• # Paper 1, Section I, 7E

Evaluate the integral

$\mathcal{P} \int_{0}^{\infty} \frac{\sin x}{x\left(x^{2}-1\right)} d x$

stating clearly any standard results involving contour integrals that you use.

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• # Paper 1, Section II, E

(a) Functions $g_{1}(z)$ and $g_{2}(z)$ are analytic in a connected open set $\mathcal{D} \subseteq \mathbb{C}$ with $g_{1}=g_{2}$ in a non-empty open subset $\tilde{\mathcal{D}} \subset \mathcal{D}$. State the identity theorem.

(b) Let $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ be connected open sets with $\mathcal{D}_{1} \cap \mathcal{D}_{2} \neq \emptyset$. Functions $f_{1}(z)$ and $f_{2}(z)$ are analytic on $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ respectively with $f_{1}=f_{2}$ on $\mathcal{D}_{1} \cap \mathcal{D}_{2}$. Explain briefly what is meant by analytic continuation of $f_{1}$ and use part (a) to prove that analytic continuation to $\mathcal{D}_{2}$ is unique.

(c) The function $F(z)$ is defined by

$F(z)=\int_{-\infty}^{\infty} \frac{e^{i t}}{(t-z)^{n}} d t$

where $\operatorname{Im} z>0$ and $n$ is a positive integer. Use the method of contour deformation to construct the analytic continuation of $F(z)$ into $\operatorname{Im} z \leqslant 0$.

(d) The function $G(z)$ is defined by

$G(z)=\int_{-\infty}^{\infty} \frac{e^{i t}}{(t-z)^{n}} d t$

where $\operatorname{Im} z \neq 0$ and $n$ is a positive integer. Prove that $G(z)$ experiences a discontinuity when $z$ crosses the real axis. Determine the value of this discontinuity. Hence, explain why $G(z)$ cannot be used as an analytic continuation of $F(z)$.

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• # Paper 2, Section I, 7E

The function $w(z)$ satisfies the differential equation

$\tag{†} \frac{d^{2} w}{d z^{2}}+p(z) \frac{d w}{d z}+q(z) w=0$

where $p(z)$ and $q(z)$ are complex analytic functions except, possibly, for isolated singularities in $\overline{\mathbb{C}}=\mathbb{C} \cup\{\infty\}$ (the extended complex plane).

(a) Given equation $(†)$, state the conditions for a point $z_{0} \in \mathbb{C}$ to be

(i) an ordinary point,

(ii) a regular singular point,

(iii) an irregular singular point.

(b) Now consider $z_{0}=\infty$ and use a suitable change of variables $z \rightarrow t$, with $y(t)=w(z)$, to rewrite $(†)$ as a differential equation that is satisfied by $y(t)$. Hence, deduce the conditions for $z_{0}=\infty$ to be

(i) an ordinary point,

(ii) a regular singular point,

(iii) an irregular singular point.

[In each case, you should express your answer in terms of the functions $p$ and $q$.]

(c) Use the results above to prove that any equation of the form ( $\dagger$ ) must have at least one singular point in $\overline{\mathbb{C}}$.

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• # Paper 2, Section II, 13E

The temperature $T(x, t)$ in a semi-infinite bar $(0 \leqslant x<\infty)$ satisfies the heat equation

$\frac{\partial T}{\partial t}=\kappa \frac{\partial^{2} T}{\partial x^{2}}, \quad \text { for } x>0 \text { and } t>0$

where $\kappa$ is a positive constant.

For $t<0$, the bar is at zero temperature. For $t \geqslant 0$, the temperature is subject to the boundary conditions

$T(0, t)=a\left(1-e^{-b t}\right),$

where $a$ and $b$ are positive constants, and $T(x, t) \rightarrow 0$ as $x \rightarrow \infty$.

(a) Show that the Laplace transform of $T(x, t)$ with respect to $t$ takes the form

$\hat{T}(x, p)=\hat{f}(p) e^{-x \sqrt{p / \kappa}}$

and find $\hat{f}(p)$. Hence write $\hat{T}(x, p)$ in terms of $a, b, \kappa, p$ and $x$.

(b) By performing the inverse Laplace transform using contour integration, show that for $t \geqslant 0$

$T(x, t)=a\left[1-e^{-b t} \cos \left(\sqrt{\frac{b}{\kappa}} x\right)\right]+\frac{2 a b}{\pi} \mathcal{P} \int_{0}^{\infty} \frac{e^{-v^{2} t} \sin (x v / \sqrt{\kappa})}{v\left(v^{2}-b\right)} d v$

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• # Paper 3 , Section I, 7E

The Beta function is defined by

$B(p, q)=\int_{0}^{1} t^{p-1}(1-t)^{q-1} d t$

for $\operatorname{Re} p>0$ and $\operatorname{Re} q>0$.

(a) Prove that $B(p, q)=B(q, p)$ and find $B(1, q)$.

(b) Show that $(p+z) B(p, z+1)=z B(p, z)$.

(c) For each fixed $p$ with $\operatorname{Re} p>0$, use part (b) to obtain the analytic continuation of $B(p, z)$ as an analytic function of $z \in \mathbb{C}$, with the exception of the points $z=$ $0,-1,-2,-3, \ldots$

(d) Use part (c) to determine the type of singularity that the function $B(p, z)$ has at $z=0,-1,-2,-3, \ldots$, for fixed $p$ with $\operatorname{Re} p>0$.

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• # Paper 4 , Section I, 7E

(a) Explain in general terms the meaning of the Papperitz symbol

$P\left\{\begin{array}{cccc} a & b & c & \\ \alpha & \beta & \gamma & z \\ \alpha^{\prime} & \beta^{\prime} & \gamma^{\prime} & \end{array}\right\}$

State a condition satisfied by $\alpha, \beta, \gamma, \alpha^{\prime}, \beta^{\prime}$ and $\gamma^{\prime}$. [You need not write down any differential equations explicitly, but should provide explicit explanation of the meaning of $a, b, c, \alpha, \beta, \gamma, \alpha^{\prime}, \beta^{\prime}$ and $\left.\gamma^{\prime} .\right]$

(b) The Papperitz symbol

$P\left\{\begin{array}{cccc} 1 & -1 & \infty & \\ -m / 2 & m / 2 & n & z \\ m / 2 & -m / 2 & 1-n \end{array}\right\}$

where $n, m$ are constants, can be transformed into

$P\left\{\begin{array}{cccc} 0 & 1 & \infty & \\ 0 & 0 & n & \frac{1-z}{2} \\ m & -m & 1-n & \end{array}\right\}$

(i) Provide an explicit description of the transformations required to obtain ( $*)$ from $(t)$.

(ii) One of the solutions to the $P$-equation that corresponds to $(*)$ is a hypergeometric function $F\left(a, b ; c ; z^{\prime}\right)$. Express $a, b, c$ and $z^{\prime}$ in terms of $n, m$ and $z$.

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• # Paper 1, Section I, 7 E

The function $I(z)$, defined by

$I(z)=\int_{0}^{\infty} t^{z-1} e^{-t} d t$

is analytic for $\operatorname{Re} z>0$.

(i) Show that $I(z+1)=z I(z)$.

(ii) Use part (i) to construct an analytic continuation of $I(z)$ into Re $z \leqslant 0$, except at isolated singular points, which you need to identify.

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• # Paper 1, Section II, E

Use the change of variable $z=\sin ^{2} x$, to rewrite the equation

$\frac{d^{2} y}{d x^{2}}+k^{2} y=0$

where $k$ is a real non-zero number, as the hypergeometric equation

$\frac{d^{2} w}{d z^{2}}+\left(\frac{C}{z}+\frac{1+A+B-C}{z-1}\right) \frac{d w}{d z}+\frac{A B}{z(z-1)} w=0$

where $y(x)=w(z)$, and $A, B$ and $C$ should be determined explicitly.

(i) Show that ( $\)$ is a Papperitz equation, with 0,1 and $\infty$ as its regular singular points. Hence, write the corresponding Papperitz symbol,

$P\left\{\begin{array}{cccc} 0 & 1 & \infty \\ 0 & 0 & A \\ 1-C & C-A-B & B \end{array}\right\}$

in terms of $k$.

(ii) By solving ( $\dagger$ ) directly or otherwise, find the hypergeometric function $F(A, B ; C ; z)$ that is the solution to $(\ddagger)$ and is analytic at $z=0$ corresponding to the exponent 0 at $z=0$, and satisfies $F(A, B ; C ; 0)=1$; moreover, write it in terms of $k$ and

(iii) By performing a suitable exponential shifting find the second solution, independent of $F(A, B ; C ; z)$, which corresponds to the exponent $1-C$, and hence write $F\left(\frac{1+k}{2}, \frac{1-k}{2} ; \frac{3}{2} ; z\right)$ in terms of $k$ and $x$.

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• # Paper 2, Section $I$, E

Evaluate

$\int_{C} \frac{d z}{\sin ^{3} z}$

where $C$ is the circle $|z|=4$ traversed in the counter-clockwise direction.

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• # Paper 2, Section II, E

A semi-infinite elastic string is initially at rest on the $x$-axis with $0 \leqslant x<\infty$. The transverse displacement of the string, $y(x, t)$, is governed by the partial differential equation

$\frac{\partial^{2} y}{\partial t^{2}}=c^{2} \frac{\partial^{2} y}{\partial x^{2}}$

where $c$ is a positive real constant. For $t \geqslant 0$ the string is subject to the boundary conditions $y(0, t)=f(t)$ and $y(x, t) \rightarrow 0$ as $x \rightarrow \infty$.

(i) Show that the Laplace transform of $y(x, t)$ takes the form

$\hat{y}(x, p)=\hat{f}(p) e^{-p x / c}$

(ii) For $f(t)=\sin \omega t$, with $\omega \in \mathbb{R}^{+}$, find $\hat{f}(p)$ and hence write $\hat{y}(x, p)$ in terms of $\omega, c, p$ and $x$. Obtain $y(x, t)$ by performing the inverse Laplace transform using contour integration. Provide a physical interpretation of the result.

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• # Paper 3, Section I, E

The Weierstrass elliptic function is defined by

$\mathcal{P}(z)=\frac{1}{z^{2}}+\sum_{m, n}\left[\frac{1}{\left(z-\omega_{m, n}\right)^{2}}-\frac{1}{\omega_{m, n^{2}}}\right]$

where $\omega_{m, n}=m \omega_{1}+n \omega_{2}$, with non-zero periods $\left(\omega_{1}, \omega_{2}\right)$ such that $\omega_{1} / \omega_{2}$ is not real, and where $(m, n)$ are integers not both zero.

(i) Show that, in a neighbourhood of $z=0$,

$\mathcal{P}(z)=\frac{1}{z^{2}}+\frac{1}{20} g_{2} z^{2}+\frac{1}{28} g_{3} z^{4}+O\left(z^{6}\right)$

where

$g_{2}=60 \sum_{m, n}\left(\omega_{m, n}\right)^{-4}, \quad g_{3}=140 \sum_{m, n}\left(\omega_{m, n}\right)^{-6}$

(ii) Deduce that $\mathcal{P}$ satisfies

$\left(\frac{d \mathcal{P}}{d z}\right)^{2}=4 \mathcal{P}^{3}-g_{2} \mathcal{P}-g_{3}$

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• # Paper 4, Section I, E

The Hilbert transform of a function $f(x)$ is defined by

$\mathcal{H}(f)(y):=\frac{1}{\pi} \mathcal{P} \int_{-\infty}^{+\infty} \frac{f(x)}{y-x} d x$

Calculate the Hilbert transform of $f(x)=\cos \omega x$, where $\omega$ is a non-zero real constant.

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• # Paper 1, Section I, A

The Beta function is defined by

$B(p, q):=\int_{0}^{1} t^{p-1}(1-t)^{q-1} d t=\frac{\Gamma(p) \Gamma(q)}{\Gamma(p+q)}$

where $\operatorname{Re} p>0, \operatorname{Re} q>0$, and $\Gamma$ is the Gamma function.

(a) By using a suitable substitution and properties of Beta and Gamma functions, show that

$\int_{0}^{1} \frac{d x}{\sqrt{1-x^{4}}}=\frac{[\Gamma(1 / 4)]^{2}}{\sqrt{32 \pi}}$

(b) Deduce that

$K(1 / \sqrt{2})=\frac{4[\Gamma(5 / 4)]^{2}}{\sqrt{\pi}}$

where $K(k)$ is the complete elliptic integral, defined as

$K(k):=\int_{0}^{1} \frac{d t}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2} t^{2}\right)}}$

[Hint: You might find the change of variable $x=t\left(2-t^{2}\right)^{-1 / 2}$ helpful in part (b).]

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• # Paper 1, Section II, A

(a) Consider the Papperitz symbol (or P-symbol):

$\tag{†} P\left\{\begin{array}{cccc} a & b & c & \\ \alpha & \beta & \gamma & z \\ \alpha^{\prime} & \beta^{\prime} & \gamma^{\prime} & \end{array}\right\}$

Explain in general terms what this $P$-symbol represents.

[You need not write down any differential equations explicitly, but should provide an explanation of the meaning of $a, b, c, \alpha, \beta, \gamma, \alpha^{\prime}, \beta^{\prime}$ and $\left.\gamma^{\prime} .\right]$

(b) Prove that the action of $[(z-a) /(z-b)]^{\delta}$ on $(†)$ results in the exponential shifting,

$\tag{‡} P\left\{\begin{array}{cccc} a & b & c \\ \alpha+\delta & \beta-\delta & \gamma & z \\ \alpha^{\prime}+\delta & \beta^{\prime}-\delta & \gamma^{\prime} \end{array}\right\}$

[Hint: It may prove useful to start by considering the relationship between two solutions, $\omega$ and $\omega_{1}$, which satisfy the $P$-equations described by the respective $P$-symbols ($†$) and $‡$]

(c) Explain what is meant by a Möbius transformation of a second order differential equation. By using suitable transformations acting on $(†)$, show how to obtain the $P$ symbol

$\tag{*} P\left\{\begin{array}{cccc} 0 & 1 & \infty \\ 0 & 0 & a & z \\ 1-c & c-a-b & b \end{array}\right\}$

which corresponds to the hypergeometric equation.

(d) The hypergeometric function $F(a, b, c ; z)$ is defined to be the solution of the differential equation corresponding to $(\star)$ that is analytic at $z=0$ with $F(a, b, c ; 0)=1$, which corresponds to the exponent zero. Use exponential shifting to show that the second solution, which corresponds to the exponent $1-c$, is

$z^{1-c} F(a-c+1, b-c+1,2-c ; z) .$

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• # Paper 2, Section I, A

Assume that $|f(z) / z| \rightarrow 0$ as $|z| \rightarrow \infty$ and that $f(z)$ is analytic in the upper half-plane (including the real axis). Evaluate

$\mathcal{P} \int_{-\infty}^{\infty} \frac{f(x)}{x\left(x^{2}+a^{2}\right)} d x$

where $a$ is a positive real number.

[You must state clearly any standard results involving contour integrals that you use.]

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• # Paper 2, Section II, A

The Riemann zeta function is defined as

$\zeta(z):=\sum_{n=1}^{\infty} \frac{1}{n^{z}}$

for $R e(z)>1$, and by analytic continuation to the rest of $\mathbb{C}$ except at singular points. The integral representation of ( $\dagger$ ) for $\operatorname{Re}(z)>1$ is given by

$\zeta(z)=\frac{1}{\Gamma(z)} \int_{0}^{\infty} \frac{t^{z-1}}{e^{t}-1} d t$

where $\Gamma$ is the Gamma function.

(a) The Hankel representation is defined as

$\zeta(z)=\frac{\Gamma(1-z)}{2 \pi i} \int_{-\infty}^{\left(0^{+}\right)} \frac{t^{z-1}}{e^{-t}-1} d t$

Explain briefly why this representation gives an analytic continuation of $\zeta(z)$ as defined in ( $\ddagger$ ) to all $z$ other than $z=1$, using a diagram to illustrate what is meant by the upper limit of the integral in $(\star)$.

[You may assume $\Gamma(z) \Gamma(1-z)=\pi / \sin (\pi z)$.]

(b) Find

$\operatorname{Res}\left(\frac{t^{-z}}{e^{-t}-1}, t=2 \pi i n\right)$

where $n$ is an integer and the poles are simple.

(c) By considering

$\int_{\gamma} \frac{t^{-z}}{e^{-t}-1} d t$

where $\gamma$ is a suitably modified Hankel contour and using the result of part (b), derive the reflection formula:

$\zeta(1-z)=2^{1-z} \pi^{-z} \cos \left(\frac{1}{2} \pi z\right) \Gamma(z) \zeta(z)$

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• # Paper 3, Section I, A

The equation

$z w^{\prime \prime}+w=0$

has solutions of the form

$w(z)=\int_{\gamma} e^{z t} f(t) d t$

for suitably chosen contours $\gamma$ and some suitable function $f(t)$.

(a) Find $f(t)$ and determine the required condition on $\gamma$, which you should express in terms of $z$ and $t$.

(b) Use the result of part (a) to specify a possible contour with the help of a clearly labelled diagram.

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• # Paper 4, Section I, A

A single-valued function $\operatorname{Arcsin}(z)$ can be defined, for $0 \leqslant \arg z<2 \pi$, by means of an integral as:

$\operatorname{Arcsin}(z)=\int_{0}^{z} \frac{d t}{\left(1-t^{2}\right)^{1 / 2}}$

(a) Choose a suitable branch-cut with the integrand taking a value $+1$ at the origin on the upper side of the cut, i.e. at $t=0^{+}$, and describe suitable paths of integration in the two cases $0 \leqslant \arg z \leqslant \pi$ and $\pi<\arg z<2 \pi$.

(b) Construct the multivalued function $\arcsin (z)$ by analytic continuation.

(c) Express arcsin $\left(e^{2 \pi i} z\right)$ in terms of $\operatorname{Arcsin}(z)$ and deduce the periodicity property of $\sin (z)$.

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• # Paper 1, Section I, B

The Beta and Gamma functions are defined by

\begin{aligned} B(p, q) &=\int_{0}^{1} t^{p-1}(1-t)^{q-1} d t \\ \Gamma(p) &=\int_{0}^{\infty} e^{-t} t^{p-1} d t \end{aligned}

where $\operatorname{Re} p>0, \operatorname{Re} q>0$.

(a) By using a suitable substitution, or otherwise, prove that

$B(z, z)=2^{1-2 z} B\left(z, \frac{1}{2}\right)$

for $\operatorname{Re} z>0$. Extending $B$ by analytic continuation, for which values of $z \in \mathbb{C}$ does this result hold?

(b) Prove that

$B(p, q)=\frac{\Gamma(p) \Gamma(q)}{\Gamma(p+q)}$

for $\operatorname{Re} p>0, \operatorname{Re} q>0$

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• # Paper 1, Section II, B

The equation

$\tag{†} z w^{\prime \prime}+2 a w^{\prime}+z w=0,$

where $a$ is a constant with $\operatorname{Re} a>0$, has solutions of the form

$w(z)=\int_{\gamma} e^{z t} f(t) d t$

for suitably chosen contours $\gamma$ and some suitable function $f(t)$.

(a) Find $f(t)$ and determine the condition on $\gamma$, which you should express in terms of $z, t$ and $a$.

(b) Use the results of part (a) to show that $\gamma$ can be a finite contour and specify two possible finite contours with the help of a clearly labelled diagram. Hence, find the corresponding solution of the equation $(†)$ in the case $a=1$.

(c) In the case $a=1$ and real $z$, show that $\gamma$ can be an infinite contour and specify two possible infinite contours with the help of a clearly labelled diagram. [Hint: Consider separately the cases $z>0$ and $z<0$.] Hence, find a second, linearly independent solution of the equation ( $\dagger$ ) in this case.

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• # Paper 2, Section $I$, $7 \mathrm{~B}$

Show that

$\int_{-\infty}^{\infty} \frac{\cos n x-\cos m x}{x^{2}} d x=\pi(m-n),$

in the sense of Cauchy principal value, where $n$ and $m$ are positive integers. [State clearly any standard results involving contour integrals that you use.]

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• # Paper 2, Section II, B

Consider a multi-valued function $w(z)$.

(a) Explain what is meant by a branch point and a branch cut.

(b) Consider $z=e^{w}$.

(i) By writing $z=r e^{i \theta}$, where $0 \leqslant \theta<2 \pi$, and $w=u+i v$, deduce the expression for $w(z)$ in terms of $r$ and $\theta$. Hence, show that $w$ is infinitely valued and state its principal value.

(ii) Show that $z=0$ and $z=\infty$ are the branch points of $w$. Deduce that the line $\operatorname{Im} z=0, \operatorname{Re} z>0$ is a possible choice of branch cut.

(iii) Use the Cauchy-Riemann conditions to show that $w$ is analytic in the cut plane. Show that $\frac{d w}{d z}=\frac{1}{z}$.

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• # Paper 3, Section I, B

Using a suitable branch cut, show that

$\int_{-a}^{a}\left(a^{2}-x^{2}\right)^{1 / 2} d x=\frac{a^{2} \pi}{2},$

where $a>0$.

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• # Paper 4, Section I, B

State the conditions for a point $z=z_{0}$ to be a regular singular point of a linear second-order homogeneous ordinary differential equation in the complex plane.

Find all singular points of the Bessel equation

$z^{2} y^{\prime \prime}(z)+z y^{\prime}(z)+\left(z^{2}-\frac{1}{4}\right) y(z)=0$

and determine whether they are regular or irregular.

By writing $y(z)=f(z) / \sqrt{z}$, find two linearly independent solutions of $(*)$. Comment on the relationship of your solutions to the nature of the singular points.

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• # Paper 1, Section I, E

Calculate the value of the integral

$P \int_{-\infty}^{\infty} \frac{e^{-i x}}{x^{n}} d x$

where $P$ stands for Principal Value and $n$ is a positive integer.

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• # Paper 1, Section II, E

The Riemann zeta function is defined by

$\zeta_{R}(s)=\sum_{n=1}^{\infty} n^{-s}$

for $\operatorname{Re}(s)>1$.

Show that

$\zeta_{R}(s)=\frac{1}{\Gamma(s)} \int_{0}^{\infty} \frac{t^{s-1}}{e^{t}-1} d t$

Let $I(s)$ be defined by

$I(s)=\frac{\Gamma(1-s)}{2 \pi i} \int_{C} \frac{t^{s-1}}{e^{-t}-1} d t$

where $C$ is the Hankel contour.

Show that $I(s)$ provides an analytic continuation of $\zeta_{R}(s)$ for a range of $s$ which should be determined.

Hence evaluate $\zeta_{R}(-1)$.

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• # Paper 2, Section I, E

Euler's formula for the Gamma function is

$\Gamma(z)=\frac{1}{z} \prod_{n=1}^{\infty}\left(1+\frac{1}{n}\right)^{z}\left(1+\frac{z}{n}\right)^{-1}$

Use Euler's formula to show

$\frac{\Gamma(2 z)}{2^{2 z} \Gamma(z) \Gamma\left(z+\frac{1}{2}\right)}=C$

where $C$ is a constant.

Evaluate $C$.

[Hint: You may use $\Gamma(z) \Gamma(1-z)=\pi / \sin (\pi z) .]$

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• # Paper 2, Section II, E

The hypergeometric equation is represented by the Papperitz symbol

$P\left\{\begin{array}{ccc} 0 & 1 & \infty \\ 0 & 0 & a \\ 1-c & c-a-b & b \end{array}\right\}$

and has solution $y_{0}(z)=F(a, b, c ; z)$.

Functions $y_{1}(z)$ and $y_{2}(z)$ are defined by

$y_{1}(z)=F(a, b, a+b+1-c ; 1-z)$

and

$y_{2}(z)=(1-z)^{c-a-b} F(c-a, c-b, c-a-b+1 ; 1-z),$

where $c-a-b$ is not an integer.

Show that $y_{1}(z)$ and $y_{2}(z)$ obey the hypergeometric equation $(*)$.

Explain why $y_{0}(z)$ can be written in the form

$y_{0}(z)=A y_{1}(z)+B y_{2}(z)$

where $A$ and $B$ are independent of $z$ but depend on $a, b$ and $c$.

Suppose that

$F(a, b, c ; z)=\frac{\Gamma(c)}{\Gamma(b) \Gamma(c-b)} \int_{0}^{1} t^{b-1}(1-t)^{c-b-1}(1-t z)^{-a} d t$

with $\operatorname{Re}(c)>\operatorname{Re}(b)>0$ and $|\arg (1-z)|<\pi$. Find expressions for $A$ and $B$.

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• # Paper 3, Section I, E

Find all the singular points of the differential equation

$z \frac{d^{2} y}{d z^{2}}+(2-z) \frac{d y}{d z}-y=0$

and determine whether they are regular or irregular singular points.

By writing $y(z)=f(z) / z$, find two linearly independent solutions to this equation.

Comment on the relationship of your solutions to the nature of the singular points of the original differential equation.

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• # Paper 4, Section I, $7 \mathbf{E} \quad$

Consider the differential equation

$z \frac{d^{2} y}{d z^{2}}-2 \frac{d y}{d z}+z y=0$

Laplace's method finds a solution of this differential equation by writing $y(z)$ in the form

$y(z)=\int_{C} e^{z t} f(t) d t$

where $C$ is a closed contour.

Determine $f(t)$. Hence find two linearly independent real solutions of $(\star)$ for $z$ real.

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• # Paper 1, Section I, A

Evaluate the integral

$f(p)=\mathcal{P} \int_{-\infty}^{\infty} d x \frac{e^{i p x}}{x^{4}-1}$

where $p$ is a real number, for (i) $p>0$ and (ii) $p<0$.

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• # Paper 1, Section II, A

(a) Legendre's equation for $w(z)$ is

$\left(z^{2}-1\right) w^{\prime \prime}+2 z w^{\prime}-\ell(\ell+1) w=0, \quad \text { where } \quad \ell=0,1,2, \ldots$

Let $\mathcal{C}$ be a closed contour. Show by direct substitution that for $z$ within $\mathcal{C}$

$\int_{\mathcal{C}} d t \frac{\left(t^{2}-1\right)^{\ell}}{(t-z)^{\ell+1}}$

is a non-trivial solution of Legendre's equation.

(b) Now consider

$Q_{\nu}(z)=\frac{1}{4 i \sin \nu \pi} \int_{\mathcal{C}^{\prime}} d t \frac{\left(t^{2}-1\right)^{\nu}}{(t-z)^{\nu+1}}$

for real $\nu>-1$ and $\nu \neq 0,1,2, \ldots$. The closed contour $\mathcal{C}^{\prime}$ is defined to start at the origin, wind around $t=1$ in a counter-clockwise direction, then wind around $t=-1$ in a clockwise direction, then return to the origin, without encircling the point $z$. Assuming that $z$ does not lie on the real interval $-1 \leqslant x \leqslant 1$, show by deforming $\mathcal{C}^{\prime}$ onto this interval that functions $Q_{\ell}(z)$ may be defined as limits of $Q_{\nu}(z)$ with $\nu \rightarrow \ell=0,1,2, \ldots$.

Find an explicit expression for $Q_{0}(z)$ and verify that it satisfies Legendre's equation with $\ell=0$.

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• # Paper 2, Section I, A

The Euler product formula for the Gamma function is

$\Gamma(z)=\lim _{n \rightarrow \infty} \frac{n ! n^{z}}{z(z+1) \ldots(z+n)}$

Use this to show that

$\frac{\Gamma(2 z)}{2^{2 z} \Gamma(z) \Gamma\left(z+\frac{1}{2}\right)}=c,$

where $c$ is a constant, independent of $z$. Find the value of $c$.

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• # Paper 2, Section II, A

The Hurwitz zeta function $\zeta_{\mathrm{H}}(s, q)$ is defined for $\operatorname{Re}(q)>0$ by

$\zeta_{\mathrm{H}}(s, q)=\sum_{n=0}^{\infty} \frac{1}{(q+n)^{s}}$

State without proof the complex values of $s$ for which this series converges.

Consider the integral

$I(s, q)=\frac{\Gamma(1-s)}{2 \pi i} \int_{\mathcal{C}} d z \frac{z^{s-1} e^{q z}}{1-e^{z}}$

where $\mathcal{C}$ is the Hankel contour. Show that $I(s, q)$ provides an analytic continuation of the Hurwitz zeta function for all $s \neq 1$. Include in your account a careful discussion of removable singularities. [Hint: $\Gamma(s) \Gamma(1-s)=\pi / \sin (\pi s)$.]

Show that $I(s, q)$ has a simple pole at $s=1$ and find its residue.

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• # Paper 3, Section I, A

The functions $f(x)$ and $g(x)$ have Laplace transforms $F(p)$ and $G(p)$ respectively, and $f(x)=g(x)=0$ for $x \leqslant 0$. The convolution $h(x)$ of $f(x)$ and $g(x)$ is defined by

$h(x)=\int_{0}^{x} f(y) g(x-y) d y \quad \text { for } \quad x>0 \quad \text { and } \quad h(x)=0 \quad \text { for } \quad x \leqslant 0$

Express the Laplace transform $H(p)$ of $h(x)$ in terms of $F(p)$ and $G(p)$.

Now suppose that $f(x)=x^{\alpha}$ and $g(x)=x^{\beta}$ for $x>0$, where $\alpha, \beta>-1$. Find expressions for $F(p)$ and $G(p)$ by using a standard integral formula for the Gamma function. Find an expression for $h(x)$ by using a standard integral formula for the Beta function. Hence deduce that

$\frac{\Gamma(z) \Gamma(w)}{\Gamma(z+w)}=\mathrm{B}(z, w)$

for all $\operatorname{Re}(z)>0, \operatorname{Re}(w)>0$.

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• # Paper 4, Section I, 7A

Consider the equation for $w(z)$ :

$w^{\prime \prime}+p(z) w^{\prime}+q(z) w=0 .$

State necessary and sufficient conditions on $p(z)$ and $q(z)$ for $z=0$ to be (i) an ordinary point or (ii) a regular singular point. Derive the corresponding conditions for the point $z=\infty$.

Determine the most general equation of the form $(*)$ that has regular singular points at $z=0$ and $z=\infty$, with all other points being ordinary.

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• # Paper 1, Section $I$, B

Evaluate the real integral

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

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

What is the value of

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

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• # Paper 1, Section II, B

Consider the differential equation

$x y^{\prime \prime}+(a-x) y^{\prime}-b y=0$

where $a$ and $b$ are constants with $\operatorname{Re}(b)>0$ and $\operatorname{Re}(a-b)>0$. Laplace's method for finding solutions involves writing

$y(x)=\int_{C} e^{x t} f(t) d t$

for some suitable contour $C$ and some suitable function $f(t)$. Determine $f(t)$ for the equation $(*)$ and use a clearly labelled diagram to specify contours $C$ giving two independent solutions when $x$ is real in each of the cases $x>0$ and $x<0$.

Now let $a=3$ and $b=1$. Find explicit expressions for two independent solutions to $(*)$. Find, in addition, a solution $y(x)$ with $y(0)=1$.

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• # Paper 2, Section I, B

Give a brief description of what is meant by analytic continuation.

The dilogarithm function is defined by

$\mathrm{Li}_{2}(z)=\sum_{n=1}^{\infty} \frac{z^{n}}{n^{2}}, \quad|z|<1$

Let

$f(z)=-\int_{C} \frac{1}{u} \ln (1-u) d u$

where $C$ is a contour that runs from the origin to the point $z$. Show that $f(z)$ provides an analytic continuation of $\mathrm{Li}_{2}(z)$ and describe its domain of definition in the complex plane, given a suitable branch cut.

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• # Paper 2, Section II, B

The Riemann zeta function is defined by the sum

$\zeta(s)=\sum_{n=1}^{\infty} n^{-s}$

which converges for $\operatorname{Re} s>1$. Show that

$\zeta(s)=\frac{1}{\Gamma(s)} \int_{0}^{\infty} \frac{t^{s-1}}{e^{t}-1} d t, \quad \operatorname{Re} s>1$

The analytic continuation of $\zeta(s)$ is given by the Hankel contour integral

$\zeta(s)=\frac{\Gamma(1-s)}{2 \pi i} \int_{-\infty}^{0+} \frac{t^{s-1}}{e^{-t}-1} d t$

Verify that this agrees with the integral $(*)$ above when Re $s>1$ and $s$ is not an integer. [You may assume $\Gamma(s) \Gamma(1-s)=\pi / \sin \pi s$.] What happens when $s=2,3,4, \ldots$ ?

Evaluate $\zeta(0)$. Show that $\left(e^{-t}-1\right)^{-1}+\frac{1}{2}$ is an odd function of $t$ and hence, or otherwise, show that $\zeta(-2 n)=0$ for any positive integer $n$.

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• # Paper 3, Section $I$, B

Define what is meant by the Cauchy principal value in the particular case

$\mathcal{P} \int_{-\infty}^{\infty} \frac{\cos x}{x^{2}-a^{2}} d x$

where the constant $a$ is real and strictly positive. Evaluate this expression explicitly, stating clearly any standard results involving contour integrals that you use.

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• # Paper 4, Section I, B

Explain how the Papperitz symbol

$P\left\{\begin{array}{cccc} z_{1} & z_{2} & z_{3} & \\ \alpha_{1} & \beta_{1} & \gamma_{1} & z \\ \alpha_{2} & \beta_{2} & \gamma_{2} & \end{array}\right\}$

represents a differential equation with certain properties. [You need not write down the differential equation explicitly.]

The hypergeometric function $F(a, b, c ; z)$ is defined to be the solution of the equation given by the Papperitz symbol

that is analytic at $z=0$ and such that $F(a, b, c ; 0)=1$. Show that

$F(a, b, c ; z)=(1-z)^{-a} F\left(a, c-b, c ; \frac{z}{z-1}\right) \text {, }$

indicating clearly any general results for manipulating Papperitz symbols that you use.

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• # Paper 1, Section $I$, B

Show that the Cauchy-Riemann equations for $f: \mathbb{C} \rightarrow \mathbb{C}$ are equivalent to

$\frac{\partial f}{\partial \bar{z}}=0 \text {, }$

where $z=x+i y$, and $\partial / \partial \bar{z}$ should be defined in terms of $\partial / \partial x$ and $\partial / \partial y$. Use Green's theorem, together with the formula $d z d \bar{z}=-2 i d x d y$, to establish the generalised Cauchy formula

$\oint_{\gamma} f(z, \bar{z}) d z=-\iint_{D} \frac{\partial f}{\partial \bar{z}} d z d \bar{z}$

where $\gamma$ is a contour in the complex plane enclosing the region $D$ and $f$ is sufficiently differentiable.

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• # Paper 1, Section II, 14B

Obtain solutions of the second-order ordinary differential equation

$z w^{\prime \prime}-w=0$

in the form

$w(z)=\int_{\gamma} f(t) e^{-z t} d t$

where the function $f$ and the choice of contour $\gamma$ should be determined from the differential equation.

Show that a non-trivial solution can be obtained by choosing $\gamma$ to be a suitable closed contour, and find the resulting solution in this case, expressing your answer in the form of a power series.

Describe a contour $\gamma$ that would provide a second linearly independent solution for the case $\operatorname{Re}(z)>0$.

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• # Paper 2, Section I, B

Suppose $z=0$ is a regular singular point of a linear second-order homogeneous ordinary differential equation in the complex plane. Define the monodromy matrix $M$ around $z=0$.

Demonstrate that if

$M=\left(\begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right)$

then the differential equation admits a solution of the form $a(z)+b(z) \log z$, where $a(z)$ and $b(z)$ are single-valued functions.

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• # Paper 2, Section II, 14B

Use the Euler product formula

$\Gamma(z)=\lim _{n \rightarrow \infty} \frac{n ! n^{z}}{z(z+1) \ldots(z+n)}$

to show that:

(i) $\Gamma(z+1)=z \Gamma(z)$;

(ii) $\frac{1}{\Gamma(z)}=z e^{\gamma z} \prod_{k=1}^{\infty}\left(1+\frac{z}{k}\right) e^{-z / k}$, where $\gamma=\lim _{n \rightarrow \infty}\left(1+\frac{1}{2}+\cdots+\frac{1}{n}-\log n\right)$.

Deduce that

$\frac{d}{d z} \log (\Gamma(z))=-\gamma-\frac{1}{z}+z \sum_{k=1}^{\infty} \frac{1}{k(z+k)}$

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• # Paper 3, Section I, B

State the conditions for a point $z=z_{0}$ to be a regular singular point of a linear second-order homogeneous ordinary differential equation in the complex plane.

Find all singular points of the Airy equation

$w^{\prime \prime}(z)-z w(z)=0$

and determine whether they are regular or irregular.

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• # Paper 4, Section I, B

Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a function such that

$f\left(z+\omega_{1}\right)=f(z), \quad f\left(z+\omega_{2}\right)=f(z)$

where $\omega_{1}, \omega_{2} \in \mathbb{C} \backslash\{0\}$ and $\omega_{1} / \omega_{2}$ is not real. Show that if $f$ is analytic on $\mathbb{C}$ then it is a constant. [Liouville's theorem may be used if stated.] Give an example of a non-constant meromorphic function which satisfies (1).

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• # Paper 1, Section I, E

Prove that there are no second order linear ordinary homogeneous differential equations for which all points in the extended complex plane are analytic.

Find all such equations which have one regular singular point at $z=0$.

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• # Paper 1, Section II, E

Show that the equation

$(z-1) w^{\prime \prime}-z w^{\prime}+(4-2 z) w=0$

has solutions of the form $w(z)=\int_{\gamma} \exp (z t) f(t) d t$, where

$f(t)=\frac{\exp (-t)}{(t-a)(t-b)^{2}}$

and the contour $\gamma$ is any closed curve in the complex plane, where $a$ and $b$ are real constants which should be determined.

Use this to find the general solution, evaluating the integrals explicitly.

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• # Paper 2, Section I, E

(i) Find all branch points of $\left(z^{3}-1\right)^{1 / 4}$ on an extended complex plane.

(ii) Use a branch cut to evaluate the integral

$\int_{-2}^{2}\left(4-x^{2}\right)^{1 / 2} d x$

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• # Paper 2, Section II, E

The Beta function is defined for $\operatorname{Re}(z)>0$ as

$B(z, q)=\int_{0}^{1} t^{q-1}(1-t)^{z-1} d t, \quad(\operatorname{Re}(q)>0)$

and by analytic continuation elsewhere in the complex $z$-plane.

Show that:

(i) $(z+q) B(z+1, q)=z B(z, q)$;

(ii) $\Gamma(z)^{2}=B(z, z) \Gamma(2 z)$.

By considering $\Gamma\left(z / 2^{m}\right)$ for all positive integers $m$, deduce that $\Gamma(z) \neq 0$ for all $z$ with $\operatorname{Re}(z)>0$.

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• # Paper 3, Section I, E

Let a real-valued function $u=u(x, y)$ be the real part of a complex-valued function $f=f(z)$ which is analytic in the neighbourhood of a point $z=0$, where $z=x+i y .$ Derive a formula for $f$ in terms of $u$ and use it to find an analytic function $f$ whose real part is

$\frac{x^{3}+x^{2}-y^{2}+x y^{2}}{(x+1)^{2}+y^{2}}$

and such that $f(0)=0$.

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• # Paper 4, Section I, E

Let the function $f(z)$ be analytic in the upper half-plane and such that $|f(z)| \rightarrow 0$ as $|z| \rightarrow \infty$. Show that

$\mathcal{P} \int_{-\infty}^{\infty} \frac{f(x)}{x} d x=i \pi f(0),$

where $\mathcal{P}$ denotes the Cauchy principal value.

Use the Cauchy integral theorem to show that

$\mathcal{P} \int_{-\infty}^{\infty} \frac{u(x, 0)}{x-t} d x=-\pi v(t, 0), \quad \mathcal{P} \int_{-\infty}^{\infty} \frac{v(x, 0)}{x-t} d x=\pi u(t, 0),$

where $u(x, y)$ and $v(x, y)$ are the real and imaginary parts of $f(z)$.

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• # Paper 1, Section I, E

Recall that if $f(z)$ is analytic in a neighbourhood of $z_{0} \neq 0$, then

$f(z)+\overline{f\left(z_{0}\right)}=2 u\left(\frac{z+\overline{z_{0}}}{2}, \frac{z-\overline{z_{0}}}{2 i}\right)$

where $u(x, y)$ is the real part of $f(z)$. Use this fact to construct the imaginary part of an analytic function whose real part is given by

$u(x, y)=y \int_{-\infty}^{\infty} \frac{g(t) d t}{(t-x)^{2}+y^{2}}, \quad x, y \in \mathbb{R}, y \neq 0$

where $g(t)$ is real and has sufficient smoothness and decay.

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• # Paper 1, Section II, E

(a) Suppose that $F(z), z=x+i y, x, y \in \mathbb{R}$, is analytic in the upper-half complex $z$-plane and $O(1 / z)$ as $z \rightarrow \infty, y \geqslant 0$. Show that the real and imaginary parts of $F(x)$, denoted by $U(x)$ and $V(x)$ respectively, satisfy the so-called Kramers-Kronig formulae:

$U(x)=H V(x), \quad V(x)=-H U(x), \quad x \in \mathbb{R} .$

Here, $H$ denotes the Hilbert transform, i.e.,

$(H f)(x)=\frac{1}{\pi} \mathrm{PV} \int_{-\infty}^{\infty} \frac{f(\xi)}{\xi-x} d \xi$

where $\mathrm{PV}$ denotes the principal value integral.

(b) Let the real function $u(x, y)$ satisfy the Laplace equation in the upper-half complex z-plane, i.e.,

$\frac{\partial^{2} u(x, y)}{\partial x^{2}}+\frac{\partial^{2} u(x, y)}{\partial y^{2}}=0, \quad-\infty0$

Assuming that $u(x, y)$ decays for large $|x|$ and for large $y$, show that $F=u_{z}$ is an analytic function for $\operatorname{Im} z>0, z=x+i y$. Then, find an expression for $u_{y}(x, 0)$ in terms of $u_{x}(x, 0)$.

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• # Paper 2, Section I, E

The hypergeometric function $F(a, b ; c ; z)$ is defined as the particular solution of the second order linear ODE characterised by the Papperitz symbol

$\mathrm{P}\left\{\begin{array}{cccc} 0 & 1 & \infty \\ 0 & 0 & a & z \\ 1-c & c-a-b & b \end{array}\right\}$

that is analytic at $z=0$ and satisfies $F(a, b ; c ; 0)=1$.

Using the fact that a second solution $w(z)$ of the above ODE is of the form

$w(z)=z^{1-c} u(z)$

where $u(z)$ is analytic in the neighbourhood of the origin, express $w(z)$ in terms of $F$.

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• # Paper 2, Section II, $14 \mathrm{E}$

Let the complex function $q(x, t)$ satisfy

$i \frac{\partial q(x, t)}{\partial t}+\frac{\partial^{2} q(x, t)}{\partial x^{2}}=0, \quad 0