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Tie’s Extra Questions: Fall 2015 #complex/exercise/work

• Let \(f(z) \in H({\mathbb D})\), \(\text{Re}(f(z)) >0\) and \(f(0)= a>0\). Show that \begin{align*} {\left\lvert { \frac{f(z)-a}{f(z)+a}} \right\rvert} \leq |z|, \; \; \; |f'(0)| \leq 2a .\end{align*}

• Show that the above is still true if \(\text{Re}(f(z)) >0\) is replaced with \(\text{Re}(f(z)) \geq 0\).

Tie’s Extra Questions: Spring 2015 #complex/exercise/work

• Let \(p(z)\) be a polynomial, \(R>0\) any positive number, and \(m \geq 1\) an integer. Let \(M_R = \sup \{ |z^{m} p(z) - 1|: |z| = R \}\). Show that \(M_R>1\).

• Let \(m \geq 1\) be an integer and \(K = \{z \in {\mathbb C}: r \leq |z| \leq R \}\) where \(r<R\). Show (i) using (1) as well as, (ii) without using (1) that there exists a positive number \(\varepsilon_0>0\) such that for each polynomial \(p(z)\), \begin{align*}\sup \{|p(z) - z^{-m}|: z \in K \} \geq \varepsilon_0 \, .\end{align*}

Tie’s Extra Questions: Spring 2015 #complex/exercise/work

• Explicitly write down an example of a non-zero analytic function in \(|z|<1\) which has infinitely zeros in \(|z|<1\).

• Why does not the phenomenon in (1) contradict the uniqueness theorem?

Tie’s Extra Questions: Spring 2015 #complex/exercise/work

Let \(f\) be holomorphic in a neighborhood of \(D_r(z_0)\). Show that for any \(s<r\), there exists a constant \(c>0\) such that \begin{align*}||f||_{(\infty, s)} \leq c ||f||_{(1, r)},\end{align*} where \(\displaystyle |f||_{(\infty, s)} = \text{sup}_{z \in D_s(z_0)}|f(z)|\) and \(\displaystyle ||f||_{(1, r)} = \int_{D_r(z_0)} |f(z)|dx dy\).

Note: Exercise 3.8.20 on p.107 in Stein et al is a straightforward consequence of this stronger result using the integral form of the Cauchy-Schwarz inequality in real analysis.

Tie’s Extra Questions: Spring 2015 #complex/exercise/work

Let \(f\) be an analytic function on a region \(\Omega\). Show that \(f\) is a constant if there is a simple closed curve \(\gamma\) in \(\Omega\) such that its image \(f(\gamma)\) is contained in the real axis.

Tie’s Extra Questions: Spring 2015 #complex/exercise/work

• Show that \(\displaystyle \frac{\pi^2}{\sin^2 \pi z}\) and \(\displaystyle g(z) = \sum_{n = - \infty}^{ \infty} \frac{1}{(z-n)^2}\) have the same principal part at each integer point.

• Show that \(\displaystyle h(z) = \frac{\pi^2}{\sin^2 \pi z} - g(z)\) is bounded on \(\mathbb C\) and conclude that \(\displaystyle \frac{\pi^2}{\sin^2 \pi z} = \sum_{n = - \infty}^{ \infty} \frac{1}{(z-n)^2} \, .\)

Tie’s Extra Questions: Spring 2015 #complex/exercise/work

Assume \(f(z)\) is analytic in \({\mathbb D}: |z|<1\) and \(f(0)=0\) and is not a rotation (i.e. \(f(z) \neq e^{i \theta} z\)). Show that \(\displaystyle \sum_{n=1}^\infty f^{n}(z)\) converges uniformly to an analytic function on compact subsets of \({\mathbb D}\), where \(f^{n+1}(z) = f(f^{n}(z))\).

Tie’s Extra Questions: Spring 2015 #complex/exercise/work

Let \(f\) be a non-constant analytic function on \(\mathbb D\) with \(f(\mathbb D) \subseteq \mathbb D\). Use \(\psi_{a} (f(z))\) (where \(a=f(0)\), \(\displaystyle \psi_a(z) = \frac{a - z}{1 - \mkern 1.5mu\overline{\mkern-1.5mua\mkern-1.5mu}\mkern 1.5muz}\)) to prove that \(\displaystyle \frac{|f(0)| - |z|}{1 + |f(0)||z|} \leq |f(z)| \leq \frac{|f(0)| + |z|}{1 - |f(0)||z|}\).

Tie’s Extra Questions: Spring 2015 #complex/exercise/work

Let \(f\) be holomorphic in a neighborhood of \(D_r(z_0)\). Show that for any \(s<r\), there exists a constant \(c>0\) such that \begin{align*} \|f\|_{(\infty, s)} \leq c \|f\|_{(1, r)} ,\end{align*} where \(\displaystyle \|f\|_{(\infty, s)} = \text{sup}_{z \in D_s(z_0)}|f(z)|\) and \(\displaystyle \|f\|_{(1, r)} = \int_{D_r(z_0)} |f(z)|dx dy\).

Tie’s Extra Questions: Spring 2015 #complex/exercise/work

Let \(\Omega\) be a simply connected open set and let \(\gamma\) be a simple closed contour in \(\Omega\) and enclosing a bounded region \(U\) anticlockwise. Let \(f: \ \Omega \to {\mathbb C}\) be a holomorphic function and \(|f(z)|\leq M\) for all \(z\in \gamma\). Prove that \(|f(z)|\leq M\) for all \(z\in U\).

Tie’s Extra Questions: Spring 2015 #complex/exercise/work

Let \(f\) be holomorphic in a neighborhood of \(D_r(z_0)\). Show that for any \(s<r\), there exists a constant \(c>0\) such that \begin{align*}\|f\|_{(\infty, s)} \leq c \|f\|_{(1, r)},\end{align*} where \(\displaystyle \|f\|_{(\infty, s)} = \text{sup}_{z \in D_s(z_0)}|f(z)|\) and \(\displaystyle \|f\|_{(1, r)} = \int_{D_r(z_0)} |f(z)|dx dy\).

Tie’s Extra Questions: Fall 2016 #complex/exercise/work

• \(f(z)= u(x,y) +i v(x,y)\) be analytic in a domain \(D\subset {\mathbb C}\). Let \(z_0=(x_0,y_0)\) be a point in \(D\) which is in the intersection of the curves \(u(x,y)= c_1\) and \(v(x,y)=c_2\), where \(c_1\) and \(c_2\) are constants. Suppose that \(f'(z_0)\neq 0\). Prove that the lines tangent to these curves at \(z_0\) are perpendicular.

• Let \(f(z)=z^2\) be defined in \({\mathbb{C}}\).

• Describe the level curves of \(\mbox{\textrm Re}{(f)}\) and of \(\mbox{Im}{(f)}\).

• What are the angles of intersections between the level curves \(\mbox{\textrm Re}{(f)}=0\) and \(\mbox{\textrm Im}{(f)}\)? Is your answer in agreement with part a) of this question?

Tie’s Extra Questions: Fall 2016 #complex/exercise/work

• \(f: D\rightarrow {\mathbb C}\) be a continuous function, where \(D\subset {\mathbb C}\) is a domain.Let \(\alpha:[a,b]\rightarrow D\) be a smooth curve. Give a precise definition of the complex line integral \begin{align*}\int_{\alpha} f.\end{align*}

• Assume that there exists a constant \(M\) such that \(|f(\tau)|\leq M\) for all \(\tau\in \mbox{\textrm Image}(\alpha\)). Prove that \begin{align*}\big | \int_{\alpha} f \big |\leq M \times \mbox{\textrm length}(\alpha).\end{align*}

• Let \(C_R\) be the circle \(|z|=R\), described in the counterclockwise direction, where \(R>1\). Provide an upper bound for \(\big | \int_{C_R} \dfrac{\log{(z)} }{z^2} \big |,\) which depends only on \(R\) and other constants.

Tie’s Extra Questions: Fall 2016 #complex/exercise/work

• Let \(F\) be an analytic function inside and on a simple closed curve \(C\), except for a pole of order \(m\geq 1\) at \(z=a\) inside \(C\). Prove that

\begin{align*}\frac{1}{2 \pi i}\oint_{C} F(\tau) d\tau = \lim_{\tau\rightarrow a} \frac{d^{m-1}}{d\tau^{m-1}}\big((\tau-a)^m F(\tau))\big).\end{align*}

• Evaluate \begin{align*}\oint_{C}\frac{e^{\tau}}{(\tau^2+\pi^2)^2}d\tau\end{align*} where \(C\) is the circle \(|z|=4\).

Tie’s Extra Questions: Spring 2014, Fall 2009, Fall 2011 #complex/exercise/work

For \(s>0\), the gamma function is defined by \(\displaystyle{\Gamma(s)=\int_0^{\infty} e^{-t}t^{s-1} dt}\).

• Show that the gamma function is analytic in the half-plane \(\Re (s)>0\), and is still given there by the integral formula above.

• Apply the formula in the previous question to show that \begin{align*} \Gamma(s)\Gamma(1-s)=\frac{\pi}{\sin \pi s} .\end{align*}

Hint: You may need \(\displaystyle{\Gamma(1-s)=t \int_0^{\infty}e^{-vt}(vt)^{-s} dv}\) for \(t>0\).

Tie’s Extra Questions: Fall 2011 #complex/exercise/work

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