Elementary Properties of Holomorphic Functions

The following fact was tacitly used in this chapter: If $A$ and $B$ are disjoint subsets of the plane, if $A$ is compact, and if $B$ is closed, then there exists a $\delta > 0$ such that $\left|\alpha -\beta \right| \geq \delta$ for all $\alpha \in A$ and $\beta \in B$. Prove this, with an arbitrary metric space in place of the plane.

Suppose that $f$ is an entire function, and that in every power series

at least one coefficient is 0. Prove that $f$ is a polynomial. Hint: $n!c_{n} = f^{(n)}(a)$.

Suppose $f$ and $g$ are entire functions, and $\left|f(z)\right| \leq \left|g(z)\right|$ for every $z$. What conclusion can you draw?

Suppose $f$ is an entire function, and

for all $z$, where $A, B$, and $k$ are positive numbers. Prove that $f$ must be a polynomial.

Suppose $\left\{ f_{n}\right\}$ is a uniformly bounded sequence of holomorphic functions in $\Omega$ such that $\left\{ f_{n}(z)\right\}$ converges for every $z \in \Omega$. Prove that the convergence is uniform on every compact subset of $\Omega$.

Hint: Apply the dominated convergence theorem to the Cauchy formula for $f_{n}-f_{m}$.

There is a region $\Omega$ that $\exp (\Omega ) = D(1 ; 1)$. Show that $\exp$ is one-to-one in $\Omega$, but that there are many such $\Omega$. Fix one, and define $\log z$, for $\left|z-1\right| < 1$, to be that $\dot{w} \in \Omega$ for which $e^{w} = z$. Prove that $\log^{\prime }(z) = 1 / z$. Find the coefficients $a_{n}$ in

and hence find the coefficients $c_{n}$ in the expansion

In what other discs can this be done?

If $f \in H(\Omega )$, the Cauchy formula for the derivatives of $f$,

is valid under certain conditions on $z$ and $\Gamma$. State these, and prove the formula.

Suppose $P$ and $Q$ are polynomials, the degree of $Q$ exceeds that of $P$ by at least 2, and the rational function $R = P / Q$ has no pole on the real axis. Prove that the integral of $R$ over $(-\infty , \infty )$ is $2 \pi i$ times the sum of the residues of $R$ in the upper half plane. What is the analogous statement for the lower half plane? Use this method to compute

Compute $\int_{-\infty }^{\infty } e^{i t x} /\left(1+x^{2}\right) d x$ for real $t$, by the method described in Exercise 8. Check your answer against the inversion theorem for Fourier transforms.

Let $\gamma$ be the positively oriented unit circle, and compute

Suppose $\alpha$ is a complex number, $\left|\alpha \right| \neq 1$, and compute

by integrating $(z-\alpha )^{-1}(z-1 / \alpha )^{-1}$ over the unit circle.

Compute

Compute

Suppose $\Omega_{1}$ and $\Omega_{2}$ are plane regions, $f$ and $g$ are nonconstant complex functions defined in $\Omega_{1}$ and $\Omega_{2}$, respectively, and $f\left(\Omega_{1}\right) \subset \Omega_{2}$. Put $h = g \circ f$. If $f$ and $g$ are holomorphic, we know that $h$ is holomorphic. Suppose we know that $f$ and $h$ are holomorphic. Can we conclude anything about $g$? What if we know that $g$ and $h$ are holomorphic?

Suppose $\Omega$ is a region, $\varphi \in H(\Omega ), \varphi^{\prime }$ has no zero in $\Omega , f \in H(\varphi (\Omega )), g = f \circ \varphi , z_{0} \in \Omega$, and $w_{0} = \varphi \left(z_{0}\right)$. Prove that if $f$ has a zero of order $m$ at $w_{0}$, then $g$ also has a zero of order $m$ at $z_{0}$. How is this modified if $\varphi^{\prime }$ has a zero of order $k$ at $z_{0}$?

Suppose $\mu$ is a complex measure on a measure space $X, \Omega$ is an open set in the plane, $\varphi$ is a bounded function on $\Omega \times X$ such that $\varphi (z, t)$ is a measurable function of $t$, for each $z \in \Omega$, and $\varphi (z, t)$ is holomorphic in $\Omega$, for each $t \in X$. Define

for $z \in \Omega$. Prove that $f \in H(\Omega )$. Hint: Show that to every compact $K \subset \Omega$ there corresponds a constant $M < \infty$ such that

Determine the regions in which the following functions are defined and holomorphic:

Hint: Either use Exercise 16, or combine Morera's theorem with Fubini's.

Suppose $f \in H(\Omega ), \bar{D}(a ; r) \subset \Omega$, $\gamma$ is the positively oriented circle with center at $a$ and radius $r$, and $f$ has no zero on $\gamma^{*}$. For $p = 0$, the integral

is equal to the number of zeros of $f$ in $D(a ; r)$. What is the value of this integral (in terms of the zeros of $f$) for $p = 1,2,3, \ldots$? What is the answer if $z^{p}$ is replaced by any $\varphi \in H(\Omega )$?

Suppose $f \in H(U), g \in H(U)$, and neither $f$ nor $g$ has a zero in $U$. If

find another simple relation between $f$ and $g$.

Suppose $\Omega$ is a region, $f_{n} \in H(\Omega )$ for $n = 1,2,3, \ldots$, none of the functions $f_{n}$ has a zero in $\Omega$, and $\left\{ f_{n}\right\}$ converges to $f$ uniformly on compact subsets of $\Omega$. Prove that either $f$ has no zero in $\Omega$ or $f(z) = 0$ for all $z \in \Omega$.

If $\Omega^{\prime }$ is a region that contains every $f_{n}(\Omega )$, and if $f$ is not constant, prove that $f(\Omega ) \subset \Omega^{\prime }$.

Suppose $f \in H(\Omega ), \Omega$ contains the closed unit disc, and $\left|f(z)\right| < 1$ if $\left|z\right| = 1$. How many fixed points must $f$ have in the disc? That is, how many solutions does the equation $f(z) = z$ have there?

Suppose $f \in H(\Omega ), \Omega$ contains the closed unit disc, $\left|f(z)\right| > 2$ if $\left|z\right| = 1$, and $f(0) = 1$. Must $f$ have a zero in the unit disc?

Suppose $P_{n}(z) = 1+z / 1!+\cdots +z^{n} / n!, Q_{n}(z) = P_{n}(z)-1$, where $n = 1,2,3, \ldots$. What can you say about the location of the zeros of $P_{n}$ and $Q_{n}$ for large $n$? Be as specific as you can.

Prove the following general form of Rouche's theorem: Let $\Omega$ be the interior of a compact set $K$ in the plane. Suppose $f$ and $g$ are continuous on $K$ and holomorphic in $\Omega$, and $\left|f(z)-g(z)\right| < \left|f(z)\right|$ for all $z \in K-\Omega$. Then $f$ and $g$ have the same number of zeros in $\Omega$.

Let $A$ be the annulus $\left\{ z: r_{1} < \left|z\right| < r_{2}\right\}$, where $r_{1}$ and $r_{2}$ are given positive numbers.

It is required to expand the function

in a series of the form $\sum_{-\infty }^{\infty } c_{n} z^{n}$. How many such expansions are there? In which region is each of them valid? Find the coefficients $c_{n}$ explicitly for each of these expansions.

Suppose $\Omega$ is a horizontal strip, determined by the inequalities $a < y < b$, say. Suppose $f \in H(\Omega )$, and $f(z) = f(z+1)$ for all $z \in \Omega$. Prove that $f$ has a Fourier expansion in $\Omega$,

which converges uniformly in $\left\{ z: a+\epsilon \leq y \leq b-\epsilon \right\}$, for every $\epsilon > 0$. Hint: The map $z \rightarrow e^{2 \pi i z}$ converts $f$ to a function in an annulus.

Find the integral formulas by means of which the coefficients $c_{n}$ can be computed from $f$.

Suppose $\Gamma$ is a closed curve in the plane, with parameter interval $[0,2 \pi ]$. Take $\alpha \notin \Gamma^{*}$. Approximate $\Gamma$ uniformly by trigonometric polynomials $\Gamma_{n}$. Show that $\operatorname {Ind}_{\Gamma_{n}}(\alpha ) = \operatorname {Ind}_{\Gamma_{m}}(\alpha )$ if $m$ and $n$ are sufficiently large. Define this common value to be $\operatorname {Ind}_{\Gamma }(\alpha )$. Prove that the result does not depend on the choice of $\left\{ \Gamma_{n}\right\}$; prove that Lemma 10.39 is now true for closed curves, and use this to give a different proof of Theorem 10.40.

Define

Show that $f(z) = \bar{z}$ if $\left|z\right| < 1$ and that $f(z) = 1 / z$ if $\left|z\right| \geq 1$. Thus $f$ is not holomorphic in the unit disc, although the integrand is a holomorphic function of z. Note the contrast between this, on the one hand, and Theorem 10.7 and Exercise 16 on the other.

Suggestion: Compute the inner integral separately for $r < \left|z\right|$ and for $r > \left|z\right|$.

Let $\Omega$ be the plane minus two points, and show that some closed paths $\Gamma$ in $\Omega$ satisfy assumption (1) of Theorem 10.35 without being null-homotopic in $\Omega$.