# Unsorted

## 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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• Show that the function $$u=u(x,y)$$ given by \begin{align*}u(x,y)=\frac{e^{ny}-e^{-ny}}{2n^2}\sin nx\quad \text{for}\ n\in {\mathbf N}\end{align*} is the solution on $$D=\{(x,y)\ | x^2+y^2<1\}$$ of the Cauchy problem for the Laplace equation \begin{align*}\frac{\partial ^2u}{\partial x^2}+\frac{\partial ^2u}{\partial y^2}=0,\quad u(x,0)=0,\quad \frac{\partial u}{\partial y}(x,0)=\frac{\sin nx}{n}.\end{align*}

• Show that there exist points $$(x,y)\in D$$ such that $$\displaystyle{\limsup_{n\to\infty} |u(x,y)|=\infty}$$.

#complex/exercise/work