Additional Problems for Chapter 3 \( \newcommand{\ds}{\displaystyle} \newcommand{\diam}{\operatorname{diam}} \newcommand{\area}{\operatorname{area}} \newcommand{\dist}{\operatorname{dist}} \newcommand{\ord}{\operatorname{ord}} \newcommand{\res}{\operatorname{res}} \newcommand{\wind}{\operatorname{W}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\iso}{\stackrel{\cong}{\longrightarrow}} \newcommand{\myint}{\operatorname{int}} \newcommand{\ve}{\varepsilon} \newcommand{\es}{\emptyset} \newcommand{\sm}{\smallsetminus} \newcommand{\bd}{\partial} \newcommand{\Chat}{\hat{\mathbb C}} \newcommand{\Cstar}{{\mathbb C}^*} \newcommand{\Dstar}{{\mathbb D}^*} \newcommand{\myre}{\operatorname{Re}} \newcommand{\myim}{\operatorname{Im}} \newcommand{\ov}{\overline} \newcommand{\io}{\iota} \newcommand{\con}{\operatorname{const.}} \newcommand{\OO}{{\mathcal O}} \newcommand{\MM}{{\mathcal M}} \newcommand{\FF}{{\mathcal F}} \newcommand{\CC}{{\mathbb C}} \newcommand{\PP}{{\mathbb P}} \newcommand{\RR}{{\mathbb R}} \newcommand{\HH}{{\mathbb H}} \newcommand{\TT}{{\mathbb T}} \newcommand{\II}{{\mathbb I}} \newcommand{\ZZ}{{\mathbb Z}} \newcommand{\NN}{{\mathbb N}} \newcommand{\DD}{{\mathbb D}} \newcommand{\QQ}{{\mathbb Q}} \)




Additional Problems for Chapter 3

  1. Suppose $p$ is an isolated singularity of $f$ and $\myre(f) \geq 0$ in some punctured neighborhood of $p$. Show that $p$ is a removable singularity of $f$.
  2. Suppose $p$ is an isolated singularity of $f$. Assume there is an integer $n \geq 1$ and a constant $C>0$ such that $|f^{(n)}(z)| \leq C |z-p|^{-n}$ for all $z$ in a punctured neighborhood of $p$. Show that $p$ is a removable singularity of $f$.
  3. The algebraic degree of a non-constant rational function $f=P/Q$ is defined as the integer $d:= \max \{ \deg(P), \deg (Q) \}$ (as usual, $P,Q$ are complex polynomials without a common factor). Show that $d$ is the mapping degree of $f$, i.e., for every $w \in \Chat$ the equation $f(z)=w$ has precisely $d$ solutions in $z \in \Chat$ counting multiplicities. The integer $d$ is simply called the degree of $f$.
  4. Prove that every rational function of degree $d \geq 1$ has $d+1$ fixed points counting multiplicities. (Hint: If $f=P/Q$, the case where $d=\deg(P)=\deg(Q)+1$ needs special care.)
  5. Let $\gamma$ be a positively oriented Jordan curve in a domain $U$ such that $D:=\operatorname{int}(\gamma) \subset U$.

    (i) If $R$ is a rational function with poles in $D$, show that $\int_\gamma R(\zeta)/(\zeta-z) \ d\zeta=0$ for all $z \in D$.

    (ii) Suppose $f$ is meromorphic in $U$ with no poles on $|\gamma|$. Let $R$ be the sum of the principal parts of $f$ at its poles in $D$, and $g:=f-R$. Show that \[ g(z)= \frac{1}{2\pi i} \int_\gamma \frac{f(\zeta)}{\zeta-z} \, d\zeta \qquad \text{for all} \ z \in D. \]

  6. Show that there is a meromorphic function $f$ in $\DD$ which satisfies the equation \[ z^2 \ (f(z))^2 + \sin z \ f(z) + 1 =0 \] for all $z \in \DD^{\ast}$. Verify that any such $f$ has a simple pole at $z=0$.
  7. Suppose $f$ is non-constant and holomorphic is some neighborhood of $p \in \CC$. Compute the integral \[ \frac{1}{2 \pi i} \int_{\TT(p,r)} \frac{dz}{f(z)-f(p)} \] for small $r>0$ when $\deg(f,p)=1$ and when $\deg(f,p)=2$.
  8. Show that $\ds{\int_{-\infty}^{\infty} \frac{\cos x}{x^2+1} \, dx = \frac{\pi}{e}}$.
  9. Suppose $f \in \OO(\CC^{\ast})$ satisfies \[ f(z)= \frac{1}{z} \ \ov{f \left( \frac{1}{\bar{z}}\right) } \qquad \text{for all} \ z \neq 0. \] Show that $\res(f^2,0) \geq 0$.
  10. Suppose $f \in \OO(\Cstar)$ is nowhere vanishing. Prove the following statements:

    (i) There are $n \in \ZZ$ and $g \in \OO(\Cstar)$ such that $f(z)=z^n \exp(g)$.

    (ii) More precisely, we can write \[ f(z)=c z^n \exp(h(z)+k(1/z)), \] where $c \neq 0$, $n \in \ZZ$, and $h,k$ are entire functions with $h(0)=k(0)=0$. Moreover, the data $c,n,h,k$ are uniquely determined by $f$.

  11. How many roots of the equation $z^8-4z^5+z^2-1=0$ lie in $\DD$? How many in $\DD(0,2)$?
  12. Prove that for every monic polynomial $P$ of degree $\geq 1$ there is a point $z$ with $|z|=1$ and $|P(z)| \geq 1$.
  13. Let $f$ be holomorphic in a neighborhood of the closed unit disk and satisfies $\myre(\bar{z}f(z))>0$ whenever $|z|=1$. Show that $f$ has a unique zero in $\DD$.
  14. Let $0 < a_0 < a_1 < a_2 < \cdots < a_n$.

    (i) Show that all zeros of the polynomial $P(z)=\sum_{k=0}^n a_k z^k$ are in $\DD$.

    (ii) Show that the equation $\sum_{k=0}^n a_k \cos(kt)=0$ has exactly $2n$ roots in $(0,2\pi)$.

    (Hint: For (i), consider $Q(z):=(z-1)P(z)$ and show that $z^{-(n+1)}Q(z) \neq 0$ if $|z|\geq 1$. For (ii), use the argument principle to find at least $2n$ roots. Then use the polynomial $z^n(P(z)+P(1/z))$ to show that there are at most $2n$ roots.)

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