# 1 Preface

I’d like to thank the following individuals for their contributions to this document:

• Edward Azoff, for supplying a problem sheet broken out by topic.
• Mentzelos Melistas, for explaining and documenting many solutions to these questions.
• Jingzhi Tie, for supplying many additional problems and solutions.

# 2 Topology and Functions of One Variable (8155a)

## 2.1 1 $$\work$$

Let $$x_0 = a, x_1 = b$$, and set \begin{align*} x_n \mathrel{\vcenter{:}}={x_{n-1} + x_{n-2} \over 2} \quad n\geq 2 .\end{align*}

Show that $$\left\{{x_n}\right\}$$ is a Cauchy sequence and find its limit in terms of $$a$$ and $$b$$.

## 2.2 2 $$\work$$

Suppose $$f:{\mathbb{R}}\to{\mathbb{R}}$$ is continuous and $$\lim_{x\to \pm \infty} f(x) = 0$$. Prove that $$f$$ is uniformly continuous.

## 2.3 3 $$\work$$

Give an example of a function $$f:{\mathbb{R}}\to {\mathbb{R}}$$ that is everywhere differentiable but $$f'$$ is not continuous at 0.

## 2.4 4 $$\work$$

Suppose $$\left\{{g_n}\right\}$$ is a uniformly convergent sequence of functions from $${\mathbb{R}}$$ to $${\mathbb{R}}$$ and $$f:{\mathbb{R}}\to {\mathbb{R}}$$ is uniformly continuous. Prove that the sequence $$\left\{{f\circ g_n}\right\}$$ is uniformly convergent.

## 2.5 5 $$\work$$

Let $$f$$ be differentiable on $$[a, b]$$. Say that $$f$$ is uniformly differentiable iff

\begin{align*} \forall \varepsilon > 0,\, \exists \delta > 0 \text{ such that } \quad {\left\lvert {x-y} \right\rvert} < \delta \implies {\left\lvert { {f(x) - f(y) \over x-y} - f'(y)} \right\rvert} < \varepsilon .\end{align*}

Prove that $$f$$ is uniformly differentiable on $$[a, b] \iff f'$$ is continuous on $$[a, b]$$.

## 2.6 6 $$\work$$

Suppose $$A, B \subseteq {\mathbb{R}}^n$$ are disjoint and compact. Prove that there exist $$a\in A, b\in B$$ such that \begin{align*} {\left\lVert {a - b} \right\rVert} = \inf\left\{{{\left\lVert {x-y} \right\rVert} {~\mathrel{\Big|}~}x\in A,\, y\in B}\right\} .\end{align*}

## 2.7 7 $$\work$$

Suppose $$A, B\subseteq {\mathbb{R}}^n$$ are connected and not disjoint. Prove that $$A\cup B$$ is also connected.

## 2.8 8 $$\work$$

Suppose $$\left\{{f_n}\right\}_{n\in {\mathbb{N}}}$$ is a sequence of continuous functions $$f_n: [0, 1]\to {\mathbb{R}}$$ such that \begin{align*} f_n(x) \geq f_{n+1}(x) \geq 0 \quad \forall n\in {\mathbb{N}},\, \forall x\in [0, 1] .\end{align*} Prove that if $$\left\{{f_n}\right\}$$ converges pointwise to $$0$$ on $$[0, 1]$$ then it converges to $$0$$ uniformly on $$[0, 1]$$.

## 2.9 9 $$\work$$

Show that if $$E\subset [0, 1]$$ is uncountable, then there is some $$t\in {\mathbb{R}}$$ such that $$E\cap(-\infty ,t)$$ and $$E\cap(t, \infty)$$ are also uncountable.

# 3 Several Variables (8155h)

## 3.1 1 $$\work$$

Is the following function continuous, differentiable, continuously differentiable? \begin{align*} f: {\mathbb{R}}^2 &\to {\mathbb{R}}\\ f(x, y) &= \begin{cases} {xy \over \sqrt{x^2 + y^2}} & (x, y) \neq (0, 0) \\ 0 & \text{else}. \end{cases} \end{align*}

## 3.2 2 $$\work$$

### 3.2.1 a $$\work$$

Complete this definition: “$$f: {\mathbb{R}}^n\to {\mathbb{R}}^m$$ is real-differentiable a point $$p\in {\mathbb{R}}^n$$ iff there exists a linear transformation…”

### 3.2.2 b $$\work$$

Give an example of a function $$f:{\mathbb{R}}^2\to {\mathbb{R}}$$ whose first-order partial derivatives exist everywhere but $$f$$ is not differentiable at $$(0, 0)$$.

### 3.2.3 c $$\work$$

Give an example of a function $$f: {\mathbb{R}}^2 \to {\mathbb{R}}$$ which is real-differentiable everywhere but nowhere complex-differentiable.

## 3.3 3 $$\work$$

Let $$f:{\mathbb{R}}^2\to {\mathbb{R}}$$.

### 3.3.1 a $$\work$$

Define in terms of linear transformations what it means for $$f$$ to be differentiable at a point $$(a, b) \in {\mathbb{R}}^2$$.

### 3.3.2 b $$\work$$

State a version of the inverse function theorem in this setting.

### 3.3.3 c $$\work$$

Identify $${\mathbb{R}}^2$$ with $${\mathbb{C}}$$ and give a necessary and sufficient condition for a real-differentiable function at $$(a, b)$$ to be complex differentiable at the point $$a+ib$$.

## 3.4 4 $$\work$$

Let $$f = u+iv$$ be complex-differentiable with continuous partial derivatives at a point $$z = re^{i\theta}$$ with $$r\neq 0$$. Show that \begin{align*} {\frac{\partial u}{\partial r}\,} = {1\over r}{\frac{\partial v}{\partial \theta}\,} \qquad {\frac{\partial v}{\partial r}\,} = -{1\over r}{\frac{\partial u}{\partial \theta}\,} .\end{align*}

## 3.5 5 $$\work$$

Let $$P = (1, 3) \in {\mathbb{R}}^2$$ and define \begin{align*} f(s, t) \mathrel{\vcenter{:}}= ps^3 -6st + t^2 .\end{align*}

### 3.5.1 a $$\work$$

State the conclusion of the implicit function theorem concerning $$f(s, t) = 0$$ when $$f$$ is considered a function $${\mathbb{R}}^2\to{\mathbb{R}}$$.

### 3.5.2 b $$\work$$

State the above conclusion when $$f$$ is considered a function $${\mathbb{C}}^2\to {\mathbb{C}}$$.

### 3.5.3 c $$\work$$

Use the implicit function theorem for a function $${\mathbb{R}}\times{\mathbb{R}}^2 \to {\mathbb{R}}^2$$ to prove (b).

There are various approaches: using the definition of the complex derivative, the Cauchy-Riemann equations, considering total derivatives, etc.

## 3.6 6 $$\work$$

Let $$F:{\mathbb{R}}^2\to {\mathbb{R}}$$ be continuously differentiable with $$F(0, 0) = 0$$ and $${\left\lVert {\nabla F(0, 0)} \right\rVert} < 1$$.

Prove that there is some real number $$r> 0$$ such that $${\left\lvert {F(x, y)} \right\rvert} < r$$ whenever $${\left\lVert {(x, y)} \right\rVert} < r$$.

## 3.7 7 $$\work$$

State the most general version of the implicit function theorem for real functions and outline how it can be proved using the inverse function theorem.

# 4 Several Variables: Extra Questions

## 4.1 ?

Let $$f=u+iv$$ be differentiable (i.e. $$f'(z)$$ exists) with continuous partial derivatives at a point $$z=re^{i\theta}$$, $$r\not= 0$$.

Show that \begin{align*} \frac{\partial u}{\partial r}=\frac{1}{r}\frac{\partial v}{\partial \theta},\quad \frac{\partial v}{\partial r}=-\frac{1}{r}\frac{\partial u}{\partial \theta} .\end{align*}

## 4.2 ?

Give an example of a

Show that $$f(z) = z^2$$ is uniformly continuous in any open disk $$|z| < R$$, where $$R>0$$ is fixed, but it is not uniformly continuous on $$\mathbb C$$.

### 4.2.1 1

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*}

### 4.2.2 2

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

# 5 Conformal Maps (8155c)

Notation: $${\mathbb{D}}$$ is the open unit disc, $${\mathbb{H}}$$ is the open upper half-plane.

## 5.1 1 $$\work$$

Find a conformal map from $${\mathbb{D}}$$ to $${\mathbb{H}}$$.

## 5.2 2 $$\work$$

Find a conformal map from the strip $$\left\{{z\in {\mathbb{C}}{~\mathrel{\Big|}~}0 < \Im(z) < 1}\right\}$$ to $${\mathbb{H}}$$.

## 5.3 3 $$\work$$

Find a fractional linear transformation $$T$$ which maps $${\mathbb{H}}$$ to $${\mathbb{D}}$$, and explicitly describe the image of the first quadrant under $$T$$.

## 5.4 4 $$\work$$

Find a conformal map from $$\left\{{z\in {\mathbb{C}}{~\mathrel{\Big|}~}{\left\lvert {z-i} \right\rvert} > 1,\, \Re(z) > 0}\right\}$$ to $${\mathbb{H}}$$.

## 5.5 5 $$\work$$

Find a conformal map from $$\left\{{z\in {\mathbb{C}}{~\mathrel{\Big|}~}{\left\lvert {z} \right\rvert} < 1,\, {\left\lvert {z - {1\over 2}} \right\rvert} > {1\over 2} }\right\}$$ to $${\mathbb{D}}$$.

## 5.6 6 $$\work$$

Find a conformal map from $$\left\{{{\left\lvert {z-1} \right\rvert} < 2}\right\} \cap\left\{{{\left\lvert {z+1} \right\rvert} < 2}\right\}$$ to $${\mathbb{H}}$$.

## 5.7 7 $$\work$$

Let $$\Omega$$ be the region inside the unit circle $${\left\lvert {z} \right\rvert} = 1$$ and outside the circle $${\left\lvert {z-{1\over 4}} \right\rvert} = {1\over 4}$$.

Find an injective conformal map from $$\Omega$$ onto some annulus $$\left\{{r < {\left\lvert {z} \right\rvert} < 1}\right\}$$ for gonstant $$r$$.

## 5.8 8 $$\work$$

Let $$D$$ be the region obtained by deleting the real interval $$[0, 1)$$ from $${\mathbb{D}}$$; find a conformal map from $$D$$ to $${\mathbb{D}}$$.

## 5.9 9 $$\work$$

Find a conformal map from $${\mathbb{C}}\setminus\left\{{x\in {\mathbb{R}}{~\mathrel{\Big|}~}x\leq 0}\right\}$$ to $${\mathbb{D}}$$.

## 5.10 10 $$\work$$

Find a conformal map from $${\mathbb{C}}\setminus\left\{{x\in {\mathbb{R}}{~\mathrel{\Big|}~}x\geq 1}\right\}$$ to $${\mathbb{D}}$$.

## 5.11 11 $$\work$$

Find a bijective conformal map from $$G$$ to $${\mathbb{H}}$$, where \begin{align*} G \mathrel{\vcenter{:}}=\left\{{z\in {\mathbb{C}}{~\mathrel{\Big|}~}{\left\lvert {z-1} \right\rvert} < \sqrt 2,\, {\left\lvert {z+1} \right\rvert} < \sqrt 2}\right\} \setminus [0, i) .\end{align*}

## 5.12 12 $$\work$$

Prove that TFAE for a Möbius transformation $$T$$ given by $$T(z) = {az + b \over cz + d}$$:

1. $$T$$ maps $${\mathbb{R}}\cup\left\{{\infty}\right\}$$ to itself.
2. It is possible to choose $$a,b,c,d$$ to be real numbers.
3. $$\mkern 1.5mu\overline{\mkern-1.5muT(z)\mkern-1.5mu}\mkern 1.5mu = T(\mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu)$$ for every $$z\in {\mathbb{CP}}^1$$.
4. There exist $$\alpha\in {\mathbb{R}}, \beta \in {\mathbb{C}}\setminus {\mathbb{R}}$$ such that $$T(\alpha) = \alpha$$ and $$T(\mkern 1.5mu\overline{\mkern-1.5mu\beta\mkern-1.5mu}\mkern 1.5mu) = \mkern 1.5mu\overline{\mkern-1.5muT(\beta)\mkern-1.5mu}\mkern 1.5mu$$.

# 6 Extra Questions

## 6.1 12.

Find a conformal map from $$D = \{z :\ |z| < 1,\ |z - 1/2| > 1/2\}$$ to the unit disk $$\Delta=\{z: \ |z|<1\}$$.

# 7 Integrals and Cauchy’s Theorem (8155d)

Some interesting problems: 3, 4, 9, 10.

## 7.1 1 $$\work$$

Suppose $$f, g: [0, 1] \to {\mathbb{R}}$$ where $$f$$ is Riemann integrable and for $$x, y\in [0, 1]$$, \begin{align*} {\left\lvert {g(x) - g(y)} \right\rvert} \leq {\left\lvert {f(x) - f(y)} \right\rvert} .\end{align*}

Prove that $$g$$ is Riemann integrable.

## 7.2 2 $$\work$$

State and prove Green’s Theorem for rectangles.

Then use it to prove Cauchy’s Theory for functions that are analytic in a rectangle.

## 7.3 3 $$\work$$

Suppose $$\left\{{f_n}\right\}_{n\in {\mathbb{N}}}$$ is a sequence of analytic functions on $${\mathbb{D}}\mathrel{\vcenter{:}}=\left\{{z\in {\mathbb{C}}{~\mathrel{\Big|}~}{\left\lvert {z} \right\rvert} < 1}\right\}$$.

Show that if $$f_n\to g$$ for some $$g: {\mathbb{D}}\to {\mathbb{C}}$$ uniformly on every compact $$K\subset {\mathbb{D}}$$, then $$g$$ is analytic on $${\mathbb{D}}$$.

## 7.4 4 $$\work$$

Suppose $$\left\{{f_n}\right\}_{n\in {\mathbb{N}}}$$ is a sequence of entire functions where

• $$f_n \to g$$ pointwise for some $$g:{\mathbb{C}}\to{\mathbb{C}}$$.
• On every line segment in $${\mathbb{C}}$$, $$f_n \to g$$ uniformly.

Show that

• $$g$$ is entire, and
• $$f_n\to g$$ uniformly on every compact subset of $${\mathbb{C}}$$.

## 7.5 5 $$\done$$

Prove that there is no sequence of polynomials that uniformly converge to $$f(z) = {1\over z}$$ on $$S^1$$.

(Click to expand)
• By Cauchy’s integral formula, $$\int_{S^1} f = 2\pi i$$
• If $$p_j$$ is any polynomial, then $$p_j$$ is holomorphic in $${\mathbb{D}}$$, so $$\int_{S^1} p_j = 0$$.
• Contradiction: compact sets in $${\mathbb{C}}$$ are bounded, so \begin{align*} {\left\lvert {\int f - \int p_j} \right\rvert} \leq \int {\left\lvert {p_j - f} \right\rvert} \leq \int {\left\lVert {p_j - f} \right\rVert}_\infty = {\left\lVert {p_j - f} \right\rVert}_\infty \int_{S^1} 1 \,dz = {\left\lVert {p_j-f} \right\rVert}_\infty \cdot 2\pi \to 0 \end{align*} which forces $$\int f = \int p_j = 0$$.

## 7.6 6 $$\work$$

Suppose that $$f: {\mathbb{R}}\to{\mathbb{R}}$$ is a continuous function that vanishes outside of some finite interval. For each $$z\in {\mathbb{C}}$$, define \begin{align*} g(z) = \int_{-\infty}^\infty f(t) e^{-izt} \,dt .\end{align*}

Show that $$g$$ is entire.

## 7.7 7 $$\work$$

Suppose $$f: {\mathbb{C}}\to {\mathbb{C}}$$ is entire and \begin{align*} {\left\lvert {f(z)} \right\rvert} \leq {\left\lvert {z} \right\rvert}^{1\over 2} \quad\text{ when } {\left\lvert {z} \right\rvert} > 10 .\end{align*}

Prove that $$f$$ is constant.

## 7.8 8 $$\work$$

Let $$\gamma$$ be a smooth curve joining two distinct points $$a, b\in {\mathbb{C}}$$.

Prove that the function \begin{align*} f(z) \mathrel{\vcenter{:}}=\int_\gamma {g(w) \over w-z} \,dw \end{align*} is analytic in $${\mathbb{C}}\setminus\gamma$$.

## 7.9 9 $$\done$$

Suppose that $$f: {\mathbb{C}}\to{\mathbb{C}}$$ is continuous everywhere and analytic on $${\mathbb{C}}\setminus {\mathbb{R}}$$ and prove that $$f$$ is entire.

(Click to expand)
• Note $$f$$ is continuous on $${\mathbb{C}}$$ since analytic implies continuous ($$f$$ equals its power series, where the partials sums uniformly converge to it, and uniform limit of continuous is continuous).
• Strategy: take $$D$$ a disc centered at a point $$x\in {\mathbb{R}}$$, show $$f$$ is holomorphic in $$D$$ by Morera’s theorem.
• Let $$\Delta \subset D$$ be a triangle in $$D$$.
• Case 1: If $$\Delta \cap{\mathbb{R}}= 0$$, then $$f$$ is holomorphic on $$\Delta$$ and $$\int_\Delta f = 0$$.
• Case 2: one side or vertex of $$\Delta$$ intersects $${\mathbb{R}}$$, and wlog the rest of $$\Delta$$ is in $${\mathbb{H}}^+$$.
• Then let $$\Delta_\varepsilon$$ be the perturbation $$\Delta + i\varepsilon= \left\{{z+ i\varepsilon{~\mathrel{\Big|}~}z\in \Delta}\right\}$$; then $$\Delta_\varepsilon\cap{\mathbb{R}}= 0$$ and $$\int_{\Delta_\varepsilon} f = 0$$.
• Now let $$\varepsilon\to 0$$ and conclude by continuity of $$f$$ (???)
• We want \begin{align*} \int_{\Delta_\varepsilon} f = \int_a^b f(\gamma_\varepsilon(t)) \gamma_\varepsilon'(t)\,dt \overset{\varepsilon\to 0}\to \int_a^b f(\gamma(t)) \gamma_\varepsilon'(t)\,dt =\int_{\Delta} f \end{align*} where $$\gamma_\varepsilon, \gamma$$ are curves parametrizing $$\Delta_\varepsilon, \Delta$$ respectively.
• Since $$\gamma, \gamma_\varepsilon$$ are closed and bounded in $${\mathbb{C}}$$, they are compact subsets. Thus it suffices to show that $$f(\gamma_\varepsilon(t)) \gamma_\varepsilon'(t)$$ converges uniformly to $$f(\gamma(t))\gamma'(t)$$.
• ??
• Case 3: $$\Delta$$ intersects both $${\mathbb{H}}^+$$ and $${\mathbb{H}}^-$$.
• Break into smaller triangles, each of which falls into one of the previous two cases.

## 7.10 10 $$\done$$

Prove Liouville’s theorem: suppose $$f:{\mathbb{C}}\to{\mathbb{C}}$$ is entire and bounded. Use Cauchy’s formula to prove that $$f'\equiv 0$$ and hence $$f$$ is constant.

(Click to expand)
• Suffices to prove $$f' = 0$$ because $${\mathbb{C}}$$ is connected (see Stein Ch 1, 3.4)
• Idea: Fix $$w_0$$, show $$f(w) = f(w_0)$$ for any $$w\neq w_0$$
• Connected = Path connected in $${\mathbb{C}}$$, so take $$\gamma$$ joining $$w$$ to $$w_0$$.
• $$f$$ is a primitive for $$f'$$, and $$\int_\gamma f' = f(w) - f(w_0)$$, but $$f'=0$$.
• Fix $$z_0\in {\mathbb{C}}$$, let $$B$$ be the bound for $$f$$, so $${\left\lvert {f(z)} \right\rvert} \leq B$$ for all $$z$$.
• Apply Cauchy inequalities: if $$f$$ is holomorphic on $$U\supset \mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_R(z_0)$$ then setting $${\left\lVert {f} \right\rVert}_C \mathrel{\vcenter{:}}=\sup_{z\in C} {\left\lvert {f(z)} \right\rvert}$$, \begin{align*} {\left\lvert {f^{(n)} (z_0)} \right\rvert} \leq {n! {\left\lVert {f} \right\rVert}_C \over R^n} .\end{align*}
• Yields $${\left\lvert { f'(z_0) } \right\rvert} \leq B/R$$
• Take $$R\to \infty$$, QED.

# 8 Extra

## 8.1 ?

Assume $$f$$ is continuous in the region: $$0< |z-a| \leq R, \; 0 \leq \arg(z-a) \leq \beta_0$$ ($$0 < \beta_0 \leq 2 \pi$$) and the limit $$\displaystyle \lim_{z \rightarrow a} (z-a) f(z) = A$$ exists. Show that \begin{align*}\lim_{r \rightarrow 0} \int_{\gamma_r} f(z) dz = i A \beta_0 \; , \; \;\end{align*} where $$\gamma_r : = \{ z \; | \; z = a + r e^{it}, \; 0 \leq t \leq \beta_0 \}.$$

# 9 Liouville’s Theorem, Power Series (8155e)

## 9.1 1 $$\done$$

Suppose $$f$$ is analytic on a region $$\Omega$$ such that $${\mathbb{D}}\subseteq \Omega \subseteq {\mathbb{C}}$$ and $$f(z) = \sum_{n=0}^\infty a_n z^n$$ is a power series with radius of convergence exactly 1.

### 9.1.1 a $$\done$$

Give an example of such an $$f$$ that converges at every point of $$S^1$$.

### 9.1.2 b $$\work$$

Give an example of such an $$f$$ which is analytic at $$1$$ but $$\sum_{n=0}^\infty a_n$$ diverges.

### 9.1.3 c $$\work$$

Prove that $$f$$ can not be analytic at every point of $$S^1$$.

(Click to expand)

### 9.1.4 a $$\done$$

• Take $$\sum {z^n \over n^2}$$
• Then \begin{align*}{\left\lvert {z} \right\rvert}\leq 1 \implies {\left\lvert {z^n\over n^2} \right\rvert} \leq {1\over n^2}\end{align*} which is summable
• So the series converges for $${\left\lvert {z} \right\rvert}\leq 1$$.

### 9.1.5 b $$\work$$

• Take $$\sum {z^n \over n}$$;

• Then $$z=1$$ yields the harmonic series, which diverges.

• For $$z\in S^1\setminus\left\{{1}\right\}$$, we have $$z = e^{2\pi it}$$ for $$0<t<2\pi$$.

• So fix $$t$$.

• Toward applying the Dirichlet test, set $$a_n = 1/n, b_n = z^n$$.

• Then for all $$N$$, \begin{align*} {\left\lvert {\sum_{n=1}^N b_n} \right\rvert} = {\left\lvert {\sum_{n=1}^N b_n} \right\rvert} = {\left\lvert {\sum_{n=1}^N z^n} \right\rvert} = {\left\lvert { {z-z^{N+1} \over {\left\lvert {1 - z} \right\rvert}} } \right\rvert} \leq {2 \over 1-z} < \infty .\end{align*}

• Thus $$\sum a_n b_n < \infty$$ and $$\sum z^n/n$$ converges.

1. ?

## 9.2 2 $$\work$$

Suppose $$f$$ is entire and has Taylor series $$\sum a_n z^n$$ about 0.

### 9.2.1 a $$\work$$

Express $$a_n$$ as a contour integral along the circle $${\left\lvert {z} \right\rvert} = R$$.

### 9.2.2 b $$\work$$

Apply (a) to show that the above Taylor series converges uniformly on every bounded subset of $${\mathbb{C}}$$.

### 9.2.3 c $$\work$$

Determine those functions $$f$$ for which the above Taylor series converges uniformly on all of $${\mathbb{C}}$$.

## 9.3 3 $$\work$$

Suppose $$D$$ is a domain and $$f, g$$ are analytic on $$D$$.

Prove that if $$fg = 0$$ on $$D$$, then either $$f \equiv 0$$ or $$g\equiv 0$$ on $$D$$.

## 9.4 4 $$\work$$

Suppose $$f$$ is analytic on $${\mathbb{D}}^\circ$$. Determine with proof which of the following are possible:

1. $$f\qty{1\over n} = (-1)^n$$ for each $$n>1$$.

2. $$f\qty{1\over n} = e^{-n}$$ for each even integer $$n>1$$ while $$f\qty{1\over n} = 0$$ for each odd integer $$n>1$$.

3. $$f\qty{1\over n^2} = {1\over n}$$ for each integer $$n>1$$.

4. $$f\qty{1\over n} = {n-2 \over n-1}$$ for each integer $$n>1$$.

## 9.5 5 $$\done$$

Prove the Fundamental Theorem of Algebra (using complex analysis).

(Click to expand)
• Strategy: By contradiction with Liouville’s Theorem
• Suppose $$p$$ is non-constant and has no roots.
• Claim: $$1/p(z)$$ is a bounded holomorphic function on $${\mathbb{C}}$$.
• Holomorphic: clear? Since $$p$$ has no roots.

• Bounded: for $$z\neq 0$$, write \begin{align*} \frac{P(z)}{z^{n}}=a_{n}+\left(\frac{a_{n-1}}{z}+\cdots+\frac{a_{0}}{z^{n}}\right) .\end{align*}

• The term in parentheses goes to 0 as $${\left\lvert {z} \right\rvert}\to \infty$$

• Thus there exists an $$R>0$$ such that \begin{align*} {\left\lvert {z} \right\rvert} > R \implies {\left\lvert {P(z) \over z^n} \right\rvert} \geq c \coloneqq{{\left\lvert {a_n} \right\rvert} \over 2} .\end{align*}

• So $$p$$ is bounded below when $${\left\lvert {z} \right\rvert} > R$$

• Since $$p$$ is continuous and has no roots in $${\left\lvert {z} \right\rvert} \leq R$$, it is bounded below when $${\left\lvert {z} \right\rvert} \leq R$$.

• Thus $$p$$ is bounded below on $${\mathbb{C}}$$ and thus $$1/p$$ is bounded above on $${\mathbb{C}}$$.

• By Liouville’s theorem, $$1/p$$ is constant and thus $$p$$ is constant, a contradiction.

## 9.6 6 $$\done$$

Find all entire functions that satisfy \begin{align*} {\left\lvert {f(z)} \right\rvert} \geq {\left\lvert {z} \right\rvert} \quad \forall z\in {\mathbb{C}} .\end{align*} Prove this list is complete.

(Click to expand)
• Suppose $$f$$ is entire and define $$g(z) \mathrel{\vcenter{:}}={z \over f(z)}$$.
• By the inequality, $${\left\lvert {g(z)} \right\rvert} \leq 1$$, so $$g$$ is bounded.
• $$g$$ potentially has singularities at the zeros $$Z_f \mathrel{\vcenter{:}}= f^{-1}(0)$$, but since $$f$$ is entire, $$g$$ is holomorphic on $${\mathbb{C}}\setminus Z_f$$.
• Claim: $$Z_f = \left\{{0}\right\}$$.
• If $$f(z) = 0$$, then $${\left\lvert {z} \right\rvert} \leq {\left\lvert {f(z)} \right\rvert} = 0$$ which forces $$z=0$$.
• We can now apply Riemann’s removable singularity theorem:
• Check $$g$$ is bounded on some open subset $$D\setminus\left\{{0}\right\}$$, clear since it’s bounded everywhere
• Check $$g$$ is holomorphic on $$D\setminus\left\{{0}\right\}$$, clear since the only singularity of $$g$$ is $$z=0$$.
• By Riemann’s removable singularity theorem, the singularity $$z = 0$$ is removable and $$g$$ has an extension to an entire function $$\tilde g$$.
• By continuity, we have $${\left\lvert {\tilde g(z)} \right\rvert} \leq 1$$ on all of $${\mathbb{C}}$$
• If not, then $${\left\lvert {\tilde g(0)} \right\rvert} = 1+\varepsilon> 1$$, but then there would be a domain $$\Omega \subseteq {\mathbb{C}}\setminus\left\{{0}\right\}$$ such that $$1 < {\left\lvert {\tilde g(z)} \right\rvert} \leq 1 +\varepsilon$$ on $$\Omega$$, a contradiction.
• By Liouville, $$\tilde g$$ is constant, so $$\tilde g(z) = c_0$$ with $${\left\lvert {c_0} \right\rvert} \leq 1$$
• Thus $$f(z) = c_0^{-1}z \mathrel{\vcenter{:}}= cz$$ where $${\left\lvert {c} \right\rvert}\geq 1$$

Thus all such functions are of the form $$f(z) = cz$$ for some $$c\in {\mathbb{C}}$$ with $${\left\lvert {c} \right\rvert}\geq 1$$.

## 9.7 7 $$\work$$

Suppose $$\sum_{n=0}^\infty a_n z^n$$ converges for some $$z_0 \neq 0$$.

### 9.7.1 a $$\work$$

Prove that the series converges absolutely for each $$z$$ with $${\left\lvert {z} \right\rvert} < {\left\lvert {z} \right\rvert}_0$$.

### 9.7.2 b $$\work$$

Suppose $$0 < r < {\left\lvert {z_0} \right\rvert}$$ and show that the series converges uniformly on $${\left\lvert {z} \right\rvert} \leq r$$.

## 9.8 8 $$\work$$

Suppose $$f$$ is entire and suppose that for some integer $$n\geq 1$$, \begin{align*} \lim_{z\to \infty} {f(z) \over z^n} = 0 .\end{align*}

Prove that $$f$$ is a polynomial of degree at most $$n-1$$.

## 9.9 9 $$\work$$

Find all entire functions satisfying \begin{align*} {\left\lvert {f(z)} \right\rvert} \leq {\left\lvert {z} \right\rvert}^{1\over 2} \quad\text{ for } {\left\lvert {z} \right\rvert} > 10 .\end{align*}

## 9.10 10 $$\work$$

Prove that the following series converges uniformly on the set $$\left\{{z {~\mathrel{\Big|}~}\Im(z) < \ln 2}\right\}$$: \begin{align*} \sum_{n=1}^\infty {\sin(nz) \over 2^n} .\end{align*}

# 10 Extra

## 10.1 ?

Let $$f(z)$$ be entire and assume values of $$f(z)$$ lie outside a bounded open set $$\Omega$$. Show without using Picard’s theorems that $$f(z)$$ is a constant.

Let $$f(z)$$ be entire and assume values of $$f(z)$$ lie outside a bounded open set $$\Omega$$.

Show without using Picard’s theorems that $$f(z)$$ is a constant.

## 10.2 ?

### 10.2.1 1

Assume $$\displaystyle f(z) = \sum_{n=0}^\infty c_n z^n$$ converges in $$|z| < R$$.

Show that for $$r <R$$, \begin{align*} \frac{1}{2 \pi} \int_0^{2 \pi} |f(r e^{i \theta})|^2 d \theta = \sum_{n=0}^\infty |c_n|^2 r^{2n} .\end{align*}

### 10.2.2 2

Deduce Liouville’s theorem from (1).

# 11 Laurent Expansions and Singularities (8155f)

## 11.1 1 $$\done$$

Find the Laurent expansion of \begin{align*} f(z) = {z + 1 \over z(z-1)} \end{align*}

(Click to expand)

Let $$f(z) = {z+1\over z(z-1)}$$.

About $$z=0$$:

\begin{align*} f(z) &= (z+1) \qty{- {1 \over z} + {1\over z-1} } \\ &= -(z+1) \qty{{1\over z} + \sum_{n=0}^\infty z^n } \\ &= -(z+1)\sum_{n=-1}^\infty z^n \\ &= {1\over z} + 2\sum_{n=0}^\infty z^n \\ &= -{1\over z} -2 - 2z - 2z^2 - \cdots .\end{align*}

About $$z=1$$:

\begin{align*} f(z) &= \qty{(1-z) -2 \over 1-z} \qty{1 \over 1 - (1-z)} \\ &= \qty{1 - {2\over 1-z}} \sum_{n=0}^\infty (1-z)^n \\ &= \sum_{n=0}^\infty (1-z)^n - 2 \sum_{n=-1}^\infty (1-z)^n \\ &= -{2\over 1-z} - \sum_{n=0}^\infty (1-z)^n \\ &= {2\over z-1} + \sum_{n=0}^\infty (-1)^{n+1} (z-1)^n \\ &= {2\over z-1} - 1 + (z-1) - (z-1)^2 + \cdots .\end{align*}

about $$z=0$$ and $$z=1$$ respectively.

## 11.2 2 $$\done$$

Find the Laurent expansions about $$z=0$$ of the following functions: \begin{align*} \exp{1\over z} \hspace{8em} \cos \qty{1\over z} .\end{align*}

(Click to expand)

Let $$f(z) = {z+1\over z(z-1)}$$.

About $$z=0$$:

\begin{align*} f(z) &= (z+1) \qty{- {1 \over z} + {1\over z-1} } \\ &= -(z+1) \qty{{1\over z} + \sum_{n=0}^\infty z^n } \\ &= -(z+1)\sum_{n=-1}^\infty z^n \\ &= {1\over z} + 2\sum_{n=0}^\infty z^n \\ &= -{1\over z} -2 - 2z - 2z^2 - \cdots .\end{align*}

About $$z=1$$:

\begin{align*} f(z) &= \qty{(1-z) -2 \over 1-z} \qty{1 \over 1 - (1-z)} \\ &= \qty{1 - {2\over 1-z}} \sum_{n=0}^\infty (1-z)^n \\ &= \sum_{n=0}^\infty (1-z)^n - 2 \sum_{n=-1}^\infty (1-z)^n \\ &= -{2\over 1-z} - \sum_{n=0}^\infty (1-z)^n \\ &= {2\over z-1} + \sum_{n=0}^\infty (-1)^{n+1} (z-1)^n \\ &= {2\over z-1} - 1 + (z-1) - (z-1)^2 + \cdots .\end{align*}

## 11.3 3 $$\work$$

Find the Laurent expansion of \begin{align*} f(z) = {z+1 \over z(z-1)^2} \end{align*} about $$z=0$$ and $$z=1$$ respectively.

Hint: recall that power series can be differentiated.

## 11.4 4 $$\work$$

For the following functions, find the Laurent series about $$0$$ and classify their singularities there: \begin{align*} {\sin^2(z) \over z} \\ z \exp{1\over z^2} \\ {1 \over z(4-z)} .\end{align*}

## 11.5 5 $$\work$$

Find all entire functions with have poles at $$\infty$$.

## 11.6 6 $$\work$$

Find all functions on the Riemann sphere that have a simple pole at $$z=2$$ and a double pole at $$z=\infty$$, but are analytic elsewhere.

## 11.7 7 $$\work$$

Let $$f$$ be entire, and discuss (with proofs and examples) the types of singularities $$f$$ might have (removable, pole, or essential) at $$z=\infty$$ in the following cases:

1. $$f$$ has at most finitely many zeros in $${\mathbb{C}}$$.
2. $$f$$ has infinitely many zeros in $${\mathbb{C}}$$.

## 11.8 8 $$\work$$

Define \begin{align*} f(z) &= {\pi^2 \over \sin^2 \qty{\pi z} } \\ g(z) &= \sum_{n\in {\mathbb{Z}}} {1\over (z-n)^2} .\end{align*}

1. Show that $$f$$ and $$g$$ have the same singularities in $${\mathbb{C}}$$.
2. Show that $$f$$ and $$g$$ have the same singular parts at each of their singularities.
3. Show that $$f, g$$ each have period one and approach zero uniformly on $$0\leq x \leq 1$$ as $${\left\lvert {y} \right\rvert}\to \infty$$.
4. Conclude that $$f = g$$.
(Click to expand)

Idea: show their $$f-g$$ is analytic by taking away all of the negative powers, and bounded by (c).

# 12 Residues (8155g)

## 12.1 1 $$\work$$

Calculate \begin{align*} \int_0^\infty {1 \over (1+z)^2 (z+9x^2)} \, dx .\end{align*}

## 12.2 2 $$\work$$

Let $$a>0$$ and calculate \begin{align*} \int_0^\infty {x\sin(x) \over x^2 + a^2} \,dx .\end{align*}

## 12.3 3 $$\work$$

Calculate \begin{align*} \int_0^\infty {\sqrt x \over (x+1)^2} \,dx .\end{align*}

## 12.4 4 $$\work$$

Calculate \begin{align*} \int_0^\infty {\cos(x) - \cos(4x) \over x^2} \, dx .\end{align*}

## 12.5 5 $$\work$$

Let $$a>0$$ and calculate \begin{align*} \int_0^\infty {x^2 \over (x^2 + a^2)^2} \, dx .\end{align*}

## 12.6 6 $$\work$$

Calculate \begin{align*} \int_0^\infty {\sin(x) \over x}\, dx .\end{align*}

## 12.7 7 $$\work$$

Calculate \begin{align*} \int_0^\infty {\sin(x) \over x(x^2+1)}\, dx .\end{align*}

## 12.8 8 $$\work$$

Calculate \begin{align*} \int_0^\infty {\sqrt x \over 1 + x^2} \, dx .\end{align*}

## 12.9 9 $$\work$$

Calculate \begin{align*} \int_{-\infty}^\infty {1+x^2 \over 1+x^4}\, dx .\end{align*}

## 12.10 10 $$\work$$

Let $$a>0$$ and calculate \begin{align*} \int_0^\infty {\cos(x) \over (x^2 + a^2)^2}\, dx .\end{align*}

## 12.11 11 $$\work$$

Calculate \begin{align*} \int_0^\infty {\sin^3(x) \over x^3} \, dx .\end{align*}

## 12.12 12 $$\work$$

Let $$n\in {\mathbb{Z}}^{\geq 1}$$ and $$0<\theta<\pi$$ and show that \begin{align*} {1\over 2\pi i} \int_{{\left\lvert {z} \right\rvert} = 2} {z^n \over 1 -3z\cos(\theta) + z^2} \,dz = {\sin(n\theta) \over \sin(\theta)} .\end{align*}

## 12.13 13 $$\work$$

Suppose $$a>b>0$$ and calculate \begin{align*} \int_0^{2\pi} {1 \over (a+b\cos(\theta))^2} \,d\theta .\end{align*}

# 13 Residue Theorem: Extra Questions

## 13.1 ?

Suppose that $$f$$ is an analytic function in the region $$D$$ which contains the point $$a$$. Let \begin{align*}F(z)= z-a-qf(z),\quad \text{where}\quad q \ \text{is a complex parameter}.\end{align*}

### 13.1.1 1

Let $$K\subset D$$ be a circle with the center at point $$a$$ and also we assume that $$f(z)\not =0$$ for $$z\in K$$. Prove that the function $$F$$ has one and only one zero $$z=w$$ on the closed disc $$\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu$$ whose boundary is the circle $$K$$ if $$\displaystyle{ |q|<\min_{z\in K} \frac{|z-a|}{|f(z)|}.}$$

### 13.1.2 2

Let $$G(z)$$ be an analytic function on the disk $$\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu$$. Apply the residue theorem to prove that \begin{align*} \displaystyle{ \frac{G(w)}{F'(w)}=\frac{1}{2\pi i}\int_K \frac{G(z)}{F(z)} dz,} \end{align*} where $$w$$ is the zero from (1).

## 13.2 ?

Evaluate \begin{align*} \displaystyle{ \int_{0}^{\infty}\frac{x\sin x}{x^2+a^2} \, dx } .\end{align*}

## 13.3 ?

Show that \begin{align*} \displaystyle \int_0^\infty \frac{x^{a-1}}{1+x^n} dx=\frac{\pi}{n\sin \frac{a\pi}{n}} \end{align*} using complex analysis, $$0< a < n$$. Here $$n$$ is a positive integer.

# 14 Rouche’s Theorem (8155h)

## 14.1 1 $$\done$$

Prove that for every $$n\in {\mathbb{Z}}^{\geq 0}$$ the following polynomial has no roots in the open unit disc: \begin{align*} f_n(x) \mathrel{\vcenter{:}}=\sum_{k=0}^n {z^k \over k!} .\end{align*}

Hint: check $$n=1,2$$ directly.

(Click to expand)

Note

• $$f_1(z) = 1+z$$, which has the single root $$z=-1$$ which is not inside $${\left\lvert {z} \right\rvert} < 1$$.

• $$f_2(z) = 1 + z + {1\over 2}z^2 = (z - (1+i))(z- (1-i))$$, and $${\left\lvert {1\pm i} \right\rvert} = \sqrt 2 >1$$.

• Note that $$p_n(z) \overset{n\to\infty}e^z$$ uniformly on any compact set.

• Let $$r$$ be arbitrary and fix $$N \mathrel{\vcenter{:}}={\mathbb{D}}_r(0)$$, then $$p_n(z) \to e^z$$ uniformly on $$\mkern 1.5mu\overline{\mkern-1.5muN\mkern-1.5mu}\mkern 1.5mu$$.

• Set $$g_n(z) \mathrel{\vcenter{:}}= p_n(z) / e^z$$, then $$g_n \to 1$$ uniformly on $$\mkern 1.5mu\overline{\mkern-1.5muN\mkern-1.5mu}\mkern 1.5mu$$.

• Choose $$n\gg 0$$ so that $${\left\lvert {f(z) - 1} \right\rvert} < \varepsilon< 1$$ for all $$z\in \mkern 1.5mu\overline{\mkern-1.5muN\mkern-1.5mu}\mkern 1.5mu$$.

• So take $$h(z) = 1$$, then on $${{\partial}}N$$,?

## 14.2 2 $$\done$$

Assume that $${\left\lvert {b} \right\rvert} < 1$$ and show that the following polynomial has exactly two roots (counting multiplicity) in $${\left\lvert {z} \right\rvert} < 1$$: \begin{align*} f(z) \mathrel{\vcenter{:}}= z^3 + 3z^2 + bz + b^2 .\end{align*}

(Click to expand)

Multiple versions of Rouches theorem!

• Set $$h(z) = 3z^2$$ and $$g(z) = z^3 + bz + b^2$$.

• Then on $${\left\lvert {z} \right\rvert} = 1$$, \begin{align*} {\left\lvert {g(z)} \right\rvert} \leq 1 + b + b^2 < 3 = 3{\left\lvert {z} \right\rvert}^2 = {\left\lvert {3z^2} \right\rvert} = {\left\lvert {h} \right\rvert} ,\end{align*} so $$g, h$$ have the same number of roots in $${\left\lvert {z} \right\rvert} \leq_? 1$$.

• But $$h$$ evidently has two roots in this region.

## 14.3 3 $$\work$$

Let $$c\in {\mathbb{C}}$$ with $${\left\lvert {c} \right\rvert} < {1\over 3}$$. Show that on the open set $$\left\{{z\in {\mathbb{C}}{~\mathrel{\Big|}~}\Re(z) < 1}\right\}$$, the function $$f(z) \mathrel{\vcenter{:}}= ce^z$$ has exactly one fixed point.

## 14.4 4 $$\done$$

How many roots does the following polynomial have in the open disc $${\left\lvert {z} \right\rvert} < 1$$? \begin{align*} f(z) = z^7 - 4z^3 - 1 .\end{align*}

(Click to expand)
• Set $$h(z) = -4z^3$$ and $$g(z) = z^7 - 1$$, then on $${\left\lvert {z} \right\rvert} = 1$$, \begin{align*} {\left\lvert {g(z)} \right\rvert} = {\left\lvert {z^7 - 1} \right\rvert} \leq 1 + 1 = 2 < 4 = {\left\lvert {-4z^3} \right\rvert} = {\left\lvert {h(z)} \right\rvert} .\end{align*}

• So $$h$$ and $$h+g$$ have the same number of roots, but $$h$$ has three roots here.

## 14.5 5 $$\done$$

Let $$n\in {\mathbb{Z}}^{\geq 0}$$ and show that the equation \begin{align*} e^z = az^n \end{align*} has $$n$$ solutions in the open unit disc if $${\left\lvert {a} \right\rvert} > e$$, and no solutions if $${\left\lvert {a} \right\rvert} < {1\over e}$$.

## 14.6 6 $$\work$$

Let $$f$$ be analytic in a domain $$D$$ and fix $$z_0 \in D$$ with $$w_0 \mathrel{\vcenter{:}}= f(z_0)$$. Suppose $$z_0$$ is a zero of $$f(z) - w_0$$ with finite multiplicity $$m$$. Show that there exists $$\delta >0$$ and $$\varepsilon> 0$$ such that for each $$w$$ such that $$0 < {\left\lvert {w-w_0} \right\rvert} < \varepsilon$$, the equation $$f(z) - w = 0$$ has exactly $$m$$ distinct solutions inside the disc $${\left\lvert {z-z_0} \right\rvert} < \delta$$.

## 14.7 7 $$\work$$

For $$k=1,2,\cdots, n$$, suppose $${\left\lvert {a_k} \right\rvert} < 1$$ and \begin{align*} f(z) \mathrel{\vcenter{:}}=\qty{z - a_1 \over 1 - \mkern 1.5mu\overline{\mkern-1.5mua\mkern-1.5mu}\mkern 1.5mu_q z} \qty{z-a_2 \over 1 - \mkern 1.5mu\overline{\mkern-1.5mua\mkern-1.5mu}\mkern 1.5mu_2 z} \cdots \qty{z - a_n \over 1 - \mkern 1.5mu\overline{\mkern-1.5mua\mkern-1.5mu}\mkern 1.5mu_n z} .\end{align*} Show that $$f(z) = b$$ has $$n$$ solutions in $${\left\lvert {z} \right\rvert} < 1$$.

## 14.8 8 $$\work$$

For each $$n\in {\mathbb{Z}}^{\geq 1}$$, let \begin{align*} P_n(z) = 1 + z + {1\over 2!} z^2 + \cdots + {1\over n!}z^n .\end{align*} Show that for sufficiently large $$n$$, the polynomial $$P_n$$ has no zeros in $${\left\lvert {z} \right\rvert} < 10$$, while the polynomial $$P_n(z) - 1$$ has precisely 3 zeros there.

## 14.9 9 $$\work$$

Prove that \begin{align*} \max_{{\left\lvert {z} \right\rvert} = 1} {\left\lvert {a_0 + a_1 z + \cdots + a_{n-1}z^{n-1} + z^n} \right\rvert} \geq 1 .\end{align*}

Hint: the first part of the problem asks for a statement of Rouche’s theorem.

## 14.10 10 $$\work$$

Use Rouche’s theorem to prove the Fundamental Theorem of Algebra.

# 15 Extras

## 15.1 ?

Apply Rouché’s Theorem to prove the Fundamental Theorem of Algebra:

If \begin{align*} P_n(z) = a_0 + a_1z + \cdots + a_{n-1}z^{n-1} + a_nz^n\quad (a_n \neq 0) \end{align*} is a polynomial of degree $$n$$, then it has $$n$$ zeros in $$\mathbb{C}$$.

## 15.2 ?

Suppose $$f$$ is entire and there exist $$A, R >0$$ and natural number $$N$$ such that \begin{align*}|f(z)| \geq A |z|^N\ \text{for}\ |z| \geq R.\end{align*}

Show that (i) $$f$$ is a polynomial and (ii) the degree of $$f$$ is at least $$N$$.

# 16 Schwarz Lemma and Reflection Principle (8155i)

## 16.1 1 $$\work$$

Suppose $$f:{\mathbb{D}}\to{\mathbb{D}}$$ is analytic and admits a continuous extension $$\tilde f: \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{D}}\mkern-1.5mu}\mkern 1.5mu\to \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{D}}\mkern-1.5mu}\mkern 1.5mu$$ such that $${\left\lvert {z} \right\rvert} = 1 \implies {\left\lvert {f(z)} \right\rvert} = 1$$.

### 16.1.1 a $$\work$$

Prove that $$f$$ is a rational function.

### 16.1.2 b $$\work$$

Suppose that $$z=0$$ is the unique zero of $$f$$. Show that \begin{align*} \exists n\in {\mathbb{N}}, \lambda \in S^1 {\quad \operatorname{ such that } \quad}f(z) = \lambda z^n .\end{align*}

### 16.1.3 c $$\work$$

Suppose that $$a_1, \cdots, a_n \in {\mathbb{D}}$$ are the zeros of $$f$$ and prove that \begin{align*} \exists \lambda \in S^1 {\quad \operatorname{such that} \quad} f(z) = \lambda \prod_{j=1}^n {z - a_j \over 1 - \mkern 1.5mu\overline{\mkern-1.5mua_j\mkern-1.5mu}\mkern 1.5mu z} .\end{align*}

## 16.2 2 $$\work$$

Let $$\mkern 1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu(a, r)$$ denote the closed disc of radius $$r$$ about $$a\in {\mathbb{C}}$$. Let $$f$$ be holomorphic on an open set containing $$\mkern 1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu(a, r)$$ and let \begin{align*} M \mathrel{\vcenter{:}}=\sup_{z\in \mkern 1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu(a, r)} {\left\lvert {f(z)} \right\rvert} .\end{align*}

Prove that \begin{align*} z\in \mkern 1.5mu\overline{\mkern-1.5muB\mkern-1.5mu}\mkern 1.5mu\qty{a, {r\over 2}},\,z\neq a, \qquad {{\left\lvert { f(z) - f(a)} \right\rvert} \over {\left\lvert {z-a} \right\rvert}} \leq {2M \over r} .\end{align*}

## 16.3 3 $$\work$$

Define \begin{align*} G \mathrel{\vcenter{:}}=\left\{{z\in {\mathbb{C}}{~\mathrel{\Big|}~}\Re(z) > 0, \, {\left\lvert {z-1} \right\rvert} > 1}\right\} .\end{align*}

Find all of the injective conformal maps $$G\to {\mathbb{D}}$$. These may be expressed as compositions of maps, but explain why this list is complete.

## 16.4 4 $$\work$$

Suppose $$f: {\mathbb{H}}\cup{\mathbb{R}}\to {\mathbb{C}}$$ satisfies the following:

• $$f(i) = i$$
• $$f$$ is continuous
• $$f$$ is analytic on $${\mathbb{H}}$$
• $$f(z) \in {\mathbb{R}}\iff z\in {\mathbb{R}}$$.

Show that $$f({\mathbb{H}})$$ is a dense subset of $${\mathbb{H}}$$.

## 16.5 5 $$\work$$

Suppose $$f: {\mathbb{D}}\to {\mathbb{H}}$$ is analytic and satisfies $$f(0) = 2$$. Find a sharp upper bound for $${\left\lvert {f'(0)} \right\rvert}$$, and prove it is sharp by example.

## 16.6 6 $$\work$$

Suppose $$f:{\mathbb{D}}\to{\mathbb{D}}$$ is analytic, has a single zero of order $$k$$ at $$z=0$$, and satsifies $$\lim_{{\left\lvert {z} \right\rvert} \to 1} {\left\lvert {f(z)} \right\rvert} = 1$$. Give with proof a formula for $$f(z)$$.

## 16.7 7 $$\work$$

### 16.7.1 a $$\work$$

State the standard Schwarz reflection principle involving reflection across the real axis.

### 16.7.2 b $$\work$$

Give a linear fractional transformation $$T$$ mapping $${\mathbb{D}}$$ to $${\mathbb{H}}$$. Let $$g(z) = \mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu$$, and show \begin{align*} (T^{-1} \circ g \circ T)(z) = 1/\mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu .\end{align*}

### 16.7.3 c $$\work$$

Suppose that $$f$$ is holomorphic on $${\mathbb{D}}$$, continuous on $$\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{D}}\mkern-1.5mu}\mkern 1.5mu$$, and real on $$S^1$$. Show that $$f$$ must be constant.

## 16.8 8 $$\work$$

Suppose $$f, g: {\mathbb{D}}\to \Omega$$ are holomorphic with $$f$$ injective and $$f(0) = g(0)$$.

Show that \begin{align*} \mathop{\mathrm{\forall}}0 < r < 1,\qquad g\qty{\left\{{{\left\lvert {z} \right\rvert} < r}\right\}} \subseteq f\qty{\left\{{{\left\lvert {z} \right\rvert} < r}\right\}} .\end{align*}

The first part of this problem asks for a statement of the Schwarz lemma.

## 16.9 9 $$\work$$

Let $$S\mathrel{\vcenter{:}}=\left\{{z\in {\mathbb{D}}{~\mathrel{\Big|}~}\Im(z) \geq 0}\right\}$$. Suppose $$f:S\to {\mathbb{C}}$$ is continuous on $$S$$, real on $$S\cap{\mathbb{R}}$$, and holomorphic on $$S^\circ$$.

Prove that $$f$$ is the restriction of a holomorphic function on $${\mathbb{D}}$$.

## 16.10 10 $$\work$$

Suppose $$f:{\mathbb{D}}\to {\mathbb{D}}$$ is analytic. Prove that \begin{align*} \forall a\in {\mathbb{D}}, \qquad {{\left\lvert {f'(a)} \right\rvert} \over 1 - {\left\lvert {f(a)} \right\rvert}^2 } \leq {1 \over 1 - {\left\lvert {a} \right\rvert}^2} .\end{align*}

# 17 Unsorted/Unknown

## 17.2 ?

Let $$f$$ be a continuous function in the region \begin{align*} D=\{z\ | |z|>R, 0\leq \arg Z\leq \theta\}\quad\text{where}\quad 0\leq \theta \leq 2\pi .\end{align*} If there exists $$k$$ such that $$\displaystyle{\lim_{z\to\infty} zf(z)=k}$$ for $$z$$ in the region $$D$$.

Show that \begin{align*}\lim_{R'\to\infty} \int_{L} f(z) dz=i\theta k,\end{align*} where $$L$$ is the part of the circle $$|z|=R'$$ which lies in the region $$D$$.

### 17.2.1 3

If $$z\in K$$, prove that the function $$\displaystyle{\frac{1}{F(z)}}$$ can be represented as a convergent series with respect to $$q$$: \begin{align*} \displaystyle{ \frac{1}{F(z)}=\sum_{n=0}^{\infty} \frac{(qf(z))^n}{(z-a)^{n+1}}.}\end{align*}

## 17.3 ?

Show that \begin{align*} \displaystyle \int_0^\infty \frac{x^{a-1}}{1+x^n} dx=\frac{\pi}{n\sin \frac{a\pi}{n}} \end{align*} using complex analysis, $$0< a < n$$.

Here $$n$$ is a positive integer.

## 17.4 11.

Let $$g$$ be analytic for $$|z|\leq 1$$ and $$|g(z)| < 1$$ for $$|z| = 1$$.

### 17.4.1 a

Show that $$g$$ has a unique fixed point in $$|z| < 1$$.

### 17.4.2 b

What happens if we replace $$|g(z)| < 1$$ with $$|g(z)|\leq 1$$ for $$|z|=1$$?

Give an example if (a) is not true or give an proof if (a) is still true.

### 17.4.3 c

What happens if we simply assume that $$f$$ is analytic for $$|z| < 1$$ and $$|f(z)| < 1$$ for $$|z| < 1$$? Suppose that $$f(z)\not\equiv z$$.

Can $$f$$ have more than one fixed point in$$|z| < 1$$?

Hint: The map \begin{align*}\displaystyle{\psi_{\alpha}(z)=\frac{\alpha-z}{1-\mkern 1.5mu\overline{\mkern-1.5mu\alpha\mkern-1.5mu}\mkern 1.5muz}}\end{align*} may be useful.

## 17.5 ?

Let $$f(z)$$ be entire and assume that $$f(z) \leq M |z|^2$$ outside some disk for some constant $$M$$.

Show that $$f(z)$$ is a polynomial in $$z$$ of degree $$\leq 2$$.

## 17.6 ?

Let $$a_n(z)$$ be an analytic sequence in a domain $$D$$ such that $$\displaystyle \sum_{n=0}^\infty |a_n(z)|$$ converges uniformly on bounded and closed sub-regions of $$D$$.

Show that $$\displaystyle \sum_{n=0}^\infty |a'_n(z)|$$ converges uniformly on bounded and closed sub-regions of $$D$$.

## 17.7 16

Let $$f(z)$$ be analytic in an open set $$\Omega$$ except possibly at a point $$z_0$$ inside $$\Omega$$.

Show that if $$f(z)$$ is bounded in near $$z_0$$, then $$\displaystyle \int_\Delta f(z) dz = 0$$ for all triangles $$\Delta$$ in $$\Omega$$.

# 18 Riemann Mapping and Casorati-Weierstrass

## 18.1 10.

Let $$f: {\mathbb C} \rightarrow {\mathbb C}$$ be an injective analytic (also called univalent) function. Show that there exist complex numbers $$a \neq 0$$ and $$b$$ such that $$f(z) = az + b$$.

# 19 Spring 2020 Homework 1

## 19.1 1

Geometrically describe the following subsets of $${\mathbb{C}}$$:

1. $${\left\lvert {z-1} \right\rvert} = 1$$
2. $${\left\lvert {z-1} \right\rvert} = 2{\left\lvert {z-2} \right\rvert}$$
3. $$1/z = \mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu$$
4. $$\Re(z) = 3$$
5. $$\Im(z) = a$$ with $$a\in {\mathbb{R}}$$.
6. $$\Re(z) > a$$ with $$a\in {\mathbb{R}}$$.
7. $${\left\lvert {z-1} \right\rvert} < 2{\left\lvert {z-2} \right\rvert}$$

## 19.2 2

Prove the following inequality, and explain when equality holds: \begin{align*} {\left\lvert {z+w} \right\rvert} \geq {\left\lvert { {\left\lvert {z} \right\rvert} - {\left\lvert {w} \right\rvert} } \right\rvert} .\end{align*}

## 19.3 3

Prove that the following polynomial has its roots outside of the unit circle: \begin{align*} p(z) = z^3 + 2z + 4 .\end{align*}

Hint: What is the maximum value of the modulus of the first two terms if $${\left\lvert {z} \right\rvert} \leq 1$$?

## 19.4 4

1. Prove that if $$c>0$$, \begin{align*} {\left\lvert {w_1} \right\rvert} = c{\left\lvert {w_2} \right\rvert} \implies {\left\lvert {w_1 - c^2 w_2} \right\rvert} = c{\left\lvert {w_1 - w_2} \right\rvert} .\end{align*}

2. Prove that if $$c>0$$ and $$c\neq 1$$, with $$z_1\neq z_2$$, then the following equation represents a circle: \begin{align*} {\left\lvert {z-z_1 \over z-z_2} \right\rvert} = c .\end{align*} Find its center and radius.

Hint: use part (a)

## 19.5 5

1. Let $$z, w \in {\mathbb{C}}$$ with $$\mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu w \neq 1$$. Prove that \begin{align*} {\left\lvert {w-z \over 1 - \mkern 1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu z} \right\rvert} < 1 \quad\text{ if } {\left\lvert {z} \right\rvert}<1,~ {\left\lvert {w} \right\rvert} < 1 \end{align*} with equality when $${\left\lvert {z} \right\rvert} = 1$$ or $${\left\lvert {w} \right\rvert} = 1$$.

2. Prove that for a fixed $$w\in {\mathbb{D}}$$, the mapping $$F: z\mapsto {w-z \over 1 - \mkern 1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu z}$$ satisfies

• $$F$$ maps $${\mathbb{D}}$$ to itself and is holomorphic.
• $$F(0) = w$$ and $$F(w) = 0$$.
• $${\left\lvert {z} \right\rvert} = 1$$ implies $${\left\lvert {F(z)} \right\rvert} = 1$$.

## 19.6 6

Use $$n$$th roots of unity to show that \begin{align*} 2^{n-1} \sin\qty{\pi \over n} \sin\qty{2\pi \over n} \cdots \sin\qty{(n-1)\pi \over n} = n .\end{align*}

Hint: \begin{align*} 1 - \cos(2\theta) &= 2\sin^2(\theta) \\ 2 \sin(2\theta) &= 2\sin(\theta) \cos(\theta) .\end{align*}

## 19.7 7

Prove that $$f(z) = {\left\lvert {z} \right\rvert}^2$$ has a derivative at $$z=0$$ and nowhere else.

## 19.8 8

Let $$f(z)$$ be analytic in a domain, and prove that $$f$$ is constant if it satisfies any of the following conditions:

1. $${\left\lvert {f(z)} \right\rvert}$$ is constant.
2. $$\Re(f(z))$$ is constant.
3. $$\arg(f(z))$$ is constant.
4. $$\mkern 1.5mu\overline{\mkern-1.5muf(z)\mkern-1.5mu}\mkern 1.5mu$$ is analytic.

How do you generalize (a) and (b)?

## 19.9 9

Prove that if $$z\mapsto f(z)$$ is analytic, then $$z \mapsto \mkern 1.5mu\overline{\mkern-1.5muf(\mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu)\mkern-1.5mu}\mkern 1.5mu$$ is analytic.

## 19.10 10

1. Show that in polar coordinates, the Cauchy-Riemann equations take the form \begin{align*} \frac{\partial u}{\partial r}=\frac{1}{r} \frac{\partial v}{\partial \theta} \text { and } \frac{\partial v}{\partial r}=-\frac{1}{r} \frac{\partial u}{\partial \theta} .\end{align*}

2. Use (a) to show that the logarithm function, defined as \begin{align*} \log z=\log r+i \theta \text { where } z=r e^{i \theta} \text { with }-\pi<\theta<\pi .\end{align*} is holomorphic on the region $$r> 0, -\pi < \theta < \pi$$.

Also show that this function is not continuous in $$r>0$$.

## 19.11 11

Prove that the distinct complex numbers $$z_1, z_2, z_3$$ are the vertices of an equilateral triangle if and only if \begin{align*} z_{1}^{2}+z_{2}^{2}+z_{3}^{2}=z_{1} z_{2}+z_{2} z_{3}+z_{3} z_{1} .\end{align*}

# 20 Spring 2020 Homework 2

Note on notation: I sometimes use $$f_x \mathrel{\vcenter{:}}={\frac{\partial f}{\partial x}\,}$$ to denote partial derivatives, and $${{\partial}}_z^n f$$ as $$f^{(n)}(z)$$.

## 20.1 Stein And Shakarchi

### 20.1.1 2.6.1

Show that \begin{align*} \int_{0}^{\infty} \sin \left(x^{2}\right) d x=\int_{0}^{\infty} \cos \left(x^{2}\right) d x=\frac{\sqrt{2 \pi}}{4} .\end{align*}

Hint: integrate $$e^{-x^2}$$ over the following contour, using the fact that $$\int_{-\infty}^{\infty} e^{-x^{2}} d x=\sqrt{\pi}$$:

### 20.1.2 2.6.2

Show that

\begin{align*} \int_{0}^{\infty} \frac{\sin x}{x} d x=\frac{\pi}{2} .\end{align*}

Hint: use the fact that this integral equals $$\frac{1}{2 i} \int_{-\infty}^{\infty} \frac{e^{i x}-1}{x} d x$$, and integrate around an indented semicircle.

### 20.1.3 2.6.5

Suppose $$f\in C_{\mathbb{C}}^1(\Omega)$$ and $$T\subset \Omega$$ is a triangle with $$T^\circ \subset \Omega$$. Apply Green’s theorem to show that $$\int_T f(z) ~dz = 0$$.

Assume that $$f'$$ is continuous and prove Goursat’s theorem.

Hint: Green’s theorem states

\begin{align*} \int_{T} F d x+G d y=\int_{T^\circ}\left(\frac{\partial G}{\partial x}-\frac{\partial F}{\partial y}\right) d x d y .\end{align*}

### 20.1.4 2.6.6

Suppose that $$f$$ is holomorphic on a punctured open set $$\Omega\setminus\left\{{w_0}\right\}$$ and let $$T\subset \Omega$$ be a triangle containing $$w_0$$. Prove that if $$f$$ is bounded near $$w_0$$, then $$\int_T f(z) ~dz = 0$$.

### 20.1.5 2.6.7

Suppose $$f: {\mathbb{D}}\to {\mathbb{C}}$$ is holomorphic and let $$d \mathrel{\vcenter{:}}=\sup_{z, w\in {\mathbb{D}}}{\left\lvert {f(z) - f(w)} \right\rvert}$$ be the diameter of the image of $$f$$. Show that $$2 {\left\lvert {f'(0)} \right\rvert} \leq d$$, and that equality holds iff $$f$$ is linear, so $$f(z) = a_1 z + a_2$$.

Hint: $$2f'(0) = \frac{1}{2\pi i} \int_{{\left\lvert {\xi } \right\rvert}= r} \frac{ f(\xi) - f(-\xi) }{\xi^2} ~d\xi$$ whenever $$0<r<1$$.

### 20.1.6 2.6.8

Suppose that $$f$$ is holomorphic on the strip $$S = \left\{{x+iy {~\mathrel{\Big|}~}x\in {\mathbb{R}},~ -1<y<1}\right\}$$ with $${\left\lvert {f(z)} \right\rvert} \leq A \qty{1 + {\left\lvert {z} \right\rvert}}^\nu$$ for $$\nu$$ some fixed real number. Show that for all $$z\in S$$, for each integer $$n\geq 0$$ there exists an $$A_n \geq 0$$ such that $${\left\lvert {f^{(n)}(x)} \right\rvert} \leq A_n (1 + {\left\lvert {x} \right\rvert})^\nu$$ for all $$x\in {\mathbb{R}}$$.

Hint: Use the Cauchy inequalities.

### 20.1.7 2.6.9

Let $$\Omega \subset {\mathbb{C}}$$ be open and bounded and $$\phi: \Omega \to \Omega$$ holomorphic. Prove that if there exists a point $$z_0 \in \Omega$$ such that $$\phi(z_0) = z_0$$ and $$\phi'(z_0) = 1$$, then $$\phi$$ is linear.

Hint: assume $$z_0 = 0$$ (explain why this can be done) and write $$\phi(z) = z + a_n z^n + O(z^{n+1})$$ near $$0$$. Let $$\phi_k = \phi \circ \phi \circ \cdots \circ \phi$$ and prove that $$\phi_k(z) = z + ka_nz^n + O(z^{n+1})$$. Apply Cauchy’s inequalities and let $$k\to \infty$$ to conclude.

### 20.1.8 2.6.10

Can every continuous function on $$\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{D}}\mkern-1.5mu}\mkern 1.5mu$$ be uniformly approximated by polynomials in the variable $$z$$?

Hint: compare to Weierstrass for the real interval.

### 20.1.9 2.6.13

Suppose $$f$$ is analytic, defined on all of $${\mathbb{C}}$$, and for each $$z_0 \in {\mathbb{C}}$$ there is at least one coefficient in the expansion $$f(z) = \sum_{n=0}^\infty c_n(z-z_0)^n$$ is zero. Prove that $$f$$ is a polynomial.

Hint: use the fact that $$c_n n! = f^{(n)}(z_0)$$ and use a countability argument.

### 20.1.10 2.6.14

Suppose that $$f$$ is holomorphic in an open set containing $${\mathbb{D}}$$ except for a pole $$z_0 \in {{\partial}}{\mathbb{D}}$$. Let $$\sum_{n=0}^\infty a_n z^n$$ be the power series expansion of $$f$$ in $${\mathbb{D}}$$, and show that $$\lim \frac{a_n}{a_{n+1}} = z_0$$.

### 20.1.11 2.6.15

Suppose $$f$$ is continuous and nonvanishing on $$\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{D}}\mkern-1.5mu}\mkern 1.5mu$$, and holomorphic in $${\mathbb{D}}$$. Prove that if $${\left\lvert {z} \right\rvert} = 1 \implies {\left\lvert {f(z)} \right\rvert} = 1$$, then $$f$$ is constant.

Hint: Extend $$f$$ to all of $${\mathbb{C}}$$ by $$f(z) = 1/ \mkern 1.5mu\overline{\mkern-1.5muf(1/\mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu)\mkern-1.5mu}\mkern 1.5mu$$ for any $${\left\lvert {z} \right\rvert} > 1$$, and argue as in the Schwarz reflection principle.

## 20.2 Additional Problems

### 20.2.1 1

Let $$a_n\neq 0$$ and show that \begin{align*} \lim_{n\to \infty} {{\left\lvert {a_{n+1}} \right\rvert} \over {\left\lvert {a_n} \right\rvert}} = L \implies \lim_{n\to\infty} {\left\lvert {a_n} \right\rvert}^{1\over n} = L .\end{align*}

In particular, this shows that when applicable, the ratio test can be used to calculate the radius of convergence of a power series.

### 20.2.2 2

Let $$f$$ be a power series centered at the origin. Prove that $$f$$ has a power series expansion about any point in its disc of convergence.

### 20.2.3 3

Prove the following:

1. $$\sum_{n} nz^n$$ does not converge at any point of $$S^1$$

2. $$\sum_n {z^n \over n^2}$$ converges at every point of $$S^1$$.

3. $$\sum_n {z^n \over n}$$ converges at every point of $$S^1$$ except $$z=1$$.

### 20.2.4 4

Without using Cauchy’s integral formula, show that if $${\left\lvert {a} \right\rvert} < r < {\left\lvert {b} \right\rvert}$$, then \begin{align*} \int_{\gamma} \frac{d z}{(z-\alpha)(z-\beta)}=\frac{2 \pi i}{\alpha-\beta} \end{align*} where $$\gamma$$ denotes the circle centered at the origin of radius $$r$$ with positive orientation.

### 20.2.5 5

Assume $$f$$ is continuous in the region $$\left\{{x+iy {~\mathrel{\Big|}~}x\geq x_0, ~ 0\leq y \leq b}\right\}$$, and the following limit exists independent of $$y$$: \begin{align*} \lim_{x\to +\infty}f(x+iy) = A .\end{align*}

Show that if $$\gamma_x \mathrel{\vcenter{:}}=\left\{{z = x+it {~\mathrel{\Big|}~}0 \leq t \leq b}\right\}$$, then \begin{align*} \lim_{x\to +\infty} \int_{\gamma_x} f(z) \,dz = iAb .\end{align*}

### 20.2.6 6

Show by example that there exists a function $$f(z)$$ that is holomorphic on $$\left\{{z\in {\mathbb{C}}{~\mathrel{\Big|}~}0 < {\left\lvert {z} \right\rvert} < 1}\right\}$$ and for all $$r<1$$, \begin{align*} \int_{{\left\lvert {z} \right\rvert} = r} f(z) \, dz = 0 ,\end{align*} but $$f$$ is not holomorphic at $$z=0$$.

### 20.2.7 7

Let $$f$$ be analytic on a region $$R$$ and suppose $$f'(z_0) \neq 0$$ for some $$z_0 \in R$$. Show that if $$C$$ is a circle of sufficiently small radius centered at $$z_0$$, then \begin{align*} \frac{2 \pi i}{f^{\prime}\left(z_{0}\right)}=\int_{C} \frac{d z}{f(z)-f\left(z_{0}\right)} .\end{align*}

Hint: use the inverse function theorem.

### 20.2.8 8

Assume two functions $$u, b: {\mathbb{R}}^2 \to {\mathbb{R}}$$ have continuous partial derivatives at $$(x_0 ,y_0)$$. Show that $$f \mathrel{\vcenter{:}}= u + iv$$ has derivative $$f'(z_0)$$ at $$z_0 = =x_0 + iy_0$$ if and only if \begin{align*} \lim _{r \rightarrow 0} \frac{1}{\pi r^{2}} \int_{\left|z-z_{0}\right|=r} f(z) d z=0 .\end{align*}

### 20.2.9 9 (Cauchy’s Formula for Exterior Regions)

Let $$\gamma$$ be a piecewise smooth simple closed curve with interior $$\Omega_1$$ and exterior $$\Omega_2$$. Assume $$f'$$ exists in an open set containing $$\gamma$$ and $$\Omega_2$$ with $$\lim_{z\to \infty} f(z) = A$$. Show that \begin{align*} \frac{1}{2 \pi i} \int_{\gamma} \frac{f(\xi)}{\xi-z} d \xi=\left\{\begin{array}{ll} A, & \text { if } z \in \Omega_{1} \\ -f(z)+A, & \text { if } z \in \Omega_{2} \end{array}\right. .\end{align*}

### 20.2.10 10

Let $$f(z)$$ be bounded and analytic in $${\mathbb{C}}$$. Let $$a\neq b$$ be any fixed complex numbers. Show that the following limit exists: \begin{align*} \lim_{R\to \infty} \int_{{\left\lvert {z} \right\rvert} = R} {f(z) \over (z-a)(z-b)} \,dz .\end{align*}

Use this to show that $$f(z)$$ must be constant.

### 20.2.11 11

Suppose $$f(z)$$ is entire and \begin{align*} \lim_{z\to\infty} {f(z) \over z} = 0 .\end{align*}

Show that $$f(z)$$ is a constant.

### 20.2.12 12

Let $$f$$ be analytic in a domain $$D$$ and $$\gamma$$ be a closed curve in $$D$$. For any $$z_0\in D$$ not on $$\gamma$$, show that \begin{align*} \int_{\gamma} \frac{f^{\prime}(z)}{\left(z-z_{0}\right)} d z=\int_{\gamma} \frac{f(z)}{\left(z-z_{0}\right)^{2}} d z .\end{align*} Give a generalization of this result.

### 20.2.13 13

Compute \begin{align*} \int_{{\left\lvert {z} \right\rvert} = 1} \qty{z + {1\over z}}^{2n} {dz \over z} \end{align*} and use it to show that \begin{align*} \int_0^{2\pi} \cos^{2n}(\theta) \, d\theta = 2\pi \qty{1\cdot 3 \cdot 5 \cdots (2n-1) \over 2 \cdot 4 \cdot 6 \cdots (2n)} .\end{align*}

# 21 Spring 2020 Homework 3

## 21.1 Stein and Shakarchi

### 21.1.1 3.8.1

Use the following formula to show that the complex zeros of $$\sin(\pi z)$$ are exactly the integers, and they are each of order 1: \begin{align*} \sin \pi z=\frac{e^{i \pi z}-e^{-i \pi z}}{2 i} .\end{align*}

Calculate the residue of $${1\over \sin(\pi z)}$$ at $$z=n\in {\mathbb{Z}}$$.

### 21.1.2 3.8.2

Evaluate the integral \begin{align*} \int_{\mathbb{R}}{dx \over 1 + x^4} .\end{align*}

What are the poles of $${1\over 1 + z^4}$$ ?

### 21.1.3 3.8.4

Show that \begin{align*} \int_{-\infty}^{\infty} \frac{x \sin x}{x^{2}+a^{2}} d x=\pi e^{-a}, \quad \text { for all } a>0 .\end{align*}

### 21.1.4 3.8.5

Show that if $$\xi\in {\mathbb{R}}$$, then \begin{align*} \int_{-\infty}^{\infty} \frac{e^{-2 \pi i x \xi}}{\left(1+x^{2}\right)^{2}} d x=\frac{\pi}{2}(1+2 \pi|\xi|) e^{-2 \pi|\xi|} .\end{align*}

### 21.1.5 3.8.6

Show that \begin{align*} \int_{-\infty}^{\infty} \frac{d x}{\left(1+x^{2}\right)^{n+1}}=\frac{1 \cdot 3 \cdot 5 \cdots(2 n-1)}{2 \cdot 4 \cdot 6 \cdots(2 n)} \cdot \pi .\end{align*}

### 21.1.6 3.8.7

Show that \begin{align*} \int_{0}^{2 \pi} \frac{d \theta}{(a+\cos \theta)^{2}}=\frac{2 \pi a}{\left(a^{2}-1\right)^{3 / 2}}, \quad \text { whenever } a>1 .\end{align*}

### 21.1.7 3.8.8

Show that if $$a,b\in {\mathbb{R}}$$ with $$a > {\left\lvert {b} \right\rvert}$$, then \begin{align*} \int_{0}^{2 \pi} \frac{d \theta}{a+b \cos \theta}=\frac{2 \pi}{\sqrt{a^{2}-b^{2}}} .\end{align*}

### 21.1.8 3.8.9

Show that \begin{align*} \int_{0}^{1} \log (\sin \pi x) d x=-\log 2 .\end{align*}

Hint: use the following contour.

### 21.1.9 3.8.10

Show that if $$a>0$$, then \begin{align*} \int_{0}^{\infty} \frac{\log x}{x^{2}+a^{2}} d x=\frac{\pi}{2 a} \log a .\end{align*}

Hint: use the following contour.

### 21.1.10 3.8.14

Prove that all entire functions that are injective are of the form $$f(z) = az + b$$ with $$a,b\in {\mathbb{C}}$$ and $$a\neq 0$$.

Hint: Apply the Casorati-Weierstrass theorem to $$f(1/z)$$.

### 21.1.11 3.8.15

Use the Cauchy inequalities or the maximum modulus principle to solve the following problems:

1. Prove that if $$f$$ is an entire function that satisfies \begin{align*} \sup _{|z|=R}|f(z)| \leq A R^{k}+B \end{align*} for all $$R>0$$, some integer $$k\geq 0$$, and some constants $$A, B > 0$$, then $$f$$ is a polynomial of degree $$\leq k$$.

2. Show that if $$f$$ is holomorphic in the unit disc, is bounded, and converges uniformly to zero in the sector $$\theta < \arg(z) < \phi$$ as $${\left\lvert {z} \right\rvert} \to 0$$, then $$f \equiv 0$$.

3. Let $$w_1, \cdots w_n$$ be points on $$S^1 \subset {\mathbb{C}}$$. Prove that there exists a point $$z\in S^1$$ such that the product of the distances from $$z$$ to the points $$w_j$$ is at least 1.

Conclude that there exists a point $$w\in S^1$$ such that the product of the above distances is exactly 1.

4. Show that if the real part of an entire function is bounded, then $$f$$ is constant.

### 21.1.12 3.8.17

Let $$f$$ be non-constant and holomorphic in an open set containing the closed unit disc.

1. Show that if $${\left\lvert {f(z)} \right\rvert} = 1$$ whenever $${\left\lvert {z} \right\rvert} = 1$$, then the image of $$f$$ contains the unit disc.

Hint: Show that $$f(z) = w_0$$ has a root for every $$w_0 \in {\mathbb{D}}$$, for which it suffices to show that $$f(z) = 0$$ has a root. Conclude using the maximum modulus principle.

2. If $${\left\lvert {f(z)} \right\rvert} \geq 1$$ whenever $${\left\lvert {z} \right\rvert} = 1$$ and there exists a $$z_0\in {\mathbb{D}}$$ such that $${\left\lvert {f(z_0)} \right\rvert} < 1$$, then the image of $$f$$ contains the unit disc.

### 21.1.13 3.8.19

Prove that maximum principle for harmonic functions, i.e.

1. If $$u$$ is a non-constant real-valued harmonic function in a region $$\Omega$$, then $$u$$ can not attain a maximum or a minimum in $$\Omega$$.

2. Suppose $$\Omega$$ is a region with compact closure $$\mkern 1.5mu\overline{\mkern-1.5mu\Omega\mkern-1.5mu}\mkern 1.5mu$$. If $$u$$ is harmonic in $$\Omega$$ and continuous in $$\mkern 1.5mu\overline{\mkern-1.5mu\Omega\mkern-1.5mu}\mkern 1.5mu$$, then \begin{align*} \sup _{z \in \Omega}|u(z)| \leq \sup _{z \in \mkern 1.5mu\overline{\mkern-1.5mu\Omega \mkern-1.5mu}\mkern 1.5mu-\Omega}|u(z)| .\end{align*}

Hint: to prove (a), assume $$u$$ attains a local maximum at $$z_0$$. Let $$f$$ be holomorphic near $$z_0$$ with $$\Re(f) = u$$, and show that $$f$$ is not an open map. Then (a) implies (b).

## 21.2 Extra Problems

### 21.2.1 1

Problem
Prove that if $$f$$ has two Laurent series expansions, \begin{align*} f(z) = \sum c_n(z-a)^n \quad\text{and}\quad f(z) = \sum c_n'(z-a)^n \end{align*} then $$c_n = c_n'$$.

### 21.2.2 2

Problem
Find Laurent series expansions of \begin{align*} \frac{1}{1-z^2} + \frac{1}{3-z} \end{align*} How many such expansions are there? In what domains are each valid?

### 21.2.3 3

Problem
Let $$P, Q$$ be polynomials with no common zeros. Assume $$a$$ is a root of $$Q$$. Find the principal part of $$P/Q$$ at $$z=a$$ in terms of $$P$$ and $$Q$$ if $$a$$ is (1) a simple root, and (2) a double root.

### 21.2.4 4

Problem

Let $$f$$ be non-constant, analytic in $${\left\lvert {z} \right\rvert} > 0$$, where $$f(z_n) = 0$$ for infinitely many points $$z_n$$ with $$\lim_{n\to\infty} z_n = 0$$.

Show that $$z=0$$ is an essential singularity for $$f$$.

Example: $$f(z) = \sin(1/z)$$.

### 21.2.5 5

Problem
Show that if $$f$$ is entire and $$\lim_{z\to\infty}f(z) = \infty$$, then $$f$$ is a polynomial.

### 21.2.6 6

Problem
1. Show (without using 3.8.9 in the S&S) that \begin{align*} \int_0^{2\pi} \log{\left\lvert {1 - e^{i\theta}} \right\rvert}~d\theta = 0 \end{align*}
2. Show that this identity is equivalent to S&S 3.8.9: \begin{align*} \int_0^1 \log(\sin(\pi x)) ~dx = -\log 2 .\end{align*}

### 21.2.7 7

Problem
Let $$0<a<4$$ and evaluate \begin{align*} \int_0^\infty \frac{x^{\alpha-1}}{1+x^3} ~dx \end{align*}

### 21.2.8 8

Problem

Prove the fundamental theorem of Algebra using

1. Rouche’s Theorem.
2. The maximum modulus principle.

### 21.2.9 9

Problem

Let $$f$$ be analytic in a region $$D$$ and $$\gamma$$ a rectifiable curve in $$D$$ with interior in $$D$$.

Prove that if $$f(z)$$ is real for all $$z\in \gamma$$, then $$f$$ is constant.

### 21.2.10 10

Problem
For $$a> 0$$, evaluate \begin{align*} \int_0^{\pi/2} \frac{d\theta}{a + \sin^2 \theta} \end{align*}

### 21.2.11 11

Problem
Find the number of roots of $$p(z) = 4z^4 - 6z + 3$$ in $${\left\lvert {z} \right\rvert} < 1$$ and $$1 < {\left\lvert {z} \right\rvert} < 2$$ respectively.

### 21.2.12 12

Problem
Prove that $$z^4 + 2z^3 -2z + 10$$ has exactly one root in each open quadrant.

### 21.2.13 13

Problem
Prove that for $$a> 0$$, $$z\tan z - a$$ has only real roots.

### 21.2.14 14

Problem

Let $$f$$ be nonzero, analytic on a bounded region $$\Omega$$ and continuous on its closure $$\overline \Omega$$.

Show that if $${\left\lvert {f(z)} \right\rvert} \equiv M$$ is constant for $$z\in \partial \Omega$$, then $$f(z) \equiv Me^{i\theta}$$ for some real constant $$\theta$$.

# 22 Extra Questions from Jingzhi Tie

## 22.1 Fall 2009

### 22.1.1 ?

1. Assume $$\displaystyle f(z) = \sum_{n=0}^\infty c_n z^n$$ converges in $$|z| < R$$. Show that for $$r <R$$, \begin{align*}\frac{1}{2 \pi} \int_0^{2 \pi} |f(r e^{i \theta})|^2 d \theta = \sum_{n=0}^\infty |c_n|^2 r^{2n} \; .\end{align*}

2. Deduce Liouville’s theorem from (1).

### 22.1.2 ?

Let $$f$$ be a continuous function in the region \begin{align*}D=\{z {~\mathrel{\Big|}~}{\left\lvert {z} \right\rvert}>R, 0\leq \arg z\leq \theta\}\quad\text{where}\quad 1\leq \theta \leq 2\pi.\end{align*} If there exists $$k$$ such that $$\displaystyle{\lim_{z\to\infty} zf(z)=k}$$ for $$z$$ in the region $$D$$. Show that \begin{align*}\lim_{R'\to\infty} \int_{L} f(z) dz=i\theta k,\end{align*} where $$L$$ is the part of the circle $$|z|=R'$$ which lies in the region $$D$$.

### 22.1.3 ?

Suppose that $$f$$ is an analytic function in the region $$D$$ which contains the point $$a$$. Let \begin{align*}F(z)= z-a-qf(z),\quad \text{where}~ q \ \text{is a complex parameter}.\end{align*}

1. Let $$K\subset D$$ be a circle with the center at point $$a$$ and also we assume that $$f(z)\not =0$$ for $$z\in K$$. Prove that the function $$F$$ has one and only one zero $$z=w$$ on the closed disc $$\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu$$ whose boundary is the circle $$K$$ if $$\displaystyle{ |q|<\min_{z\in K} \frac{|z-a|}{|f(z)|}.}$$

2. Let $$G(z)$$ be an analytic function on the disk $$\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu$$. Apply the residue theorem to prove that $$\displaystyle{ \frac{G(w)}{F'(w)}=\frac{1}{2\pi i}\int_K \frac{G(z)}{F(z)} dz,}$$ where $$w$$ is the zero from (1).

3. If $$z\in K$$, prove that the function $$\displaystyle{\frac{1}{F(z)}}$$ can be represented as a convergent series with respect to $$q$$: $$\displaystyle{ \frac{1}{F(z)}=\sum_{n=0}^{\infty} \frac{(qf(z))^n}{(z-a)^{n+1}}.}$$

### 22.1.4 ?

Evaluate \begin{align*}\displaystyle{ \int_{0}^{\infty}\frac{x\sin x}{x^2+a^2} \, dx }.\end{align*}

### 22.1.5 ?

Let $$f=u+iv$$ be differentiable (i.e. $$f'(z)$$ exists) with continuous partial derivatives at a point $$z=re^{i\theta}$$, $$r\not= 0$$. Show that \begin{align*}\frac{\partial u}{\partial r}=\frac{1}{r}\frac{\partial v}{\partial \theta},\quad \frac{\partial v}{\partial r}=-\frac{1}{r}\frac{\partial u}{\partial \theta}.\end{align*}

### 22.1.6 ?

Show that $$\displaystyle \int_0^\infty \frac{x^{a-1}}{1+x^n} dx=\frac{\pi}{n\sin \frac{a\pi}{n}}$$ using complex analysis, $$0< a < n$$. Here $$n$$ is a positive integer.

### 22.1.7 ?

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

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

2. 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 $$t>0$$.

### 22.1.8 ?

Apply Rouché’s Theorem to prove the Fundamental Theorem of Algebra: If \begin{align*}P_n(z) = a_0 + a_1z + \cdots + a_{n-1}z^{n-1} + a_nz^n\quad (a_n \neq 0)\end{align*} is a polynomial of degree n, then it has n zeros in $$\mathbb C$$.

### 22.1.9 ?

Suppose $$f$$ is entire and there exist $$A, R >0$$ and natural number $$N$$ such that \begin{align*}|f(z)| \geq A |z|^N\ \text{for}\ |z| \geq R.\end{align*} Show that (i) $$f$$ is a polynomial and (ii) the degree of $$f$$ is at least $$N$$.

### 22.1.10 ?

Let $$f: {\mathbb C} \rightarrow {\mathbb C}$$ be an injective analytic (also called univalent) function. Show that there exist complex numbers $$a \neq 0$$ and $$b$$ such that $$f(z) = az + b$$.

### 22.1.11 ?

Let $$g$$ be analytic for $$|z|\leq 1$$ and $$|g(z)| < 1$$ for $$|z| = 1$$.

1. Show that $$g$$ has a unique fixed point in $$|z| < 1$$.

2. What happens if we replace $$|g(z)| < 1$$ with $$|g(z)|\leq 1$$ for $$|z|=1$$? Give an example if (a) is not true or give an proof if (a) is still true.

3. What happens if we simply assume that $$f$$ is analytic for $$|z| < 1$$ and $$|f(z)| < 1$$ for $$|z| < 1$$? Suppose that $$f(z) \not\equiv z$$. Can f have more than one fixed point in $$|z| < 1$$?

Hint: The map $$\displaystyle{\psi_{\alpha}(z)=\frac{\alpha-z}{1-\mkern 1.5mu\overline{\mkern-1.5mu\alpha\mkern-1.5mu}\mkern 1.5muz}}$$ may be useful.

### 22.1.12 ?

Find a conformal map from $$D = \{z :\ |z| < 1,\ |z - 1/2| > 1/2\}$$ to the unit disk $$\Delta=\{z: \ |z|<1\}$$.

### 22.1.13 ?

Let $$f(z)$$ be entire and assume values of $$f(z)$$ lie outside a bounded open set $$\Omega$$. Show without using Picard’s theorems that $$f(z)$$ is a constant.

### 22.1.14 ?

1. Assume $$\displaystyle f(z) = \sum_{n=0}^\infty c_n z^n$$ converges in $$|z| < R$$. Show that for $$r <R$$, \begin{align*}\frac{1}{2 \pi} \int_0^{2 \pi} |f(r e^{i \theta})|^2 d \theta = \sum_{n=0}^\infty |c_n|^2 r^{2n} \; .\end{align*}

2. Deduce Liouville’s theorem from (1).

### 22.1.15 ?

Let $$f(z)$$ be entire and assume that $$f(z) \leq M |z|^2$$ outside some disk for some constant $$M$$. Show that $$f(z)$$ is a polynomial in $$z$$ of degree $$\leq 2$$.

### 22.1.16 ?

Let $$a_n(z)$$ be an analytic sequence in a domain $$D$$ such that $$\displaystyle \sum_{n=0}^\infty |a_n(z)|$$ converges uniformly on bounded and closed sub-regions of $$D$$. Show that $$\displaystyle \sum_{n=0}^\infty |a'_n(z)|$$ converges uniformly on bounded and closed sub-regions of $$D$$.

### 22.1.17 ?

Let $$f(z)$$ be analytic in an open set $$\Omega$$ except possibly at a point $$z_0$$ inside $$\Omega$$. Show that if $$f(z)$$ is bounded in near $$z_0$$, then $$\displaystyle \int_\Delta f(z) dz = 0$$ for all triangles $$\Delta$$ in $$\Omega$$.

### 22.1.18 ?

Assume $$f$$ is continuous in the region: $$0< |z-a| \leq R, \; 0 \leq \arg(z-a) \leq \beta_0$$ ($$0 < \beta_0 \leq 2 \pi$$) and the limit $$\displaystyle \lim_{z \rightarrow a} (z-a) f(z) = A$$ exists. Show that \begin{align*}\lim_{r \rightarrow 0} \int_{\gamma_r} f(z) dz = i A \beta_0 \; , \; \;\end{align*} where $$\gamma_r : = \{ z \; | \; z = a + r e^{it}, \; 0 \leq t \leq \beta_0 \}.$$

### 22.1.19 ?

Show that $$f(z) = z^2$$ is uniformly continuous in any open disk $$|z| < R$$, where $$R>0$$ is fixed, but it is not uniformly continuous on $$\mathbb C$$.

### 22.1.20 ?

1. 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*}
2. Show that there exist points $$(x,y)\in D$$ such that $$\displaystyle{\limsup_{n\to\infty} |u(x,y)|=\infty}$$.

## 22.2 Fall 2011

### 22.2.1 ?

1. Assume $$\displaystyle f(z) = \sum_{n=0}^\infty c_n z^n$$ converges in $$|z| < R$$. Show that for $$r <R$$, \begin{align*}\frac{1}{2 \pi} \int_0^{2 \pi} |f(r e^{i \theta})|^2 d \theta = \sum_{n=0}^\infty |c_n|^2 r^{2n} \; .\end{align*}

2. Deduce Liouville’s theorem from (1).

### 22.2.2 ?

Let $$f$$ be a continuous function in the region \begin{align*}D=\{z\ | |z|>R, 0\leq \arg Z\leq \theta\}\quad\text{where}\quad 0\leq \theta \leq 2\pi.\end{align*} If there exists $$k$$ such that $$\displaystyle{\lim_{z\to\infty} zf(z)=k}$$ for $$z$$ in the region $$D$$. Show that \begin{align*}\lim_{R'\to\infty} \int_{L} f(z) dz=i\theta k,\end{align*} where $$L$$ is the part of the circle $$|z|=R'$$ which lies in the region $$D$$.

### 22.2.3 ?

Suppose that $$f$$ is an analytic function in the region $$D$$ which contains the point $$a$$. Let \begin{align*}F(z)= z-a-qf(z),\quad \text{where}\quad q \ \text{is a complex parameter}.\end{align*}

1. Let $$K\subset D$$ be a circle with the center at point $$a$$ and also we assume that $$f(z)\not =0$$ for $$z\in K$$. Prove that the function $$F$$ has one and only one zero $$z=w$$ on the closed disc $$\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu$$ whose boundary is the circle $$K$$ if $$\displaystyle{ |q|<\min_{z\in K} \frac{|z-a|}{|f(z)|}.}$$

2. Let $$G(z)$$ be an analytic function on the disk $$\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu$$. Apply the residue theorem to prove that $$\displaystyle{ \frac{G(w)}{F'(w)}=\frac{1}{2\pi i}\int_K \frac{G(z)}{F(z)} dz,}$$ where $$w$$ is the zero from (1).

3. If $$z\in K$$, prove that the function $$\displaystyle{\frac{1}{F(z)}}$$ can be represented as a convergent series with respect to $$q$$: $$\displaystyle{ \frac{1}{F(z)}=\sum_{n=0}^{\infty} \frac{(qf(z))^n}{(z-a)^{n+1}}.}$$

### 22.2.4 ?

Evaluate $$\displaystyle{ \int_{0}^{\infty}\frac{x\sin x}{x^2+a^2} \, dx }$$.

### 22.2.5 ?

Let $$f=u+iv$$ be differentiable (i.e. $$f'(z)$$ exists) with continuous partial derivatives at a point $$z=re^{i\theta}$$, $$r\not= 0$$. Show that \begin{align*}\frac{\partial u}{\partial r}=\frac{1}{r}\frac{\partial v}{\partial \theta},\quad \frac{\partial v}{\partial r}=-\frac{1}{r}\frac{\partial u}{\partial \theta}.\end{align*}

### 22.2.6 ?

Show that $$\displaystyle \int_0^\infty \frac{x^{a-1}}{1+x^n} dx=\frac{\pi}{n\sin \frac{a\pi}{n}}$$ using complex analysis, $$0< a < n$$. Here $$n$$ is a positive integer.

### 22.2.7 ?

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

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

2. 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 $$t>0$$.

### 22.2.8 ?

Apply Rouché’s Theorem to prove the Fundamental Theorem of Algebra: If \begin{align*}P_n(z) = a_0 + a_1z + \cdots + a_{n-1}z^{n-1} + a_nz^n\quad (a_n \neq 0)\end{align*} is a polynomial of degree n, then it has n zeros in $$\mathbb C$$.

### 22.2.9 ?

Suppose $$f$$ is entire and there exist $$A, R >0$$ and natural number $$N$$ such that \begin{align*}|f(z)| \geq A |z|^N\ \text{for}\ |z| \geq R.\end{align*} Show that (i) $$f$$ is a polynomial and (ii) the degree of $$f$$ is at least $$N$$.

### 22.2.10 ?

Let $$f: {\mathbb C} \rightarrow {\mathbb C}$$ be an injective analytic (also called univalent) function. Show that there exist complex numbers $$a \neq 0$$ and $$b$$ such that $$f(z) = az + b$$.

### 22.2.11 ?

Let $$g$$ be analytic for $$|z|\leq 1$$ and $$|g(z)| < 1$$ for $$|z| = 1$$.

• Show that $$g$$ has a unique fixed point in $$|z| < 1$$.

• What happens if we replace $$|g(z)| < 1$$ with $$|g(z)|\leq 1$$ for $$|z|=1$$? Give an example if (a) is not true or give an proof if (a) is still true.

• What happens if we simply assume that $$f$$ is analytic for $$|z| < 1$$ and $$|f(z)| < 1$$ for $$|z| < 1$$? Suppose that $$f(z) \not\equiv z$$. Can f have more than one fixed point in $$|z| < 1$$?

Hint: The map $$\displaystyle{\psi_{\alpha}(z)=\frac{\alpha-z}{1-\mkern 1.5mu\overline{\mkern-1.5mu\alpha\mkern-1.5mu}\mkern 1.5muz}}$$ may be useful.

### 22.2.12 ?

Find a conformal map from $$D = \{z :\ |z| < 1,\ |z - 1/2| > 1/2\}$$ to the unit disk $$\Delta=\{z: \ |z|<1\}$$.

### 22.2.13 ?

Let $$f(z)$$ be entire and assume values of $$f(z)$$ lie outside a bounded open set $$\Omega$$. Show without using Picard’s theorems that $$f(z)$$ is a constant.

### 22.2.14 ?

Let $$f(z)$$ be entire and assume values of $$f(z)$$ lie outside a bounded open set $$\Omega$$. Show without using Picard’s theorems that $$f(z)$$ is a constant.

### 22.2.15 ?

1. Assume $$\displaystyle f(z) = \sum_{n=0}^\infty c_n z^n$$ converges in $$|z| < R$$. Show that for $$r <R$$, \begin{align*}\frac{1}{2 \pi} \int_0^{2 \pi} |f(r e^{i \theta})|^2 d \theta = \sum_{n=0}^\infty |c_n|^2 r^{2n} \; .\end{align*}

2. Deduce Liouville’s theorem from (1).

### 22.2.16 ?

Let $$f(z)$$ be entire and assume that $$f(z) \leq M |z|^2$$ outside some disk for some constant $$M$$. Show that $$f(z)$$ is a polynomial in $$z$$ of degree $$\leq 2$$.

### 22.2.17 ?

Let $$a_n(z)$$ be an analytic sequence in a domain $$D$$ such that $$\displaystyle \sum_{n=0}^\infty |a_n(z)|$$ converges uniformly on bounded and closed sub-regions of $$D$$. Show that $$\displaystyle \sum_{n=0}^\infty |a'_n(z)|$$ converges uniformly on bounded and closed sub-regions of $$D$$.

### 22.2.18 ?

Let $$f(z)$$ be analytic in an open set $$\Omega$$ except possibly at a point $$z_0$$ inside $$\Omega$$. Show that if $$f(z)$$ is bounded in near $$z_0$$, then $$\displaystyle \int_\Delta f(z) dz = 0$$ for all triangles $$\Delta$$ in $$\Omega$$.

### 22.2.19 ?

Assume $$f$$ is continuous in the region: $$0< |z-a| \leq R, \; 0 \leq \arg(z-a) \leq \beta_0$$ ($$0 < \beta_0 \leq 2 \pi$$) and the limit $$\displaystyle \lim_{z \rightarrow a} (z-a) f(z) = A$$ exists. Show that \begin{align*}\lim_{r \rightarrow 0} \int_{\gamma_r} f(z) dz = i A \beta_0 \; , \; \;\end{align*} where $$\gamma_r : = \{ z \; | \; z = a + r e^{it}, \; 0 \leq t \leq \beta_0 \}.$$

### 22.2.20 ?

Show that $$f(z) = z^2$$ is uniformly continuous in any open disk $$|z| < R$$, where $$R>0$$ is fixed, but it is not uniformly continuous on $$\mathbb C$$.

1. 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*}
2. Show that there exist points $$(x,y)\in D$$ such that $$\displaystyle{\limsup_{n\to\infty} |u(x,y)|=\infty}$$.

## 22.3 Spring 2014

### 22.3.1 ?

The question provides some insight into Cauchy’s theorem. Solve the problem without using the Cauchy theorem.

1. Evaluate the integral $$\displaystyle{\int_{\gamma} z^n dz}$$ for all integers $$n$$. Here $$\gamma$$ is any circle centered at the origin with the positive (counterclockwise) orientation.

2. Same question as (a), but with $$\gamma$$ any circle not containing the origin.

3. Show that if $$|a|<r<|b|$$, then $$\displaystyle{\int_{\gamma}\frac{dz}{(z-a)(z-b)} dz=\frac{2\pi i}{a-b}}$$. Here $$\gamma$$ denotes the circle centered at the origin, of radius $$r$$, with the positive orientation.

### 22.3.2 ?

1. Assume the infinite series $$\displaystyle \sum_{n=0}^\infty c_n z^n$$ converges in $$|z| < R$$ and let $$f(z)$$ be the limit. Show that for $$r <R$$, \begin{align*}\frac{1}{2 \pi} \int_0^{2 \pi} |f(r e^{i \theta})|^2 d \theta = \sum_{n=0}^\infty |c_n|^2 r^{2n} \; .\end{align*}

2. Deduce Liouville’s theorem from (1). Liouville’s theorem: If $$f(z)$$ is entire and bounded, then $$f$$ is constant.

### 22.3.3 ?

Let $$f$$ be a continuous function in the region \begin{align*}D=\{z\ | |z|>R, 0\leq \arg Z\leq \theta\}\quad\text{where}\quad 0\leq \theta \leq 2\pi.\end{align*} If there exists $$k$$ such that $$\displaystyle{\lim_{z\to\infty} zf(z)=k}$$ for $$z$$ in the region $$D$$. Show that \begin{align*}\lim_{R'\to\infty} \int_{L} f(z) dz=i\theta k,\end{align*} where $$L$$ is the part of the circle $$|z|=R'$$ which lies in the region $$D$$.

### 22.3.4 ?

Evaluate $$\displaystyle{ \int_{0}^{\infty}\frac{x\sin x}{x^2+a^2} \, dx }$$.

### 22.3.5 ?

Let $$f=u+iv$$ be differentiable (i.e. $$f'(z)$$ exists) with continuous partial derivatives at a point $$z=re^{i\theta}$$, $$r\not= 0$$. Show that \begin{align*}\frac{\partial u}{\partial r}=\frac{1}{r}\frac{\partial v}{\partial \theta},\quad \frac{\partial v}{\partial r}=-\frac{1}{r}\frac{\partial u}{\partial \theta}.\end{align*}

### 22.3.6 ?

Show that $$\displaystyle \int_0^\infty \frac{x^{a-1}}{1+x^n} dx=\frac{\pi}{n\sin \frac{a\pi}{n}}$$ using complex analysis, $$0< a < n$$. Here $$n$$ is a positive integer.

### 22.3.7 ?

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 $$t>0$$.

### 22.3.8 ?

Apply Rouché’s Theorem to prove the Fundamental Theorem of Algebra: If \begin{align*}P_n(z) = a_0 + a_1z + \cdots + a_{n-1}z^{n-1} + a_nz^n\quad (a_n \neq 0)\end{align*} is a polynomial of degree n, then it has n zeros in $$\mathbf C$$.

### 22.3.9 ?

Suppose $$f$$ is entire and there exist $$A, R >0$$ and natural number $$N$$ such that \begin{align*}|f(z)| \geq A |z|^N\ \text{for}\ |z| \geq R.\end{align*} Show that (i) $$f$$ is a polynomial and (ii) the degree of $$f$$ is at least $$N$$.

### 22.3.10 ?

Let $$f: {\mathbb C} \rightarrow {\mathbb C}$$ be an injective analytic (also called univalent) function. Show that there exist complex numbers $$a \neq 0$$ and $$b$$ such that $$f(z) = az + b$$.

### 22.3.11 ?

Let $$g$$ be analytic for $$|z|\leq 1$$ and $$|g(z)| < 1$$ for $$|z| = 1$$.

• Show that $$g$$ has a unique fixed point in $$|z| < 1$$.

• What happens if we replace $$|g(z)| < 1$$ with $$|g(z)|\leq 1$$ for $$|z|=1$$? Give an example if (a) is not true or give an proof if (a) is still true.

• What happens if we simply assume that $$f$$ is analytic for $$|z| < 1$$ and $$|f(z)| < 1$$ for $$|z| < 1$$? Suppose that $$f(z) \not\equiv z$$. Can f have more than one fixed point in $$|z| < 1$$?

Hint: The map $$\displaystyle{\psi_{\alpha}(z)=\frac{\alpha-z}{1-\mkern 1.5mu\overline{\mkern-1.5mu\alpha\mkern-1.5mu}\mkern 1.5muz}}$$ may be useful.

### 22.3.12 ?

Find a conformal map from $$D = \{z :\ |z| < 1,\ |z - 1/2| > 1/2\}$$ to the unit disk $$\Delta=\{z: \ |z|<1\}$$.

## 22.4 Fall 2015

### 22.4.1 ?

Let $$a_n \neq 0$$ and assume that $$\displaystyle \lim_{n \rightarrow \infty} \frac{|a_{n+1}|}{|a_n|} = L$$. Show that $$\displaystyle \lim_{n \rightarrow \infty} \sqrt[n]{|a_n|} = L. %p_n^{\frac{1}{n}} = L.$$ In particular, this shows that when applicable, the ratio test can be used to calculate the radius of convergence of a power series.

### 22.4.2 ?

1. Let $$z, w$$ be complex numbers, such that $$\mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu w \neq 1$$. Prove that \begin{align*}{\left\lvert {\frac{w - z}{1 - \mkern 1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu z}} \right\rvert} < 1 \; \; \; \mbox{if} \; |z| < 1 \; \mbox{and}\; |w| < 1,\end{align*} and also that \begin{align*}{\left\lvert {\frac{w - z}{1 - \mkern 1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu z}} \right\rvert} = 1 \; \; \; \mbox{if} \; |z| = 1 \; \mbox{or}\; |w| = 1.\end{align*}

2. Prove that for fixed $$w$$ in the unit disk $$\mathbb D$$, the mapping \begin{align*}F: z \mapsto \frac{w - z}{1 - \mkern 1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5mu z}\end{align*} satisfies the following conditions:

3. $$F$$ maps $$\mathbb D$$ to itself and is holomorphic.

1. $$F$$ interchanges $$0$$ and $$w$$, namely, $$F(0) = w$$ and $$F(w) = 0$$.

2. $$|F(z)| = 1$$ if $$|z| = 1$$.

3. $$F: {\mathbb D} \mapsto {\mathbb D}$$ is bijective.

Hint: Calculate $$F \circ F$$.

### 22.4.3 ?

Use $$n$$-th roots of unity (i.e. solutions of $$z^n - 1 =0$$) to show that \begin{align*}2^{n-1} \sin\frac{\pi}{n} \sin\frac{2\pi}{n} \cdots \sin\frac{(n-1)\pi}{n} = n \; .\end{align*}

Hint: $$1 - \cos 2 \theta = 2 \sin^2 \theta,\; \sin 2 \theta = 2 \sin \theta \cos \theta$$.

1. Show that in polar coordinates, the Cauchy-Riemann equations take the form

\begin{align*}\frac{\partial u}{\partial r} = \frac{1}{r} \frac{\partial v}{\partial \theta} \; \; \; \text{and} \; \; \; \frac{\partial v}{\partial r} = - \frac{1}{r} \frac{\partial u}{\partial \theta}\end{align*}

1. Use these equations to show that the logarithm function defined by \begin{align*}\log z = \log r + i \theta \; \; \mbox{where} \; z = r e^{i \theta } \; \mbox{with} \; - \pi < \theta < \pi\end{align*} is a holomorphic function in the region $$r>0, \; - \pi < \theta < \pi$$. Also show that $$\log z$$ defined above is not continuous in $$r>0$$.

### 22.4.4 ?

Assume $$f$$ is continuous in the region: $$x \geq x_0, \; 0 \leq y \leq b$$ and the limit \begin{align*}\displaystyle \lim_{x \rightarrow + \infty} f(x + iy) = A\end{align*} exists uniformly with respect to $$y$$ (independent of $$y$$). Show that \begin{align*}\lim_{x \rightarrow + \infty} \int_{\gamma_x} f(z) dz = iA b \; , \; \;\end{align*} where $$\gamma_x : = \{ z \; | \; z = x + it, \; 0 \leq t \leq b\}.$$

### 22.4.5 ?

(Cauchy’s formula for “exterior” region) Let $$\gamma$$ be piecewise smooth simple closed curve with interior $$\Omega_1$$ and exterior $$\Omega_2$$. Assume $$f'(z)$$ exists in an open set containing $$\gamma$$ and $$\Omega_2$$ and $$\lim_{z \rightarrow \infty } f(z) = A$$. Show that \begin{align*}\frac{1}{2 \pi i} \int_\gamma \frac{f(\xi)}{\xi - z} \, d \xi = \begin{cases} A, & \text{if\ $z \in \Omega_1$}, \\ -f (z) + A, & \text{if\ $z \in \Omega_2$} \end{cases}\end{align*}

### 22.4.6 ?

Let $$f(z)$$ be bounded and analytic in $$\mathbb C$$. Let $$a \neq b$$ be any fixed complex numbers. Show that the following limit exists \begin{align*}\lim_{R \rightarrow \infty} \int_{|z|=R} \frac{f(z)}{(z-a)(z-b)} dz.\end{align*} Use this to show that $$f(z)$$ must be a constant (Liouville’s theorem).

### 22.4.7 ?

Prove by justifying all steps that for all $$\xi \in {\mathbb C}$$ we have $$\displaystyle e^{- \pi \xi^2} = \int_{- \infty}^\infty e^{- \pi x^2} e^{2 \pi i x \xi} dx \; .$$

Hint: You may use that fact in Example 1 on p. 42 of the textbook without proof, i.e., you may assume the above is true for real values of $$\xi$$.

### 22.4.8 ?

Suppose that $$f$$ is holomorphic in an open set containing the closed unit disc, except for a pole at $$z_0$$ on the unit circle. Let $$\displaystyle %f(z) = \sum_{n = 1}^\infty a_n z^n f(z) = \sum_{n = 1}^\infty c_n z^n$$ denote the the power series in the open disc. Show that (1) $$c_n \neq 0$$ for all large enough $$n$$’s, and (2) $$\displaystyle \lim_{n \rightarrow \infty} \frac{c_n}{c_{n+1}}= z_0$$.

### 22.4.9 ?

Let $$f(z)$$ be a non-constant analytic function in $$|z|>0$$ such that $$f(z_n) = 0$$ for infinite many points $$z_n$$ with $$\lim_{n \rightarrow \infty} z_n =0$$. Show that $$z=0$$ is an essential singularity for $$f(z)$$. (An example of such a function is $$f(z) = \sin (1/z)$$.)

### 22.4.10 ?

Let $$f$$ be entire and suppose that $$\lim_{z \rightarrow \infty} f(z) = \infty$$. Show that $$f$$ is a polynomial.

### 22.4.11 ?

Expand the following functions into Laurent series in the indicated regions:

1. $$\displaystyle f(z) = \frac{z^2 - 1}{ (z+2)(z+3)}, \; \; 2 < |z| < 3$$, $$3 < |z| < + \infty$$.

2. $$\displaystyle f(z) = \sin \frac{z}{1-z}, \; \; 0 < |z-1| < + \infty$$

### 22.4.12 ?

Assume $$f(z)$$ is analytic in region $$D$$ and $$\Gamma$$ is a rectifiable curve in $$D$$ with interior in $$D$$. Prove that if $$f(z)$$ is real for all $$z \in \Gamma$$, then $$f(z)$$ is a constant.

### 22.4.13 ?

Find the number of roots of $$z^4 - 6z + 3 =0$$ in $$|z|<1$$ and $$1 < |z| < 2$$ respectively.

### 22.4.14 ?

Prove that $$z^4 + 2 z^3 - 2z + 10 =0$$ has exactly one root in each open quadrant.

### 22.4.15 ?

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

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

### 22.4.16 ?

Assume $$f(z)$$ is analytic in $${\mathbb D}$$ 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 $$f^{n+1}(z) = f(f^{n}(z))$$.

### 22.4.17 ?

Let $$f(z) = \sum_{n=0}^\infty c_n z^n$$ be analytic and one-to-one in $$|z| < 1$$. For $$0<r<1$$, let $$D_r$$ be the disk $$|z|<r$$. Show that the area of $$f(D_r)$$ is finite and is given by \begin{align*}S = \pi \sum_{n=1}^\infty n |c_n|^2 r^{2n}.\end{align*} (Note that in general the area of $$f(D_1)$$ is infinite.)

### 22.4.18 ?

Let $$f(z) = \sum_{n= -\infty}^\infty c_n z^n$$ be analytic and one-to-one in $$r_0< |z| < R_0$$. For $$r_0<r<R<R_0$$, let $$D(r,R)$$ be the annulus $$r<|z|<R$$. Show that the area of $$f(D(r,R))$$ is finite and is given by \begin{align*}S = \pi \sum_{n=- \infty}^\infty n |c_n|^2 (R^{2n} - r^{2n}).\end{align*}

## 22.5 Spring 2015

### 22.5.1 ?

Let $$a_n(z)$$ be an analytic sequence in a domain $$D$$ such that $$\displaystyle \sum_{n=0}^\infty |a_n(z)|$$ converges uniformly on bounded and closed sub-regions of $$D$$. Show that $$\displaystyle \sum_{n=0}^\infty |a'_n(z)|$$ converges uniformly on bounded and closed sub-regions of $$D$$.

### 22.5.2 ?

Let $$f_n, f$$ be analytic functions on the unit disk $${\mathbb D}$$. Show that the following are equivalent.

1. $$f_n(z)$$ converges to $$f(z)$$ uniformly on compact subsets in $$\mathbb D$$.

2. $$\int_{|z|= r} |f_n(z) - f(z)| \, |dz|$$ converges to $$0$$ if $$0< r<1$$.

### 22.5.3 ?

Let $$f$$ and $$g$$ be non-zero analytic functions on a region $$\Omega$$. Assume $$|f(z)| = |g(z)|$$ for all $$z$$ in $$\Omega$$. Show that $$f(z) = e^{i \theta} g(z)$$ in $$\Omega$$ for some $$0 \leq \theta < 2 \pi$$.

### 22.5.4 ?

Suppose $$f$$ is analytic in an open set containing the unit disc $$\mathbb D$$ and $$|f(z)| =1$$ when $$|z|$$=1. Show that either $$f(z) = e^{i \theta}$$ for some $$\theta \in \mathbb R$$ or there are finite number of $$z_k \in \mathbb D$$, $$k \leq n$$ and $$\theta \in \mathbb R$$ such that $$\displaystyle f(z) = e^{i\theta} \prod_{k=1}^n \frac{z-z_k}{1 - \mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu_k z } \, .$$

Also cf. Stein et al, 1.4.7, 3.8.17

### 22.5.5 ?

1. 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 $$M_R>1$$.

2. 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*}

### 22.5.6 ?

Let $$\displaystyle f(z) = \frac{1}{z} + \frac{1}{z^2 -1}$$. Find all the Laurent series of $$f$$ and describe the largest annuli in which these series are valid.

### 22.5.7 ?

Suppose $$f$$ is entire and there exist $$A, R >0$$ and natural number $$N$$ such that $$|f(z)| \leq A |z|^N$$ for $$|z| \geq R$$. Show that (i) $$f$$ is a polynomial and (ii) the degree of $$f$$ is at most $$N$$.

### 22.5.8 ?

Suppose $$f$$ is entire and there exist $$A, R >0$$ and natural number $$N$$ such that $$|f(z)| \geq A |z|^N$$ for $$|z| \geq R$$. Show that (i) $$f$$ is a polynomial and (ii) the degree of $$f$$ is at least $$N$$.

### 22.5.9 ?

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

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

### 22.5.10 ?

1. Assume $$u$$ is harmonic on open set $$O$$ and $$z_n$$ is a sequence in $$O$$ such that $$u(z_n) = 0$$ and $$\lim z_n \in O$$. Prove or disprove that $$u$$ is identically zero. What if $$O$$ is a region?

2. Assume $$u$$ is harmonic on open set $$O$$ and $$u(z) = 0$$ on a disc in $$O$$. Prove or disprove that $$u$$ is identically zero. What if $$O$$ is a region?

3. Formulate and prove a Schwarz reflection principle for harmonic functions

cf. Theorem 5.6 on p.60 of Stein et al.

Hint: Verify the mean value property for your new function obtained by Schwarz reflection principle.

### 22.5.11 ?

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.

### 22.5.12 ?

1. Let $$f$$ be analytic in $$\Omega: 0<|z-a|<r$$ except at a sequence of poles $$a_n \in \Omega$$ with $$\lim_{n \rightarrow \infty} a_n = a$$. Show that for any $$w \in \mathbb C$$, there exists a sequence $$z_n \in \Omega$$ such that $$\lim_{n \rightarrow \infty} f(z_n) = w$$.

2. Explain the similarity and difference between the above assertion and the Weierstrass-Casorati theorem.

### 22.5.13 ?

Compute the following integrals.

$$i$$ $$\displaystyle \int_0^\infty \frac{1}{(1 + x^n)^2} \, dx$$, $$n \geq 1$$ (ii) $$\displaystyle \int_0^\infty \frac{\cos x}{(x^2 + a^2)^2} \, dx$$, $$a \in \mathbb R$$ (iii) $$\displaystyle \int_0^\pi \frac{1}{a + \sin \theta} \, d \theta$$, $$a>1$$

$$iv$$ $$\displaystyle \int_0^{\frac{\pi}{2}} \frac{d \theta}{a+ \sin ^2 \theta},$$ $$a >0$$. (v) $$\displaystyle \int_{|z|=2} \frac{1}{(z^{5} -1) (z-3)} \, dz$$ (v) $$\displaystyle \int_{- \infty}^{\infty} \frac{\sin \pi a}{\cosh \pi x + \cos \pi a} e^{- i x \xi} \, d x$$, $$0< a <1$$, $$\xi \in \mathbb R$$ (vi) $$\displaystyle \int_{|z| = 1} \cot^2 z \, dz$$.

### 22.5.14 ?

Compute the following integrals.

$$i$$ $$\displaystyle \int_0^\infty \frac{\sin x}{x} \, dx$$ (ii) $$\displaystyle \int_0^\infty (\frac{\sin x}{x})^2 \, dx$$ (iii) $$\displaystyle \int_0^\infty \frac{x^{a-1}}{(1 + x)^2} \, dx$$, $$0< a < 2$$

$$i$$ $$\displaystyle \int_0^\infty \frac{\cos a x - \cos bx}{x^2} dx$$, $$a, b >0$$ (ii) $$\displaystyle \int_0^\infty \frac{x^{a-1}}{1 + x^n} \, dx$$, $$0< a < n$$

$$iii$$ $$\displaystyle \int_0^\infty \frac{\log x}{1 + x^n} \, dx$$, $$n \geq 2$$ (iv) $$\displaystyle \int_0^\infty \frac{\log x}{(1 + x^2)^2} dx$$ (v) $$\displaystyle \int_0^{\pi} \log|1 - a \sin \theta| d \theta$$, $$a \in \mathbb C$$

### 22.5.15 ?

Let $$0<r<1$$. Show that polynomials $$P_n(z) = 1 + 2z + 3 z^2 + \cdots + n z^{n-1}$$ have no zeros in $$|z|<r$$ for all sufficiently large $$n$$’s.

### 22.5.16 ?

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.

### 22.5.17 ?

1. 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.

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

### 22.5.18 ?

Let $$f(z)$$ be an analytic function on $${\mathbb C} \backslash \{ z_0 \}$$, where $$z_0$$ is a fixed point. Assume that $$f(z)$$ is bijective from $${\mathbb C} \backslash \{ z_0 \}$$ onto its image, and that $$f(z)$$ is bounded outside $$D_r(z_0)$$, where $$r$$ is some fixed positive number. Show that there exist $$a, b, c, d \in \mathbb C$$ with $$ad-bc \neq 0$$, $$c \neq 0$$ such that $$\displaystyle f(z) = \frac{az + b}{cz + d}$$.

### 22.5.19 ?

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 $$f^{n+1}(z) = f(f^{n}(z))$$.

### 22.5.20 ?

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 $$\displaystyle \frac{|f(0)| - |z|}{1 + |f(0)||z|} \leq |f(z)| \leq \frac{|f(0)| + |z|}{1 - |f(0)||z|}$$.

### 22.5.21 ?

Find a conformal map

1. from $$\{ z: |z - 1/2| > 1/2, \text{Re}(z)>0 \}$$ to $$\mathbb H$$

2. from $$\{ z: |z - 1/2| > 1/2, |z| <1 \}$$ to $$\mathbb D$$

3. from the intersection of the disk $$|z + i| < \sqrt{2}$$ with $${\mathbb H}$$ to $${\mathbb D}$$.

4. from $${\mathbb D} \backslash [a, 1)$$ to $${\mathbb D} \backslash [0, 1)$$ ($$0<a<1)$$. \begin{align*} Short solution possible using Blaschke factor\end{align*}

5. from $$\{ z: |z| < 1, \text{Re}(z) > 0 \} \backslash (0, 1/2]$$ to $$\mathbb H$$.

### 22.5.22 ?

Let $$C$$ and $$C'$$ be two circles and let $$z_1 \in C$$, $$z_2 \notin C$$, $$z'_1 \in C'$$, $$z'_2 \notin C'$$. Show that there is a unique fractional linear transformation $$f$$ with $$f(C) = C'$$ and $$f(z_1) = z'_1$$, $$f(z_2) = z'_2$$.

### 22.5.23 ?

Assume $$f_n \in H(\Omega)$$ is a sequence of holomorphic functions on the region $$\Omega$$ that are uniformly bounded on compact subsets and $$f \in H(\Omega)$$ is such that the set $$\displaystyle \{z \in \Omega: \lim_{n \rightarrow \infty} f_n(z) = f(z) \}$$ has a limit point in $$\Omega$$. Show that $$f_n$$ converges to $$f$$ uniformly on compact subsets of $$\Omega$$.

### 22.5.24 ?

Let $$\displaystyle{\psi_{\alpha}(z)=\frac{\alpha-z}{1-\mkern 1.5mu\overline{\mkern-1.5mu\alpha\mkern-1.5mu}\mkern 1.5muz}}$$ with $$|\alpha|<1$$ and $${\mathbb D}=\{z:\ |z|<1\}$$. Prove that

• $$\displaystyle{\frac{1}{\pi}\iint_{{\mathbb D}} |\psi'_{\alpha}|^2 dx dy =1}$$.

• $$\displaystyle{\frac{1}{\pi}\iint_{{\mathbb D}} |\psi'_{\alpha}| dx dy =\frac{1-|\alpha|^2}{|\alpha|^2} \log \frac{1}{1-|\alpha|^2}}$$.

### 22.5.25 ?

Prove that $$\displaystyle{f(z)=-\frac{1}{2}\left(z+\frac{1}{z}\right)}$$ is a conformal map from half disc $$\{z=x+iy:\ |z|<1,\ y>0\}$$ to upper half plane $${\mathbb H}=\{z=x+iy:\ y>0\}$$.

### 22.5.26 ?

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 $$z\in U$$.

### 22.5.27 ?

Compute the following integrals. (i) $$\displaystyle \int_0^\infty \frac{x^{a-1}}{1 + x^n} \, dx$$, $$0< a < n$$ (ii) $$\displaystyle \int_0^\infty \frac{\log x}{(1 + x^2)^2}\, dx$$

### 22.5.28 ?

Let $$0<r<1$$. Show that polynomials $$P_n(z) = 1 + 2z + 3 z^2 + \cdots + n z^{n-1}$$ have no zeros in $$|z|<r$$ for all sufficiently large $$n$$’s.

### 22.5.29 ?

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 $$\displaystyle \|f\|_{(1, r)} = \int_{D_r(z_0)} |f(z)|dx dy$$.

### 22.5.30 ?

Let $$\displaystyle{\psi_{\alpha}(z)=\frac{\alpha-z}{1-\mkern 1.5mu\overline{\mkern-1.5mu\alpha\mkern-1.5mu}\mkern 1.5muz}}$$ with $$|\alpha|<1$$ and $${\mathbb D}=\{z:\ |z|<1\}$$. Prove that

• $$\displaystyle{\frac{1}{\pi}\iint_{{\mathbb D}} |\psi'_{\alpha}|^2 dx dy =1}$$.

• $$\displaystyle{\frac{1}{\pi}\iint_{{\mathbb D}} |\psi'_{\alpha}| dx dy =\frac{1-|\alpha|^2}{|\alpha|^2} \log \frac{1}{1-|\alpha|^2}}$$.

Prove that $$\displaystyle{f(z)=-\frac{1}{2}\left(z+\frac{1}{z}\right)}$$ is a conformal map from half disc $$\{z=x+iy:\ |z|<1,\ y>0\}$$ to upper half plane $$\mathbb H=\{z=x+iy:\ y>0\}$$.

### 22.5.31 ?

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 $$z\in U$$.

### 22.5.32 ?

Compute the following integrals. (i) $$\displaystyle \int_0^\infty \frac{x^{a-1}}{1 + x^n} \, dx$$, $$0< a < n$$ (ii) $$\displaystyle \int_0^\infty \frac{\log x}{(1 + x^2)^2}\, dx$$

### 22.5.33 ?

Let $$0<r<1$$. Show that polynomials $$P_n(z) = 1 + 2z + 3 z^2 + \cdots + n z^{n-1}$$ have no zeros in $$|z|<r$$ for all sufficiently large $$n$$’s.

### 22.5.34 ?

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 $$\displaystyle \|f\|_{(1, r)} = \int_{D_r(z_0)} |f(z)|dx dy$$.

## 22.6 Fall 2016

### 22.6.1 ?

Let $$u(x,y)$$ be harmonic and have continuous partial derivatives of order three in an open disc of radius $$R>0$$.

1. Let two points $$(a,b), (x,y)$$ in this disk be given. Show that the following integral is independent of the path in this disk joining these points: \begin{align*}v(x,y) = \int_{a,b}^{x,y} ( -\frac{\partial u}{\partial y}dx + \frac{\partial u}{\partial x}dy).\end{align*}

2. {=tex} \hfill

1. Prove that $$u(x,y)+i v(x,y)$$ is an analytic function in this disc.

2. Prove that $$v(x,y)$$ is harmonic in this disc.

### 22.6.2 ?

1. $$f(z)= u(x,y) +i v(x,y)$$ 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.

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

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

1. 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?

### 22.6.3 ?

1. $$f: D\rightarrow {\mathbb C}$$ 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*}

2. 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*}

3. 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.

### 22.6.4 ?

1. Let Let $$f:{\mathbb C}\rightarrow {\mathbb C}$$ be an entire function. Assume the existence of a non-negative integer $$m$$, and of positive constants $$L$$ and $$R$$, such that for all $$z$$ with $$|z|>R$$ the inequality \begin{align*}|f(z)| \leq L |z|^m\end{align*} holds. Prove that $$f$$ is a polynomial of degree $$\leq m$$.

2. Let $$f:{\mathbb C}\rightarrow {\mathbb C}$$ be an entire function. Suppose that there exists a real number M such that for all $$z\in {\mathbb C}$$ \begin{align*}\mbox{\textrm Re} (f) \leq M.\end{align*} Prove that $$f$$ must be a constant.

### 22.6.5 ?

Prove that all the roots of the complex polynomial \begin{align*}z^7 - 5 z^3 +12 =0\end{align*} lie between the circles $$|z|=1$$ and $$|z|=2$$.

### 22.6.6 ?

1. 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*}

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

### 22.6.7 ?

Find the conformal map that takes the upper half-plane comformally onto the half-strip $$\{ w=x+iy:\ -\pi/2<x<\pi/2\ y>0\}$$.

### 22.6.8 ?

Compute the integral $$\displaystyle{\int_{-\infty}^{\infty} \frac{e^{-2\pi ix\xi}}{\cosh\pi x}dx}$$ where $$\displaystyle{\cosh z=\frac{e^{z}+e^{-z}}{2}}$$.