Complex Analysis

# Complex Analysis

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Paper 3, Section II, G

commentLet $\gamma$ be a curve (not necessarily closed) in $\mathbb{C}$ and let $[\gamma]$ denote the image of $\gamma$. Let $\phi:[\gamma] \rightarrow \mathbb{C}$ be a continuous function and define

$f(z)=\int_{\gamma} \frac{\phi(\lambda)}{\lambda-z} d \lambda$

for $z \in \mathbb{C} \backslash[\gamma]$. Show that $f$ has a power series expansion about every $a \notin[\gamma]$.

Using Cauchy's Integral Formula, show that a holomorphic function has complex derivatives of all orders. [Properties of power series may be assumed without proof.] Let $f$ be a holomorphic function on an open set $U$ that contains the closed disc $\bar{D}(a, r)$. Obtain an integral formula for the derivative of $f$ on the open disc $D(a, r)$ in terms of the values of $f$ on the boundary of the disc.

Show that if holomorphic functions $f_{n}$ on an open set $U$ converge locally uniformly to a holomorphic function $f$ on $U$, then $f_{n}^{\prime}$ converges locally uniformly to $f^{\prime}$.

Let $D_{1}$ and $D_{2}$ be two overlapping closed discs. Let $f$ be a holomorphic function on some open neighbourhood of $D=D_{1} \cap D_{2}$. Show that there exist open neighbourhoods $U_{j}$ of $D_{j}$ and holomorphic functions $f_{j}$ on $U_{j}, j=1,2$, such that $f(z)=f_{1}(z)+f_{2}(z)$ on $U_{1} \cap U_{2}$.

Paper 4, Section I, $3 G$

commentLet $f$ be a holomorphic function on a neighbourhood of $a \in \mathbb{C}$. Assume that $f$ has a zero of order $k$ at $a$ with $k \geqslant 1$. Show that there exist $\varepsilon>0$ and $\delta>0$ such that for any $b$ with $0<|b|<\varepsilon$ there are exactly $k$ distinct values of $z \in D(a, \delta)$ with $f(z)=b$.

Paper 3, Section II, F

commentDefine the winding number $n(\gamma, w)$ of a closed path $\gamma:[a, b] \rightarrow \mathbb{C}$ around a point $w \in \mathbb{C}$ which does not lie on the image of $\gamma$. [You do not need to justify its existence.]

If $f$ is a meromorphic function, define the order of a zero $z_{0}$ of $f$ and of a pole $w_{0}$ of $f$. State the Argument Principle, and explain how it can be deduced from the Residue Theorem.

How many roots of the polynomial

$z^{4}+10 z^{3}+4 z^{2}+10 z+5$

lie in the right-hand half plane?

Paper 4, Section I, $4 \mathbf{F}$

commentState the Cauchy Integral Formula for a disc. If $f: D\left(z_{0} ; r\right) \rightarrow \mathbb{C}$ is a holomorphic function such that $|f(z)| \leqslant\left|f\left(z_{0}\right)\right|$ for all $z \in D\left(z_{0} ; r\right)$, show using the Cauchy Integral Formula that $f$ is constant.

Paper 3, Section II, F

commentLet $D=\{z \in \mathbb{C}:|z|<1\}$ and let $f: D \rightarrow \mathbb{C}$ be analytic.

(a) If there is a point $a \in D$ such that $|f(z)| \leqslant|f(a)|$ for all $z \in D$, prove that $f$ is constant.

(b) If $f(0)=0$ and $|f(z)| \leqslant 1$ for all $z \in D$, prove that $|f(z)| \leqslant|z|$ for all $z \in D$.

(c) Show that there is a constant $C$ independent of $f$ such that if $f(0)=1$ and $f(z) \notin(-\infty, 0]$ for all $z \in D$ then $|f(z)| \leqslant C$ whenever $|z| \leqslant 1 / 2 .$

[Hint: you may find it useful to consider the principal branch of the map $z \mapsto z^{1 / 2}$.]

(d) Does the conclusion in (c) hold if we replace the hypothesis $f(z) \notin(-\infty, 0]$ for $z \in D$ with the hypothesis $f(z) \neq 0$ for $z \in D$, and keep all other hypotheses? Justify your answer.

Paper 4, Section I, F

comment(a) Let $\Omega \subset \mathbb{C}$ be open, $a \in \Omega$ and suppose that $D_{\rho}(a)=\{z \in \mathbb{C}:|z-a| \leqslant \rho\} \subset \Omega$. Let $f: \Omega \rightarrow \mathbb{C}$ be analytic.

State the Cauchy integral formula expressing $f(a)$ as a contour integral over $C=\partial D_{\rho}(a)$. Give, without proof, a similar expression for $f^{\prime}(a)$.

If additionally $\Omega=\mathbb{C}$ and $f$ is bounded, deduce that $f$ must be constant.

(b) If $g=u+i v: \mathbb{C} \rightarrow \mathbb{C}$ is analytic where $u, v$ are real, and if $u^{2}(z)-u(z) \geqslant v^{2}(z)$ for all $z \in \mathbb{C}$, show that $g$ is constant.

Paper 3, Section II, F

commentLet $f$ be an entire function. Prove Taylor's theorem, that there exist complex numbers $c_{0}, c_{1}, \ldots$ such that $f(z)=\sum_{n=0}^{\infty} c_{n} z^{n}$ for all $z$. [You may assume Cauchy's Integral Formula.]

For a positive real $r$, let $M_{r}=\sup \{|f(z)|:|z|=r\}$. Explain why we have

$\left|c_{n}\right| \leqslant \frac{M_{r}}{r^{n}}$

for all $n$.

Now let $n$ and $r$ be fixed. For which entire functions $f$ do we have $\left|c_{n}\right|=\frac{M_{r}}{r^{n}} ?$

Paper 4, Section I, 4F

commentLet $D$ be a star-domain, and let $f$ be a continuous complex-valued function on $D$. Suppose that for every triangle $T$ contained in $D$ we have

$\int_{\partial T} f(z) d z=0$

Show that $f$ has an antiderivative on $D$.

If we assume instead that $D$ is a domain (not necessarily a star-domain), does this conclusion still hold? Briefly justify your answer.

Paper 3, Section II, G

comment(a) Prove Cauchy's theorem for a triangle.

(b) Write down an expression for the winding number $I(\gamma, a)$ of a closed, piecewise continuously differentiable curve $\gamma$ about a point $a \in \mathbb{C}$ which does not lie on $\gamma$.

(c) Let $U \subset \mathbb{C}$ be a domain, and $f: U \rightarrow \mathbb{C}$ a holomorphic function with no zeroes in $U$. Suppose that for infinitely many positive integers $k$ the function $f$ has a holomorphic $k$-th root. Show that there exists a holomorphic function $F: U \rightarrow \mathbb{C}$ such that $f=\exp F$.

Paper 4, Section I, G

commentState carefully Rouché's theorem. Use it to show that the function $z^{4}+3+e^{i z}$ has exactly one zero $z=z_{0}$ in the quadrant

$\{z \in \mathbb{C} \mid 0<\arg (z)<\pi / 2\}$

and that $\left|z_{0}\right| \leqslant \sqrt{2}$.

Paper 3, Section II, G

commentState the argument principle.

Let $U \subset \mathbb{C}$ be an open set and $f: U \rightarrow \mathbb{C}$ a holomorphic injective function. Show that $f^{\prime}(z) \neq 0$ for each $z$ in $U$ and that $f(U)$ is open.

Stating clearly any theorems that you require, show that for each $a \in U$ and a sufficiently small $r>0$,

$g(w)=\frac{1}{2 \pi i} \int_{|z-a|=r} \frac{z f^{\prime}(z)}{f(z)-w} d z$

defines a holomorphic function on some open disc $D$ about $f(a)$.

Show that $g$ is the inverse for the restriction of $f$ to $g(D)$.

Paper 4, Section I, G

commentLet $f$ be a continuous function defined on a connected open set $D \subset \mathbb{C}$. Prove carefully that the following statements are equivalent.

(i) There exists a holomorphic function $F$ on $D$ such that $F^{\prime}(z)=f(z)$.

(ii) $\int_{\gamma} f(z) d z=0$ holds for every closed curve $\gamma$ in $D$.

Paper 3, Section II, G

commentState the Residue Theorem precisely.

Let $D$ be a star-domain, and let $\gamma$ be a closed path in $D$. Suppose that $f$ is a holomorphic function on $D$, having no zeros on $\gamma$. Let $N$ be the number of zeros of $f$ inside $\gamma$, counted with multiplicity (i.e. order of zero and winding number). Show that

$N=\frac{1}{2 \pi i} \int_{\gamma} \frac{f^{\prime}(z)}{f(z)} d z$

[The Residue Theorem may be used without proof.]

Now suppose that $g$ is another holomorphic function on $D$, also having no zeros on $\gamma$ and with $|g(z)|<|f(z)|$ on $\gamma$. Explain why, for any $0 \leqslant t \leqslant 1$, the expression

$I(t)=\int_{\gamma} \frac{f^{\prime}(z)+\operatorname{tg}^{\prime}(z)}{f(z)+\operatorname{tg}(z)} d z$

is well-defined. By considering the behaviour of the function $I(t)$ as $t$ varies, deduce Rouché's Theorem.

For each $n$, let $p_{n}$ be the polynomial $\sum_{k=0}^{n} \frac{z^{k}}{k !}$. Show that, as $n$ tends to infinity, the smallest modulus of the roots of $p_{n}$ also tends to infinity.

[You may assume any results on convergence of power series, provided that they are stated clearly.]

Paper 4, Section I, G

commentLet $f$ be an entire function. State Cauchy's Integral Formula, relating the $n$th derivative of $f$ at a point $z$ with the values of $f$ on a circle around $z$.

State Liouville's Theorem, and deduce it from Cauchy's Integral Formula.

Let $f$ be an entire function, and suppose that for some $k$ we have that $|f(z)| \leqslant|z|^{k}$ for all $z$. Prove that $f$ is a polynomial.

Paper 3, Section II, E

commentLet $D=\{z \in \mathbb{C}|| z \mid<1\}$ be the open unit disk, and let $C$ be its boundary (the unit circle), with the anticlockwise orientation. Suppose $\phi: C \rightarrow \mathbb{C}$ is continuous. Stating clearly any theorems you use, show that

$g_{\phi}(w)=\frac{1}{2 \pi i} \int_{C} \frac{\phi(z)}{z-w} d z$

is an analytic function of $w$ for $w \in D$.

Now suppose $\phi$ is the restriction of a holomorphic function $F$ defined on some annulus $1-\epsilon<|z|<1+\epsilon$. Show that $g_{\phi}(w)$ is the restriction of a holomorphic function defined on the open disc $|w|<1+\epsilon$.

Let $f_{\phi}:[0,2 \pi] \rightarrow \mathbb{C}$ be defined by $f_{\phi}(\theta)=\phi\left(e^{i \theta}\right)$. Express the coefficients in the power series expansion of $g_{\phi}$ centered at 0 in terms of $f_{\phi}$.

Let $n \in \mathbb{Z}$. What is $g_{\phi}$ in the following cases?

$\phi(z)=z^{n}$.

$\phi(z)=\bar{z}^{n}$.

$\phi(z)=(\operatorname{Re} z)^{2}$.

Paper 4, Section I, E

commentState Rouché's theorem. How many roots of the polynomial $z^{8}+3 z^{7}+6 z^{2}+1$ are contained in the annulus $1<|z|<2$ ?

Paper 3, Section II, E

commentLet $D(a, R)$ denote the disc $|z-a|<R$ and let $f: D(a, R) \rightarrow \mathbb{C}$ be a holomorphic function. Using Cauchy's integral formula show that for every $r \in(0, R)$

$f(a)=\int_{0}^{1} f\left(a+r e^{2 \pi i t}\right) d t$

Deduce that if for every $z \in D(a, R),|f(z)| \leqslant|f(a)|$, then $f$ is constant.

Let $f: D(0,1) \rightarrow D(0,1)$ be holomorphic with $f(0)=0$. Show that $|f(z)| \leqslant|z|$ for all $z \in D(0,1)$. Moreover, show that if $|f(w)|=|w|$ for some $w \neq 0$, then there exists $\lambda$ with $|\lambda|=1$ such that $f(z)=\lambda z$ for all $z \in D(0,1)$.

Paper 4, Section I, $4 \mathrm{E}$

commentLet $h: \mathbb{C} \rightarrow \mathbb{C}$ be a holomorphic function with $h(i) \neq h(-i)$. Does there exist a holomorphic function $f$ defined in $|z|<1$ for which $f^{\prime}(z)=\frac{h(z)}{1+z^{2}}$ ? Does there exist a holomorphic function $f$ defined in $|z|>1$ for which $f^{\prime}(z)=\frac{h(z)}{1+z^{2}}$ ? Justify your answers.

Paper 3, Section II, E

commentLet $g: \mathbb{C} \rightarrow \mathbb{C}$ be a continuous function such that

$\int_{\Gamma} g(z) d z=0$

for any closed curve $\Gamma$ which is the boundary of a rectangle in $\mathbb{C}$ with sides parallel to the real and imaginary axes. Prove that $g$ is analytic.

Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be continuous. Suppose in addition that $f$ is analytic at every point $z \in \mathbb{C}$ with non-zero imaginary part. Show that $f$ is analytic at every point in $\mathbb{C} .$

Let $\mathbb{H}$ be the upper half-plane of complex numbers $z$ with positive imaginary part $\Im(z)>0$. Consider a continuous function $F: \mathbb{H} \cup \mathbb{R} \rightarrow \mathbb{C}$ such that $F$ is analytic on $\mathbb{H}$ and $F(\mathbb{R}) \subset \mathbb{R}$. Define $f: \mathbb{C} \rightarrow \mathbb{C}$ by

$f(z)= \begin{cases}F(z) & \text { if } \Im(z) \geqslant 0 \\ \overline{F(\bar{z})} & \text { if } \Im(z) \leqslant 0\end{cases}$

Show that $f$ is analytic.

Paper 4, Section I, E

commentLet $f(z)$ be an analytic function in an open subset $U$ of the complex plane. Prove that $f$ has derivatives of all orders at any point $z$ in $U$. [You may assume Cauchy's integral formula provided it is clearly stated.]

Paper 3, Section II, G

commentState Morera's theorem. Suppose $f_{n}(n=1,2, \ldots)$ are analytic functions on a domain $U \subset \mathbf{C}$ and that $f_{n}$ tends locally uniformly to $f$ on $U$. Show that $f$ is analytic on $U$. Explain briefly why the derivatives $f_{n}^{\prime}$ tend locally uniformly to $f^{\prime}$.

Suppose now that the $f_{n}$ are nowhere vanishing and $f$ is not identically zero. Let $a$ be any point of $U$; show that there exists a closed disc $\bar{\Delta} \subset U$ with centre $a$, on which the convergence of $f_{n}$ and $f_{n}^{\prime}$ are both uniform, and where $f$ is nowhere zero on $\bar{\Delta} \backslash\{a\}$. By considering

$\frac{1}{2 \pi i} \int_{C} \frac{f_{n}^{\prime}(w)}{f_{n}(w)} d w$

(where $C$ denotes the boundary of $\bar{\Delta}$ ), or otherwise, deduce that $f(a) \neq 0$.

Paper 4, Section I, G

commentState the principle of the argument for meromorphic functions and show how it follows from the Residue theorem.

Paper 3, Section II, E

commentFor each positive real number $R$ write $B_{R}=\{z \in \mathbb{C}:|z| \leqslant R\}$. If $F$ is holomorphic on some open set containing $B_{R}$, we define

$J(F, R)=\frac{1}{2 \pi} \int_{0}^{2 \pi} \log \left|F\left(R e^{i \theta}\right)\right| d \theta$

If $F_{1}, F_{2}$ are both holomorphic on some open set containing $B_{R}$, show that $J\left(F_{1} F_{2}, R\right)=$ $J\left(F_{1}, R\right)+J\left(F_{2}, R\right) .$

Suppose that $F(0)=1$ and that $F$ does not vanish on some open set containing $B_{R}$. By showing that there is a holomorphic branch of logarithm of $F$ and then considering $z^{-1} \log F(z)$, prove that $J(F, R)=0$.

Suppose that $|w|<R$. Prove that the function $\psi_{W, R}(z)=R(z-w) /\left(R^{2}-\bar{w} z\right)$ has modulus 1 on $|z|=R$ and hence that it satisfies $J\left(\psi_{W, R}, R\right)=0$.

Suppose now that $F: \mathbb{C} \rightarrow \mathbb{C}$ is holomorphic and not identically zero, and let $R$ be such that no zeros of $F$ satisfy $|z|=R$. Briefly explain why there are only finitely many zeros of $F$ in $B_{R}$ and, assuming these are listed with the correct multiplicity, derive a formula for $J(F, R)$ in terms of the zeros, $R$, and $F(0)$.

Suppose that $F$ has a zero at every lattice point (point with integer coordinates) except for $(0,0)$. Show that there is a constant $c>0$ such that $\left|F\left(z_{i}\right)\right|>e^{c\left|z_{i}\right|^{2}}$ for a sequence $z_{1}, z_{2}, \ldots$ of complex numbers tending to infinity.

Paper 4, Section I, E

commentState Rouché's Theorem. How many complex numbers $z$ are there with $|z| \leqslant 1$ and $2 z=\sin z ?$

3.II.14E

commentState and prove Rouché's theorem, and use it to count the number of zeros of $3 z^{9}+8 z^{6}+z^{5}+2 z^{3}+1$ inside the annulus $\{z: 1<|z|<2\}$.

Let $\left(p_{n}\right)_{n=1}^{\infty}$ be a sequence of polynomials of degree at most $d$ with the property that $p_{n}(z)$ converges uniformly on compact subsets of $\mathbb{C}$ as $n \rightarrow \infty$. Prove that there is a polynomial $p$ of degree at most $d$ such that $p_{n} \rightarrow p$ uniformly on compact subsets of $\mathbb{C}$. [If you use any results about uniform convergence of analytic functions, you should prove them.]

Suppose that $p$ has $d$ distinct roots $z_{1}, \ldots, z_{d}$. Using Rouché's theorem, or otherwise, show that for each $i$ there is a sequence $\left(z_{i, n}\right)_{n=1}^{\infty}$ such that $p_{n}\left(z_{i, n}\right)=0$ and $z_{i, n} \rightarrow z_{i}$ as $n \rightarrow \infty$.

4.I.4E

commentSuppose that $f$ and $g$ are two functions which are analytic on the whole complex plane $\mathbb{C}$. Suppose that there is a sequence of distinct points $z_{1}, z_{2}, \ldots$ with $\left|z_{i}\right| \leqslant 1$ such that $f\left(z_{i}\right)=g\left(z_{i}\right)$. Show that $f(z)=g(z)$ for all $z \in \mathbb{C}$. [You may assume any results on Taylor expansions you need, provided they are clearly stated.]

What happens if the assumption that $\left|z_{i}\right| \leqslant 1$ is dropped?

3.II.14H

commentSay that a function on the complex plane $\mathbb{C}$ is periodic if $f(z+1)=f(z)$ and $f(z+i)=f(z)$ for all $z$. If $f$ is a periodic analytic function, show that $f$ is constant.

If $f$ is a meromorphic periodic function, show that the number of zeros of $f$ in the square $[0,1) \times[0,1)$ is equal to the number of poles, both counted with multiplicities.

Define

$f(z)=\frac{1}{z^{2}}+\sum_{w}\left[\frac{1}{(z-w)^{2}}-\frac{1}{w^{2}}\right]$

where the sum runs over all $w=a+b i$ with $a$ and $b$ integers, not both 0 . Show that this series converges to a meromorphic periodic function on the complex plane.

4.I.4H

commentState the argument principle.

Show that if $f$ is an analytic function on an open set $U \subset \mathbb{C}$ which is one-to-one, then $f^{\prime}(z) \neq 0$ for all $z \in U$.

3.II.14H

commentAssuming the principle of the argument, prove that any polynomial of degree $n$ has precisely $n$ zeros in $\mathbf{C}$, counted with multiplicity.

Consider a polynomial $p(z)=z^{4}+a z^{3}+b z^{2}+c z+d$, and let $R$ be a positive real number such that $|a| R^{3}+|b| R^{2}+|c| R+|d|<R^{4}$. Define a curve $\Gamma:[0,1] \rightarrow \mathbf{C}$ by

$\Gamma(t)= \begin{cases}p\left(R e^{\pi i t}\right) & \text { for } 0 \leqslant t \leqslant \frac{1}{2} \\ (2-2 t) p(i R)+(2 t-1) p(R) & \text { for } \frac{1}{2} \leqslant t \leqslant 1\end{cases}$

Show that the winding number $n(\Gamma, 0)=1$.

Suppose now that $p(z)$ has real coefficients, that $z^{4}-b z^{2}+d$ has no real zeros, and that the real zeros of $p(z)$ are all strictly negative. Show that precisely one of the zeros of $p(z)$ lies in the quadrant $\{x+i y: x>0, y>0\}$.

[Standard results about winding numbers may be quoted without proof; in particular, you may wish to use the fact that if $\gamma_{i}:[0,1] \rightarrow \mathbf{C}, i=1,2$, are two closed curves with $\left|\gamma_{2}(t)-\gamma_{1}(t)\right|<\left|\gamma_{1}(t)\right|$ for all $t$, then $n\left(\gamma_{1}, 0\right)=n\left(\gamma_{2}, 0\right)$.]

4.I.4H

commentState the principle of isolated zeros for an analytic function on a domain in $\mathbf{C}$.

Suppose $f$ is an analytic function on $\mathbf{C} \backslash\{0\}$, which is real-valued at the points $1 / n$, for $n=1,2, \ldots$, and does not have an essential singularity at the origin. Prove that $f(z)=\overline{f(\bar{z})}$ for all $z \in \mathbf{C} \backslash\{0\}$.

3.II.14A

commentState the Cauchy integral formula, and use it to deduce Liouville's theorem.

Let $f$ be a meromorphic function on the complex plane such that $\left|f(z) / z^{n}\right|$ is bounded outside some disc (for some fixed integer $n$ ). By considering Laurent expansions, or otherwise, show that $f$ is a rational function in $z$.

4.I.4A

commentLet $\gamma:[0,1] \rightarrow \mathbf{C}$ be a closed path, where all paths are assumed to be piecewise continuously differentiable, and let $a$ be a complex number not in the image of $\gamma$. Write down an expression for the winding number $n(\gamma, a)$ in terms of a contour integral. From this characterization of the winding number, prove the following properties:

(a) If $\gamma_{1}$ and $\gamma_{2}$ are closed paths not passing through zero, and if $\gamma:[0,1] \rightarrow \mathbf{C}$ is defined by $\gamma(t)=\gamma_{1}(t) \gamma_{2}(t)$ for all $t$, then

$n(\gamma, 0)=n\left(\gamma_{1}, 0\right)+n\left(\gamma_{2}, 0\right)$

(b) If $\eta:[0,1] \rightarrow \mathbf{C}$ is a closed path whose image is contained in $\{\operatorname{Re}(z)>0\}$, then $n(\eta, 0)=0$.

(c) If $\gamma_{1}$ and $\gamma_{2}$ are closed paths and $a$ is a complex number, not in the image of either path, such that

$\left|\gamma_{1}(t)-\gamma_{2}(t)\right|<\left|\gamma_{1}(t)-a\right|$

for all $t$, then $n\left(\gamma_{1}, a\right)=n\left(\gamma_{2}, a\right)$.

[You may wish here to consider the path defined by $\eta(t)=1-\left(\gamma_{1}(t)-\gamma_{2}(t)\right) /\left(\gamma_{1}(t)-a\right)$.]