of the complement of $\Gamma_{\alpha }^{*}$ which contains the origin, and define

Prove that $f_{\beta }$ is an analytic continuation of $f_{\alpha }$ if $\alpha < \beta$. Prove that therefore there is an entire function $f$ whose restriction to $\Omega_{\alpha }$ is $f_{\alpha }$. Prove that

for every $e^{i \theta } \neq 1$. (Here $r$ is positive and $\theta$ is real, as usual). Prove that $f$ is not constant. Hint: Look at $f(r)$. If

prove that

for every $e^{i \theta }$. Show that there exists an entire function $h$ such that

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.

Let $\Omega = \left\{ z: \frac{1}{2} < \left|z\right| < 2\right\}$. For $n = 1,2,3, \ldots$ let $X_{n}$ be the set of all $f \in H(\Omega )$ that are $n$th derivatives of some $g \in H(\Omega )$.

Suppose $\Omega$ is a region, $p \in \Omega , R < \infty$. Let $\mathscr {F}$ be the class of all $f \in H(\Omega )$ such that $\left|f(p)\right| \leq R$ and $f(\Omega )$ contains neither 0 nor 1. Prove that $\mathscr {F}$ is a normal family.

Show that Theorem 16.2 leads to a very simple proof of the special case of the monodromy theorem (16.15) in which $\Omega$ and $D$ are concentric discs. Combine this special case with the Riemann mapping theorem to prove Theorem 16.15 in the generality in which it is stated.