• # Paper 1, Section I, B

Let $x>0, x \neq 2$, and let $C_{x}$ denote the positively oriented circle of radius $x$ centred at the origin. Define

$g(x)=\oint_{C_{x}} \frac{z^{2}+e^{z}}{z^{2}(z-2)} d z$

Evaluate $g(x)$ for $x \in(0, \infty) \backslash\{2\}$.

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• # Paper 1, Section II, G

(a) State a theorem establishing Laurent series of analytic functions on suitable domains. Give a formula for the $n^{\text {th }}$Laurent coefficient.

Define the notion of isolated singularity. State the classification of an isolated singularity in terms of Laurent coefficients.

Compute the Laurent series of

$f(z)=\frac{1}{z(z-1)}$

on the annuli $A_{1}=\{z: 0<|z|<1\}$ and $A_{2}=\{z: 1<|z|\}$. Using this example, comment on the statement that Laurent coefficients are unique. Classify the singularity of $f$ at 0 .

(b) Let $U$ be an open subset of the complex plane, let $a \in U$ and let $U^{\prime}=U \backslash\{a\}$. Assume that $f$ is an analytic function on $U^{\prime}$ with $|f(z)| \rightarrow \infty$ as $z \rightarrow a$. By considering the Laurent series of $g(z)=\frac{1}{f(z)}$ at $a$, classify the singularity of $f$ at $a$ in terms of the Laurent coefficients. [You may assume that a continuous function on $U$ that is analytic on $U^{\prime}$ is analytic on $U$.]

Now let $f: \mathbb{C} \rightarrow \mathbb{C}$ be an entire function with $|f(z)| \rightarrow \infty$ as $z \rightarrow \infty$. By considering Laurent series at 0 of $f(z)$ and of $h(z)=f\left(\frac{1}{z}\right)$, show that $f$ is a polynomial.

(c) Classify, giving reasons, the singularity at the origin of each of the following functions and in each case compute the residue:

$g(z)=\frac{\exp (z)-1}{z \log (z+1)} \quad \text { and } \quad h(z)=\sin (z) \sin (1 / z)$

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• # Paper 2, Section II, B

(a) Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be an entire function and let $a>0, b>0$ be constants. Show that if

$|f(z)| \leqslant a|z|^{n / 2}+b$

for all $z \in \mathbb{C}$, where $n$ is a positive odd integer, then $f$ must be a polynomial with degree not exceeding $\lfloor n / 2\rfloor$ (closest integer part rounding down).

Does there exist a function $f$, analytic in $\mathbb{C} \backslash\{0\}$, such that $|f(z)| \geqslant 1 / \sqrt{|z|}$ for all nonzero $z ?$ Justify your answer.

(b) State Liouville's Theorem and use it to show the following.

(i) If $u$ is a positive harmonic function on $\mathbb{R}^{2}$, then $u$ is a constant function.

(ii) Let $L=\{z \mid z=a x+b, x \in \mathbb{R}\}$ be a line in $\mathbb{C}$ where $a, b \in \mathbb{C}, a \neq 0$. If $f: \mathbb{C} \rightarrow \mathbb{C}$ is an entire function such that $f(\mathbb{C}) \cap L=\emptyset$, then $f$ is a constant function.

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• # Paper 1, Section I, G

Let $D$ be the open disc with centre $e^{2 \pi i / 6}$ and radius 1 , and let $L$ be the open lower half plane. Starting with a suitable Möbius map, find a conformal equivalence (or conformal bijection) of $D \cap L$ onto the open unit disc.

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• # Paper 1, Section II, G

Let $\ell(z)$ be an analytic branch of $\log z$ on a domain $D \subset \mathbb{C} \backslash\{0\}$. Write down an analytic branch of $z^{1 / 2}$ on $D$. Show that if $\psi_{1}(z)$ and $\psi_{2}(z)$ are two analytic branches of $z^{1 / 2}$ on $D$, then either $\psi_{1}(z)=\psi_{2}(z)$ for all $z \in D$ or $\psi_{1}(z)=-\psi_{2}(z)$ for all $z \in D$.

Describe the principal value or branch $\sigma_{1}(z)$ of $z^{1 / 2}$ on $D_{1}=\mathbb{C} \backslash\{x \in \mathbb{R}: x \leqslant 0\}$. Describe a branch $\sigma_{2}(z)$ of $z^{1 / 2}$ on $D_{2}=\mathbb{C} \backslash\{x \in \mathbb{R}: x \geqslant 0\}$.

Construct an analytic branch $\varphi(z)$ of $\sqrt{1-z^{2}}$ on $\mathbb{C} \backslash\{x \in \mathbb{R}:-1 \leqslant x \leqslant 1\}$ with $\varphi(2 i)=\sqrt{5}$. [If you choose to use $\sigma_{1}$ and $\sigma_{2}$ in your construction, then you may assume without proof that they are analytic.]

Show that for $0<|z|<1$ we have $\varphi(1 / z)=-i \sigma_{1}\left(1-z^{2}\right) / z$. Hence find the first three terms of the Laurent series of $\varphi(1 / z)$ about 0 .

Set $f(z)=\varphi(z) /\left(1+z^{2}\right)$ for $|z|>1$ and $g(z)=f(1 / z) / z^{2}$ for $0<|z|<1$. Compute the residue of $g$ at 0 and use it to compute the integral

$\int_{|z|=2} f(z) d z$

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• # Paper 2, Section II, B

For the function

$f(z)=\frac{1}{z(z-2)}$

find the Laurent expansions

(i) about $z=0$ in the annulus $0<|z|<2$,

(ii) about $z=0$ in the annulus $2<|z|<\infty$,

(iii) about $z=1$ in the annulus $0<|z-1|<1$.

What is the nature of the singularity of $f$, if any, at $z=0, z=\infty$ and $z=1$ ?

Using an integral of $f$, or otherwise, evaluate

$\int_{0}^{2 \pi} \frac{2-\cos \theta}{5-4 \cos \theta} d \theta$

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• # Paper 1, Section I, F

What is the Laurent series for a function $f$ defined in an annulus $A$ ? Find the Laurent series for $f(z)=\frac{10}{(z+2)\left(z^{2}+1\right)}$ on the annuli

\begin{aligned} &A_{1}=\{z \in \mathbb{C}|0<| z \mid<1\} \quad \text { and } \\ &A_{2}=\{z \in \mathbb{C}|1<| z \mid<2\} \end{aligned}

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• # Paper 1, Section II, F

State and prove Jordan's lemma.

What is the residue of a function $f$ at an isolated singularity $a$ ? If $f(z)=\frac{g(z)}{(z-a)^{k}}$ with $k$ a positive integer, $g$ analytic, and $g(a) \neq 0$, derive a formula for the residue of $f$ at $a$ in terms of derivatives of $g$.

Evaluate

$\int_{-\infty}^{\infty} \frac{x^{3} \sin x}{\left(1+x^{2}\right)^{2}} d x$

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• # Paper 2, Section II, D

Let $C_{1}$ and $C_{2}$ be smooth curves in the complex plane, intersecting at some point $p$. Show that if the map $f: \mathbb{C} \rightarrow \mathbb{C}$ is complex differentiable, then it preserves the angle between $C_{1}$ and $C_{2}$ at $p$, provided $f^{\prime}(p) \neq 0$. Give an example that illustrates why the condition $f^{\prime}(p) \neq 0$ is important.

Show that $f(z)=z+1 / z$ is a one-to-one conformal map on each of the two regions $|z|>1$ and $0<|z|<1$, and find the image of each region.

Hence construct a one-to-one conformal map from the unit disc to the complex plane with the intervals $(-\infty,-1 / 2]$ and $[1 / 2, \infty)$ removed.

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• # Paper 1, Section I, A

(a) Show that

$w=\log (z)$

is a conformal mapping from the right half $z$-plane, $\operatorname{Re}(z)>0$, to the strip

$S=\left\{w:-\frac{\pi}{2}<\operatorname{Im}(w)<\frac{\pi}{2}\right\}$

for a suitably chosen branch of $\log (z)$ that you should specify.

(b) Show that

$w=\frac{z-1}{z+1}$

is a conformal mapping from the right half $z$-plane, $\operatorname{Re}(z)>0$, to the unit disc

$D=\{w:|w|<1\}$

(c) Deduce a conformal mapping from the strip $S$ to the disc $D$.

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• # Paper 1, Section II, A

(a) Let $C$ be a rectangular contour with vertices at $\pm R+\pi i$ and $\pm R-\pi i$ for some $R>0$ taken in the anticlockwise direction. By considering

$\lim _{R \rightarrow \infty} \oint_{C} \frac{e^{i z^{2} / 4 \pi}}{e^{z / 2}-e^{-z / 2}} d z$

show that

$\lim _{R \rightarrow \infty} \int_{-R}^{R} e^{i x^{2} / 4 \pi} d x=2 \pi e^{\pi i / 4}$

(b) By using a semi-circular contour in the upper half plane, calculate

$\int_{0}^{\infty} \frac{x \sin (\pi x)}{x^{2}+a^{2}} d x$

for $a>0$.

[You may use Jordan's Lemma without proof.]

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• # Paper 2, Section II, A

(a) Let $f(z)$ be a complex function. Define the Laurent series of $f(z)$ about $z=z_{0}$, and give suitable formulae in terms of integrals for calculating the coefficients of the series.

(b) Calculate, by any means, the first 3 terms in the Laurent series about $z=0$ for

$f(z)=\frac{1}{e^{2 z}-1}$

Indicate the range of values of $|z|$ for which your series is valid.

(c) Let

$g(z)=\frac{1}{2 z}+\sum_{k=1}^{m} \frac{z}{z^{2}+\pi^{2} k^{2}}$

Classify the singularities of $F(z)=f(z)-g(z)$ for $|z|<(m+1) \pi$.

(d) By considering

$\oint_{C_{R}} \frac{F(z)}{z^{2}} d z$

where $C_{R}=\{|z|=R\}$ for some suitably chosen $R>0$, show that

$\sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6}$

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• # Paper 1, Section I, A

Let $F(z)=u(x, y)+i v(x, y)$ where $z=x+i y$. Suppose $F(z)$ is an analytic function of $z$ in a domain $\mathcal{D}$ of the complex plane.

Derive the Cauchy-Riemann equations satisfied by $u$ and $v$.

For $u=\frac{x}{x^{2}+y^{2}}$ find a suitable function $v$ and domain $\mathcal{D}$ such that $F=u+i v$ is analytic in $\mathcal{D}$.

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• # Paper 1, Section II, A

(a) Let $f(z)$ be defined on the complex plane such that $z f(z) \rightarrow 0$ as $|z| \rightarrow \infty$ and $f(z)$ is analytic on an open set containing $\operatorname{Im}(z) \geqslant-c$, where $c$ is a positive real constant.

Let $C_{1}$ be the horizontal contour running from $-\infty-i c$ to $+\infty-i c$ and let

$F(\lambda)=\frac{1}{2 \pi i} \int_{C_{1}} \frac{f(z)}{z-\lambda} d z$

By evaluating the integral, show that $F(\lambda)$ is analytic for $\operatorname{Im}(\lambda)>-c$.

(b) Let $g(z)$ be defined on the complex plane such that $z g(z) \rightarrow 0$ as $|z| \rightarrow \infty$ with $\operatorname{Im}(z) \geqslant-c$. Suppose $g(z)$ is analytic at all points except $z=\alpha_{+}$and $z=\alpha_{-}$which are simple poles with $\operatorname{Im}\left(\alpha_{+}\right)>c$ and $\operatorname{Im}\left(\alpha_{-}\right)<-c$.

Let $C_{2}$ be the horizontal contour running from $-\infty+i c$ to $+\infty+i c$, and let

$\begin{gathered} H(\lambda)=\frac{1}{2 \pi i} \int_{C_{1}} \frac{g(z)}{z-\lambda} d z \\ J(\lambda)=-\frac{1}{2 \pi i} \int_{C_{2}} \frac{g(z)}{z-\lambda} d z . \end{gathered}$

(i) Show that $H(\lambda)$ is analytic for $\operatorname{Im}(\lambda)>-c$.

(ii) Show that $J(\lambda)$ is analytic for $\operatorname{Im}(\lambda).

(iii) Show that if $-c<\operatorname{Im}(\lambda) then $H(\lambda)+J(\lambda)=g(\lambda)$.

[You should be careful to make sure you consider all points in the required regions.]

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• # Paper 2, Section II, 13A

State the residue theorem.

By considering

$\oint_{C} \frac{z^{1 / 2} \log z}{1+z^{2}} d z$

with $C$ a suitably chosen contour in the upper half plane or otherwise, evaluate the real integrals

$\int_{0}^{\infty} \frac{x^{1 / 2} \log x}{1+x^{2}} d x$

and

$\int_{0}^{\infty} \frac{x^{1 / 2}}{1+x^{2}} d x$

where $x^{1 / 2}$ is taken to be the positive square root.

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• # Paper 1, Section I, A

Classify the singularities of the following functions at both $z=0$ and at the point at infinity on the extended complex plane:

\begin{aligned} f_{1}(z) &=\frac{e^{z}}{z \sin ^{2} z}, \\ f_{2}(z) &=\frac{1}{z^{2}(1-\cos z)}, \\ f_{3}(z) &=z^{2} \sin (1 / z) \end{aligned}

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• # Paper 1, Section II, A

Let $w=u+i v$ and let $z=x+i y$, for $u, v, x, y$ real.

(a) Let A be the map defined by $w=\sqrt{z}$, using the principal branch. Show that A maps the region to the left of the parabola $y^{2}=4(1-x)$ on the $z-$ plane, with the negative real axis $x \in(-\infty, 0]$ removed, into the vertical strip of the $w-$ plane between the lines $u=0$ and $u=1$.

(b) Let $\mathrm{B}$ be the map defined by $w=\tan ^{2}(z / 2)$. Show that $\mathrm{B}$ maps the vertical strip of the $z$-plane between the lines $x=0$ and $x=\pi / 2$ into the region inside the unit circle on the $w$-plane, with the part $u \in(-1,0]$ of the negative real axis removed.

(c) Using the results of parts (a) and (b), show that the map C, defined by $w=\tan ^{2}(\pi \sqrt{z} / 4)$, maps the region to the left of the parabola $y^{2}=4(1-x)$ on the $z$-plane, including the negative real axis, onto the unit disc on the $w$-plane.

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• # Paper 2, Section II, A

Let $a=N+1 / 2$ for a positive integer $N$. Let $C_{N}$ be the anticlockwise contour defined by the square with its four vertices at $a \pm i a$ and $-a \pm i a$. Let

$I_{N}=\oint_{C_{N}} \frac{d z}{z^{2} \sin (\pi z)}$

Show that $1 / \sin (\pi z)$ is uniformly bounded on the contours $C_{N}$ as $N \rightarrow \infty$, and hence that $I_{N} \rightarrow 0$ as $N \rightarrow \infty$.

Using this result, establish that

$\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^{2}}=\frac{\pi^{2}}{12}$

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• # Paper 1, Section I, B

Consider the analytic (holomorphic) functions $f$ and $g$ on a nonempty domain $\Omega$ where $g$ is nowhere zero. Prove that if $|f(z)|=|g(z)|$ for all $z$ in $\Omega$ then there exists a real constant $\alpha$ such that $f(z)=e^{i \alpha} g(z)$ for all $z$ in $\Omega$.

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• # Paper 1, Section II, B

(i) Show that transformations of the complex plane of the form

$\zeta=\frac{a z+b}{c z+d}$

always map circles and lines to circles and lines, where $a, b, c$ and $d$ are complex numbers such that $a d-b c \neq 0$.

(ii) Show that the transformation

$\zeta=\frac{z-\alpha}{\bar{\alpha} z-1}, \quad|\alpha|<1$

maps the unit disk centered at $z=0$ onto itself.

(iii) Deduce a conformal transformation that maps the non-concentric annular domain $\Omega=\{|z|<1,|z-c|>c\}, 0, to a concentric annular domain.

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• # Paper 2, Section II, B

(i) A function $f(z)$ has a pole of order $m$ at $z=z_{0}$. Derive a general expression for the residue of $f(z)$ at $z=z_{0}$ involving $f$ and its derivatives.

(ii) Using contour integration along a contour in the upper half-plane, determine the value of the integral

$I=\int_{0}^{\infty} \frac{(\ln x)^{2}}{\left(1+x^{2}\right)^{2}} \mathrm{~d} x$

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• # Paper 1, Section I, B

Let $f(z)$ be an analytic/holomorphic function defined on an open set $D$, and let $z_{0} \in D$ be a point such that $f^{\prime}\left(z_{0}\right) \neq 0$. Show that the transformation $w=f(z)$ preserves the angle between smooth curves intersecting at $z_{0}$. Find such a transformation $w=f(z)$ that maps the second quadrant of the unit disc (i.e. $|z|<1, \pi / 2<\arg (z)<\pi)$ to the region in the first quadrant of the complex plane where $|w|>1$ (i.e. the region in the first quadrant outside the unit circle).

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• # Paper 1, Section II, B

By choice of a suitable contour show that for $a>b>0$

$\int_{0}^{2 \pi} \frac{\sin ^{2} \theta d \theta}{a+b \cos \theta}=\frac{2 \pi}{b^{2}}\left[a-\sqrt{a^{2}-b^{2}}\right]$

Hence evaluate

$\int_{0}^{1} \frac{\left(1-x^{2}\right)^{1 / 2} x^{2} d x}{1+x^{2}}$

using the substitution $x=\cos (\theta / 2)$.

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• # Paper 2, Section II, B

By considering a rectangular contour, show that for $0 we have

$\int_{-\infty}^{\infty} \frac{e^{a x}}{e^{x}+1} d x=\frac{\pi}{\sin \pi a}$

Hence evaluate

$\int_{0}^{\infty} \frac{d t}{t^{5 / 6}(1+t)}$

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• # Paper 1, Section I, $2 \mathrm{D}$

Classify the singularities (in the finite complex plane) of the following functions: (i) $\frac{1}{(\cosh z)^{2}}$; (ii) $\frac{1}{\cos (1 / z)}$; (iii) $\frac{1}{\log z} \quad(-\pi<\arg z<\pi)$; (iv) $\frac{z^{\frac{1}{2}}-1}{\sin \pi z} \quad(-\pi<\arg z<\pi)$.

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• # Paper 1, Section II, E

Suppose $p(z)$ is a polynomial of even degree, all of whose roots satisfy $|z|. Explain why there is a holomorphic (i.e. analytic) function $h(z)$ defined on the region $R<|z|<\infty$ which satisfies $h(z)^{2}=p(z)$. We write $h(z)=\sqrt{p(z)}$

By expanding in a Laurent series or otherwise, evaluate

$\int_{C} \sqrt{z^{4}-z} d z$

where $C$ is the circle of radius 2 with the anticlockwise orientation. (Your answer will be well-defined up to a factor of $\pm 1$, depending on which square root you pick.)

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• # Paper 2, Section II, 13D Let

$I=\oint_{C} \frac{e^{i z^{2} / \pi}}{1+e^{-2 z}} d z$

where $C$ is the rectangle with vertices at $\pm R$ and $\pm R+i \pi$, traversed anti-clockwise.

(i) Show that $I=\frac{\pi(1+i)}{\sqrt{2}}$.

(ii) Assuming that the contribution to $I$ from the vertical sides of the rectangle is negligible in the limit $R \rightarrow \infty$, show that

$\int_{-\infty}^{\infty} e^{i x^{2} / \pi} d x=\frac{\pi(1+i)}{\sqrt{2}}$

(iii) Justify briefly the assumption that the contribution to $I$ from the vertical sides of the rectangle is negligible in the limit $R \rightarrow \infty$.

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• # Paper 1, Section I, A

Find a conformal transformation $\zeta=\zeta(z)$ that maps the domain $D, 0<\arg z<\frac{3 \pi}{2}$, on to the strip $0<\operatorname{Im}(\zeta)<1$.

Hence find a bounded harmonic function $\phi$ on $D$ subject to the boundary conditions $\phi=0, A$ on $\arg z=0, \frac{3 \pi}{2}$, respectively, where $A$ is a real constant.

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• # Paper 1, Section II, 13A

Using Cauchy's integral theorem, write down the value of a holomorphic function $f(z)$ where $|z|<1$ in terms of a contour integral around the unit circle, $\zeta=e^{i \theta} .$

By considering the point $1 / \bar{z}$, or otherwise, show that

$f(z)=\frac{1}{2 \pi} \int_{0}^{2 \pi} f(\zeta) \frac{1-|z|^{2}}{|\zeta-z|^{2}} \mathrm{~d} \theta$

By setting $z=r e^{i \alpha}$, show that for any harmonic function $u(r, \alpha)$,

$u(r, \alpha)=\frac{1}{2 \pi} \int_{0}^{2 \pi} u(1, \theta) \frac{1-r^{2}}{1-2 r \cos (\alpha-\theta)+r^{2}} \mathrm{~d} \theta$

if $r<1$.

Assuming that the function $v(r, \alpha)$, which is the conjugate harmonic function to $u(r, \alpha)$, can be written as

$v(r, \alpha)=v(0)+\frac{1}{\pi} \int_{0}^{2 \pi} u(1, \theta) \frac{r \sin (\alpha-\theta)}{1-2 r \cos (\alpha-\theta)+r^{2}} \mathrm{~d} \theta$

deduce that

$f(z)=i v(0)+\frac{1}{2 \pi} \int_{0}^{2 \pi} u(1, \theta) \frac{\zeta+z}{\zeta-z} \mathrm{~d} \theta$

[You may use the fact that on the unit circle, $\zeta=1 / \bar{\zeta}$, and hence

$\left.\frac{\zeta}{\zeta-1 / \bar{z}}=-\frac{\bar{z}}{\bar{\zeta}-\bar{z}} \cdot\right]$

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• # Paper 2, Section II, A

By a suitable choice of contour show that, for $-1<\alpha<1$,

$\int_{0}^{\infty} \frac{x^{\alpha}}{1+x^{2}} \mathrm{~d} x=\frac{\pi}{2 \cos (\alpha \pi / 2)}$

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• # Paper 1, Section I, A

Derive the Cauchy-Riemann equations satisfied by the real and imaginary parts of a complex analytic function $f(z)$.

If $|f(z)|$ is constant on $|z|<1$, prove that $f(z)$ is constant on $|z|<1$.

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• # Paper 1, Section II, A

(i) Let $-1<\alpha<0$ and let

\begin{aligned} &f(z)=\frac{\log (z-\alpha)}{z} \text { where }-\pi \leqslant \arg (z-\alpha)<\pi \\ &g(z)=\frac{\log z}{z} \quad \text { where }-\pi \leqslant \arg (z)<\pi \end{aligned}

Here the logarithms take their principal values. Give a sketch to indicate the positions of the branch cuts implied by the definitions of $f(z)$ and $g(z)$.

(ii) Let $h(z)=f(z)-g(z)$. Explain why $h(z)$ is analytic in the annulus $1 \leqslant|z| \leqslant R$ for any $R>1$. Obtain the first three terms of the Laurent expansion for $h(z)$ around $z=0$ in this annulus and hence evaluate

$\oint_{|z|=2} h(z) d z$

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• # Paper 2, Section II, A

(i) Let $C$ be an anticlockwise contour defined by a square with vertices at $z=x+i y$ where

$|x|=|y|=\left(2 N+\frac{1}{2}\right) \pi$

for large integer $N$. Let

$I=\oint_{C} \frac{\pi \cot z}{(z+\pi a)^{4}} d z$

Assuming that $I \rightarrow 0$ as $N \rightarrow \infty$, prove that, if $a$ is not an integer, then

$\sum_{n=-\infty}^{\infty} \frac{1}{(n+a)^{4}}=\frac{\pi^{4}}{3 \sin ^{2}(\pi a)}\left(\frac{3}{\sin ^{2}(\pi a)}-2\right) .$

(ii) Deduce the value of

$\sum_{n=-\infty}^{\infty} \frac{1}{\left(n+\frac{1}{2}\right)^{4}}$

(iii) Briefly justify the assumption that $I \rightarrow 0$ as $N \rightarrow \infty$.

[Hint: For part (iii) it is sufficient to consider, at most, one vertical side of the square and one horizontal side and to use a symmetry argument for the remaining sides.]

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• # Paper 1, Section I, A

(a) Write down the definition of the complex derivative of the function $f(z)$ of a single complex variable.

(b) Derive the Cauchy-Riemann equations for the real and imaginary parts $u(x, y)$ and $v(x, y)$ of $f(z)$, where $z=x+i y$ and

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

(c) State necessary and sufficient conditions on $u(x, y)$ and $v(x, y)$ for the function $f(z)$ to be complex differentiable.

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• # Paper 1, Section II, A

Calculate the following real integrals by using contour integration. Justify your steps carefully.

(a)

$I_{1}=\int_{0}^{\infty} \frac{x \sin x}{x^{2}+a^{2}} d x, \quad a>0$

(b)

$I_{2}=\int_{0}^{\infty} \frac{x^{1 / 2} \log x}{1+x^{2}} d x$

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• # Paper 2, Section II, A

(a) Prove that a complex differentiable map, $f(z)$, is conformal, i.e. preserves angles, provided a certain condition holds on the first complex derivative of $f(z)$.

(b) Let $D$ be the region

$D:=\{z \in \mathbb{C}:|z-1|>1 \text { and }|z-2|<2\}$

Draw the region $D$. It might help to consider the two sets

\begin{aligned} &C(1):=\{z \in \mathbb{C}:|z-1|=1\} \\ &C(2):=\{z \in \mathbb{C}:|z-2|=2\} \end{aligned}

(c) For the transformations below identify the images of $D$.

Step 1: The first map is $f_{1}(z)=\frac{z-1}{z}$,

Step 2: The second map is the composite $f_{2} f_{1}$ where $f_{2}(z)=\left(z-\frac{1}{2}\right) i$,

Step 3: The third map is the composite $f_{3} f_{2} f_{1}$ where $f_{3}(z)=e^{2 \pi z}$.

(d) Write down the inverse map to the composite $f_{3} f_{2} f_{1}$, explaining any choices of branch.

[The composite $f_{2} f_{1}$ means $f_{2}\left(f_{1}(z)\right)$.]

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• # Paper 1, Section I, D

Let $f(z)=u(x, y)+i v(x, y)$, where $z=x+i y$, be an analytic function of $z$ in a domain $D$ of the complex plane. Derive the Cauchy-Riemann equations relating the partial derivatives of $u$ and $v$.

For $u=e^{-x} \cos y$, find $v$ and hence $f(z)$.

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• # Paper 1, Section II, D

Consider the real function $f(t)$ of a real variable $t$ defined by the following contour integral in the complex $s$-plane:

$f(t)=\frac{1}{2 \pi i} \int_{\Gamma} \frac{e^{s t}}{\left(s^{2}+1\right) s^{1 / 2}} d s,$

where the contour $\Gamma$ is the line $s=\gamma+i y,-\infty, for constant $\gamma>0$. By closing the contour appropriately, show that

$f(t)=\sin (t-\pi / 4)+\frac{1}{\pi} \int_{0}^{\infty} \frac{e^{-r t} d r}{\left(r^{2}+1\right) r^{1 / 2}}$

when $t>0$ and is zero when $t<0$. You should justify your evaluation of the inversion integral over all parts of the contour.

By expanding $\left(r^{2}+1\right)^{-1} r^{-1 / 2}$ as a power series in $r$, and assuming that you may integrate the series term by term, show that the two leading terms, as $t \rightarrow \infty$, are

$f(t) \sim \sin (t-\pi / 4)+\frac{1}{(\pi t)^{1 / 2}}+\cdots$

[You may assume that $\int_{0}^{\infty} x^{-1 / 2} e^{-x} d x=\pi^{1 / 2}$.]

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• # Paper 2, Section II, D

Show that both the following transformations from the $z$-plane to the $\zeta$-plane are conformal, except at certain critical points which should be identified in both planes, and in each case find a domain in the $z$-plane that is mapped onto the upper half $\zeta$-plane:

\begin{aligned} &\text { (i) } \zeta=z+\frac{b^{2}}{z} \\ &\text { (ii) } \zeta=\cosh \frac{\pi z}{b} \end{aligned}

where $b$ is real and positive.

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• # 1.I.3C

Given that $f(z)$ is an analytic function, show that the mapping $w=f(z)$

(a) preserves angles between smooth curves intersecting at $z$ if $f^{\prime}(z) \neq 0$;

(b) has Jacobian given by $\left|f^{\prime}(z)\right|^{2}$.

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• # 1.II.13C

By a suitable choice of contour show the following:

(a)

$\int_{0}^{\infty} \frac{x^{1 / n}}{1+x^{2}} d x=\frac{\pi}{2 \cos (\pi / 2 n)}$

where $n>1$,

(b)

$\int_{0}^{\infty} \frac{x^{1 / 2} \log x}{1+x^{2}} d x=\frac{\pi^{2}}{2 \sqrt{2}}$

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• # 2.II.14C

Let $f(z)=1 /\left(e^{z}-1\right)$. Find the first three terms in the Laurent expansion for $f(z)$ valid for $0<|z|<2 \pi$.

Now let $n$ be a positive integer, and define

\begin{aligned} &f_{1}(z)=\frac{1}{z}+\sum_{r=1}^{n} \frac{2 z}{z^{2}+4 \pi^{2} r^{2}} \\ &f_{2}(z)=f(z)-f_{1}(z) \end{aligned}

Show that the singularities of $f_{2}$ in $\{z:|z|<2(n+1) \pi\}$ are all removable. By expanding $f_{1}$ as a Laurent series valid for $|z|>2 n \pi$, and $f_{2}$ as a Taylor series valid for $|z|<2(n+1) \pi$, find the coefficients of $z^{j}$ for $-1 \leq j \leq 1$ in the Laurent series for $f$ valid for $2 n \pi<|z|<2(n+1) \pi$.

By estimating an appropriate integral around the contour $|z|=(2 n+1) \pi$, show that

$\sum_{r=1}^{\infty} \frac{1}{r^{2}}=\frac{\pi^{2}}{6}$

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• # 1.I.3F

For the function

$f(z)=\frac{2 z}{z^{2}+1},$

determine the Taylor series of $f$ around the point $z_{0}=1$, and give the largest $r$ for which this series converges in the disc $|z-1|.

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• # 1.II.13F

By integrating round the contour $C_{R}$, which is the boundary of the domain

$D_{R}=\left\{z=r e^{i \theta}: 0

evaluate each of the integrals

$\int_{0}^{\infty} \sin x^{2} d x, \quad \int_{0}^{\infty} \cos x^{2} d x$

[You may use the relations $\int_{0}^{\infty} e^{-r^{2}} d r=\frac{\sqrt{\pi}}{2}$ and $\sin t \geq \frac{2}{\pi} t$ for $\left.0 \leq t \leq \frac{\pi}{2} \cdot\right]$

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• # 2.II.14F

Let $\Omega$ be the half-strip in the complex plane,

$\Omega=\left\{z=x+i y \in \mathbb{C}:-\frac{\pi}{2}0\right\}$

Find a conformal mapping that maps $\Omega$ onto the unit disc.

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• # 1.I.3D

Let $L$ be the Laplace operator, i.e., $L(g)=g_{x x}+g_{y y}$. Prove that if $f: \Omega \rightarrow \mathbf{C}$ is analytic in a domain $\Omega$, then

$L\left(|f(z)|^{2}\right)=4\left|f^{\prime}(z)\right|^{2}, \quad z \in \Omega .$

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• # 1.II.13D

By integrating round the contour involving the real axis and the line $\operatorname{Im}(z)=2 \pi$, or otherwise, evaluate

$\int_{-\infty}^{\infty} \frac{e^{a x}}{1+e^{x}} d x, \quad 0

Explain why the given restriction on the value $a$ is necessary.

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• # 2.II.14D

Let $\Omega$ be the region enclosed between the two circles $C_{1}$ and $C_{2}$, where

$C_{1}=\{z \in \mathbf{C}:|z-i|=1\}, \quad C_{2}=\{z \in \mathbf{C}:|z-2 i|=2\}$

Find a conformal mapping that maps $\Omega$ onto the unit disc.

[Hint: you may find it helpful first to map $\Omega$ to a strip in the complex plane. ]

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• # 1.I.3F

State the Cauchy integral formula.

Using the Cauchy integral formula, evaluate

$\int_{|z|=2} \frac{z^{3}}{z^{2}+1} d z$

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• # 1.II.13F

Determine a conformal mapping from $\Omega_{0}=\mathbf{C} \backslash[-1,1]$ to the complex unit disc $\Omega_{1}=\{z \in \mathbf{C}:|z|<1\} .$

[Hint: A standard method is first to map $\Omega_{0}$ to $\mathbf{C} \backslash(-\infty, 0]$, then to the complex right half-plane $\{z \in \mathbf{C}: \operatorname{Re} z>0\}$ and, finally, to $\left.\Omega_{1} .\right]$

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• # 2.II.14F

Let $F=P / Q$ be a rational function, where $\operatorname{deg} Q \geqslant \operatorname{deg} P+2$ and $Q$ has no real zeros. Using the calculus of residues, write a general expression for

$\int_{-\infty}^{\infty} F(x) e^{i x} d x$

in terms of residues and briefly sketch its proof.

Evaluate explicitly the integral

$\int_{-\infty}^{\infty} \frac{\cos x}{4+x^{4}} d x$

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