Complex Methods

• 1.I.7E

  Complex Methods

  Part IB, 2001

  State the Cauchy integral formula.

  Assuming that the function $f(z)$ is analytic in the disc $|z|<1$, prove that, for every $0<r<1$, it is true that

  $$\frac{d^{n} f(0)}{d z^{n}}=\frac{n !}{2 \pi i} \int_{|\xi|=r} \frac{f(\xi)}{\xi^{n+1}} d \xi, \quad n=0,1, \ldots$$

  [Taylor's theorem may be used if clearly stated.]

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• 1.II.16E

  Complex Methods

  Part IB, 2001

  Let the function $F$ be integrable for all real arguments $x$, such that

  $$\int_{-\infty}^{\infty}|F(x)| d x<\infty$$

  and assume that the series

  $$f(\tau)=\sum_{n=-\infty}^{\infty} F(2 n \pi+\tau)$$

  converges uniformly for all $0 \leqslant \tau \leqslant 2 \pi$.

  Prove the Poisson summation formula

  $$f(\tau)=\frac{1}{2 \pi} \sum_{n=-\infty}^{\infty} \hat{F}(n) e^{i n \tau}$$

  where $\hat{F}$ is the Fourier transform of $F$. [Hint: You may show that

  $$\frac{1}{2 \pi} \int_{0}^{2 \pi} e^{-i m x} f(x) d x=\frac{1}{2 \pi} \int_{-\infty}^{\infty} e^{-i m x} F(x) d x$$

  or, alternatively, prove that $f$ is periodic and express its Fourier expansion coefficients explicitly in terms of $\hat{F}$.]

  Letting $F(x)=e^{-|x|}$, use the Poisson summation formula to evaluate the sum

  $$\sum_{n=-\infty}^{\infty} \frac{1}{1+n^{2}}$$

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• $2 . \mathrm{I} . 7 \mathrm{E} \quad$

  Complex Methods

  Part IB, 2001

  A complex function is defined for every $z \in V$, where $V$ is a non-empty open subset of $\mathbb{C}$, and it possesses a derivative at every $z \in V$. Commencing from a formal definition of derivative, deduce the Cauchy-Riemann equations.

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• 2.II.16E

  Complex Methods

  Part IB, 2001

  Let $R$ be a rational function such that $\lim _{z \rightarrow \infty}\{z R(z)\}=0$. Assuming that $R$ has no real poles, use the residue calculus to evaluate

  $$\int_{-\infty}^{\infty} R(x) d x$$

  Given that $n \geqslant 1$ is an integer, evaluate

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

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• 4.I.8F

  Complex Methods

  Part IB, 2001

  Consider a conformal mapping of the form

  $$f(z)=\frac{a+b z}{c+d z}, \quad z \in \mathbb{C}$$

  where $a, b, c, d \in \mathbb{C}$, and $a d \neq b c$. You may assume $b \neq 0$. Show that any such $f(z)$ which maps the unit circle onto itself is necessarily of the form

  $$f(z)=e^{i \psi} \frac{a+z}{1+\bar{a} z} .$$

  [Hint: Show that it is always possible to choose $b=1$.]

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• 4.II.17F

  Complex Methods

  Part IB, 2001

  State Jordan's Lemma.

  Consider the integral

  $$I=\oint_{C} d z \frac{z \sin (x z)}{\left(a^{2}+z^{2}\right) \sin \pi z}$$

  for real $x$ and $a$. The rectangular contour $C$ runs from $+\infty+i \epsilon$ to $-\infty+i \epsilon$, to $-\infty-i \epsilon$, to $+\infty-i \epsilon$ and back to $+\infty+i \epsilon$, where $\epsilon$ is infinitesimal and positive. Perform the integral in two ways to show that

  $$\sum_{n=-\infty}^{\infty}(-1)^{n} \frac{n \sin n x}{a^{2}+n^{2}}=-\pi \frac{\sinh a x}{\sinh a \pi}$$

  for $|x|<\pi$.

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• 1.I.7B

  Complex Methods

  Part IB, 2002

  Using contour integration around a rectangle with vertices

  $$-x, x, x+i y,-x+i y,$$

  prove that, for all real $y$,

  $$\int_{-\infty}^{+\infty} e^{-(x+i y)^{2}} d x=\int_{-\infty}^{+\infty} e^{-x^{2}} d x$$

  Hence derive that the function $f(x)=e^{-x^{2} / 2}$ is an eigenfunction of the Fourier transform

  $$\widehat{f}(y)=\int_{-\infty}^{+\infty} f(x) e^{-i x y} d x$$

  i.e. $\widehat{f}$ is a constant multiple of $f$.

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• 1.II.16B

  Complex Methods

  Part IB, 2002

  (a) Show that if $f$ is an analytic function at $z_{0}$ and $f^{\prime}\left(z_{0}\right) \neq 0$, then $f$ is conformal at $z_{0}$, i.e. it preserves angles between paths passing through $z_{0}$.

  (b) Let $D$ be the disc given by $|z+i|<\sqrt{2}$, and let $H$ be the half-plane given by $y>0$, where $z=x+i y$. Construct a map of the domain $D \cap H$ onto $H$, and hence find a conformal mapping of $D \cap H$ onto the disc $\{z:|z|<1\}$. [Hint: You may find it helpful to consider a mapping of the form $(a z+b) /(c z+d)$, where ad $-b c \neq 0$.]

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• 2.I.7B

  Complex Methods

  Part IB, 2002

  Suppose that $f$ is analytic, and that $|f(z)|^{2}$ is constant in an open disk $D$. Use the Cauchy-Riemann equations to show that $f(z)$ is constant in $D$.

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• 2.II.16B

  Complex Methods

  Part IB, 2002

  A function $f(z)$ has an isolated singularity at $a$, with Laurent expansion

  $$f(z)=\sum_{n=-\infty}^{\infty} c_{n}(z-a)^{n}$$

  (a) Define res $(f, a)$, the residue of $f$ at the point $a$.

  (b) Prove that if $a$ is a pole of order $k+1$, then

  $$\operatorname{res}(f, a)=\lim _{z \rightarrow a} \frac{h^{(k)}(z)}{k !}, \quad \text { where } \quad h(z)=(z-a)^{k+1} f(z) .$$

  (c) Using the residue theorem and the formula above show that

  $$\int_{-\infty}^{\infty} \frac{d x}{\left(1+x^{2}\right)^{k+1}}=\pi \frac{(2 k) !}{(k !)^{2}} 4^{-k}, \quad k \geq 1$$

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• 4.I.8B

  Complex Methods

  Part IB, 2002

  Let $f$ be a function such that $\int_{-\infty}^{+\infty}|f(x)|^{2} d x<\infty$. Prove that

  $$\int_{-\infty}^{+\infty} f(x+k) \overline{f(x+l)} d x=0 \quad \text { for all integers } k \text { and } l \text { with } k \neq l,$$

  if and only if

  $$\int_{-\infty}^{+\infty}|\widehat{f}(t)|^{2} e^{-i m t} d t=0 \quad \text { for all integers } m \neq 0$$

  where $\widehat{f}$ is the Fourier transform of $f$.

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• 4.II.17B

  Complex Methods

  Part IB, 2002

  (a) Using the inequality $\sin \theta \geq 2 \theta / \pi$ for $0 \leq \theta \leq \frac{\pi}{2}$, show that, if $f$ is continuous for large $|z|$, and if $f(z) \rightarrow 0$ as $z \rightarrow \infty$, then

  $$\lim _{R \rightarrow \infty} \int_{\Gamma_{R}} f(z) e^{i \lambda z} d z=0 \quad \text { for } \quad \lambda>0$$

  where $\Gamma_{R}=R e^{i \theta}, 0 \leq \theta \leq \pi$.

  (b) By integrating an appropriate function $f(z)$ along the contour formed by the semicircles $\Gamma_{R}$ and $\Gamma_{r}$ in the upper half-plane with the segments of the real axis $[-R,-r]$ and $[r, R]$, show that

  $$\int_{0}^{\infty} \frac{\sin x}{x} d x=\frac{\pi}{2}$$

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• 2.I.7B

  Complex Methods

  Part IB, 2003

  (a) Using the residue theorem, evaluate

  $$\int_{|z|=1}\left(z-\frac{1}{z}\right)^{2 n} \frac{d z}{z}$$

  (b) Deduce that

  $$\int_{0}^{2 \pi} \sin ^{2 n} t d t=\frac{\pi}{2^{2 n-1}} \frac{(2 n) !}{(n !)^{2}}$$

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• 2.II.16B

  Complex Methods

  Part IB, 2003

  (a) Show that if $f$ satisfies the equation

  $$f^{\prime \prime}(x)-x^{2} f(x)=\mu f(x), \quad x \in \mathbb{R},$$

  where $\mu$ is a constant, then its Fourier transform $\widehat{f}$ satisfies the same equation, i.e.

  $$\widehat{f}^{\prime \prime}(\lambda)-\lambda^{2} \widehat{f}(\lambda)=\mu \widehat{f}(\lambda) .$$

  (b) Prove that, for each $n \geq 0$, there is a polynomial $p_{n}(x)$ of degree $n$, unique up to multiplication by a constant, such that

  $$f_{n}(x)=p_{n}(x) e^{-x^{2} / 2}$$

  is a solution of $(*)$ for some $\mu=\mu_{n}$.

  (c) Using the fact that $g(x)=e^{-x^{2} / 2}$ satisfies $\widehat{g}=c g$ for some constant $c$, show that the Fourier transform of $f_{n}$ has the form

  $$\widehat{f_{n}}(\lambda)=q_{n}(\lambda) e^{-\lambda^{2} / 2}$$

  where $q_{n}$ is also a polynomial of degree $n$.

  (d) Deduce that the $f_{n}$ are eigenfunctions of the Fourier transform operator, i.e. $\widehat{f_{n}}(x)=c_{n} f_{n}(x)$ for some constants $c_{n} .$

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• 4.I.8B

  Complex Methods

  Part IB, 2003

  Find the Laurent series centred on 0 for the function

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

  in each of the domains (a) $|z|<1$, (b) $1<|z|<2$, (c) $|z|>2$.

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• 4.II.17B

  Complex Methods

  Part IB, 2003

  Let

  $$f(z)=\frac{z^{m}}{1+z^{n}}, \quad n>m+1, \quad m, n \in \mathbb{N},$$

  and let $C_{R}$ be the boundary of the domain

  $$D_{R}=\left\{z=r e^{i \theta}: 0<r<R, \quad 0<\theta<\frac{2 \pi}{n}\right\}, \quad R>1 .$$

  (a) Using the residue theorem, determine

  $$\int_{C_{R}} f(z) d z$$

  (b) Show that the integral of $f(z)$ along the circular part $\gamma_{R}$ of $C_{R}$ tends to 0 as $R \rightarrow \infty$.

  (c) Deduce that

  $$\int_{0}^{\infty} \frac{x^{m}}{1+x^{n}} d x=\frac{\pi}{n \sin \frac{\pi(m+1)}{n}}$$

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• 1.I.7B

  Complex Methods

  Part IB, 2003

  Let $u(x, y)$ and $v(x, y)$ be a pair of conjugate harmonic functions in a domain $D$.

  Prove that

  $$U(x, y)=e^{-2 u v} \cos \left(u^{2}-v^{2}\right) \quad \text { and } \quad V(x, y)=e^{-2 u v} \sin \left(u^{2}-v^{2}\right)$$

  also form a pair of conjugate harmonic functions in $D$.

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• 1.II.16B

  Complex Methods

  Part IB, 2003

  Sketch the region $A$ which is the intersection of the discs

  $$D_{0}=\{z \in \mathbb{C}:|z|<1\} \quad \text { and } \quad D_{1}=\{z \in \mathbb{C}:|z-(1+i)|<1\} .$$

  Find a conformal mapping that maps $A$ onto the right half-plane $H=\{z \in \mathbb{C}: \operatorname{Re} z>0\}$. Also find a conformal mapping that maps $A$ onto $D_{0}$.

  [Hint: You may find it useful to consider maps of the form $w(z)=\frac{a z+b}{c z+d}$.]

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• 2.I.5A

  Complex Methods

  Part IB, 2004

  Let the functions $f$ and $g$ be analytic in an open, nonempty domain $\Omega$ and assume that $g \neq 0$ there. Prove that if $|f(z)| \equiv|g(z)|$ in $\Omega$ then there exists $\alpha \in \mathbb{R}$ such that $f(z) \equiv e^{i \alpha} g(z)$.

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• 2.II.16A

  Complex Methods

  Part IB, 2004

  Prove by using the Cauchy theorem that if $f$ is analytic in the open disc $\Omega=\{z \in \mathbb{C}:|z|<1\}$ then there exists a function $g$, analytic in $\Omega$, such that $g^{\prime}(z)=f(z)$, $z \in \Omega$.

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• 4.I.5A

  Complex Methods

  Part IB, 2004

  State and prove the Parseval formula.

  [You may use without proof properties of convolution, as long as they are precisely stated.]

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• 4.II.15A

  Complex Methods

  Part IB, 2004

  (i) Show that the inverse Fourier transform of the function

  $$\hat{g}(s)= \begin{cases}e^{s}-e^{-s}, & |s| \leqslant 1 \\ 0, & |s| \geqslant 1\end{cases}$$

  is

  $$g(x)=\frac{2 i}{\pi} \frac{1}{1+x^{2}}(x \sinh 1 \cos x-\cosh 1 \sin x)$$

  (ii) Determine, by using Fourier transforms, the solution of the Laplace equation

  $$\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0$$

  given in the strip $-\infty<x<\infty, 0<y<1$, together with the boundary conditions

  $$u(x, 0)=g(x), \quad u(x, 1) \equiv 0, \quad-\infty<x<\infty$$

  where $g$ has been given above.

  [You may use without proof properties of Fourier transforms.]

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• 1.I.5A

  Complex Methods

  Part IB, 2004

  Determine the poles of the following functions and calculate their residues there. (i) $\frac{1}{z^{2}+z^{4}}$, (ii) $\frac{e^{1 / z^{2}}}{z-1}$, (iii) $\frac{1}{\sin \left(e^{z}\right)}$.

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• 1.II.16A

  Complex Methods

  Part IB, 2004

  Let $p$ and $q$ be two polynomials such that

  $$q(z)=\prod_{l=1}^{m}\left(z-\alpha_{l}\right)$$

  where $\alpha_{1}, \ldots, \alpha_{m}$ are distinct non-real complex numbers and $\operatorname{deg} p \leqslant m-1$. Using contour integration, determine

  $$\int_{-\infty}^{\infty} \frac{p(x)}{q(x)} e^{i x} d x$$

  carefully justifying all steps.

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• 3.I.5F

  Complex Methods

  Part IB, 2005

  Define a harmonic function and state when the harmonic functions $f$ and $g$ are conjugate

  Let $\{u, v\}$ and $\{p, q\}$ be two pairs of harmonic conjugate functions. Prove that $\{p(u, v), q(u, v)\}$ are also harmonic conjugate.

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• 4.II.15F

  Complex Methods

  Part IB, 2005

  Determine the Fourier expansion of the function $f(x)=\sin \lambda x$, where $-\pi \leqslant x \leqslant \pi$, in the two cases where $\lambda$ is an integer and $\lambda$ is a real non-integer.

  Using the Parseval identity in the case $\lambda=\frac{1}{2}$, find an explicit expression for the sum

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

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• 3.I.5D

  Complex Methods

  Part IB, 2006

  The transformation

  $$w=i\left(\frac{1-z}{1+z}\right)$$

  maps conformally the interior of the unit disc $D$ onto the upper half-plane $H_{+}$, and maps the upper and lower unit semicircles $C_{+}$and $C_{-}$onto the positive and negative real axis $\mathbb{R}_{+}$and $\mathbb{R}_{-}$, respectively.

  Consider the Dirichlet problem in the upper half-plane:

  $$\frac{\partial^{2} f}{\partial u^{2}}+\frac{\partial^{2} f}{\partial v^{2}}=0 \quad \text { in } \quad H_{+} ; \quad f(u, v)= \begin{cases}1 & \text { on } \mathbb{R}_{+} \\ 0 & \text { on } \mathbb{R}_{-}\end{cases}$$

  Its solution is given by the formula

  $$f(u, v)=\frac{1}{2}+\frac{1}{\pi} \arctan \left(\frac{u}{v}\right) .$$

  Using this result, determine the solution to the Dirichlet problem in the unit disc:

  $$\frac{\partial^{2} F}{\partial x^{2}}+\frac{\partial^{2} F}{\partial y^{2}}=0 \quad \text { in } \quad D ; \quad F(x, y)= \begin{cases}1 & \text { on } C_{+} \\ 0 & \text { on } C_{-}\end{cases}$$

  Briefly explain your answer.

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• 4.II.15D

  Complex Methods

  Part IB, 2006

  Denote by $f * g$ the convolution of two functions, and by $\widehat{f}$ the Fourier transform, i.e.,

  $$[f * g](x)=\int_{-\infty}^{\infty} f(t) g(x-t) d t, \quad \widehat{f}(\lambda)=\int_{-\infty}^{\infty} f(x) e^{-i \lambda x} d x$$

  (a) Show that, for suitable functions $f$ and $g$, the Fourier transform $\widehat{F}$of the convolution $F=f * g$ is given by $\widehat{F}=\widehat{f} \cdot \widehat{g}$.

  (b) Let

  $$f_{1}(x)= \begin{cases}1 & |x| \leqslant 1 / 2 \\ 0 & \text { otherwise }\end{cases}$$

  and let $f_{2}=f_{1} * f_{1}$ be the convolution of $f_{1}$ with itself. Find the Fourier transforms of $f_{1}$ and $f_{2}$, and, by applying Parseval's theorem, determine the value of the integral

  $$\int_{-\infty}^{\infty}\left(\frac{\sin y}{y}\right)^{4} d y$$

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• $3 . \mathrm{I} . 5 \mathrm{~F} \quad$

  Complex Methods

  Part IB, 2007

  Show that the function $\phi(x, y)=\tan ^{-1} \frac{y}{x}$ is harmonic. Find its harmonic conjugate $\psi(x, y)$ and the analytic function $f(z)$ whose real part is $\phi(x, y)$. Sketch the curves $\phi(x, y)=C$ and $\psi(x, y)=K$.

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• 4.II.15F

  Complex Methods

  Part IB, 2007

  (i) Use the definition of the Laplace transform of $f(t)$ :

  $$L\{f(t)\}=F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t$$

  to show that, for $f(t)=t^{n}$,

  $$L\{f(t)\}=F(s)=\frac{n !}{s^{n+1}}, \quad L\left\{e^{a t} f(t)\right\}=F(s-a)=\frac{n !}{(s-a)^{n+1}}$$

  (ii) Use contour integration to find the inverse Laplace transform of

  $$F(s)=\frac{1}{s^{2}(s+1)^{2}}$$

  (iii) Verify the result in (ii) by using the results in (i) and the convolution theorem.

  (iv) Use Laplace transforms to solve the differential equation

  $$f^{(i v)}(t)+2 f^{\prime \prime \prime}(t)+f^{\prime \prime}(t)=0$$

  subject to the initial conditions

  $$f(0)=f^{\prime}(0)=f^{\prime \prime}(0)=0, \quad f^{\prime \prime \prime}(0)=1$$

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• 3.I.5C

  Complex Methods

  Part IB, 2008

  Using the contour integration formula for the inversion of Laplace transforms find the inverse Laplace transforms of the following functions: (a) $\frac{s}{s^{2}+a^{2}} \quad(a$ real and non-zero $)$, (b) $\frac{1}{\sqrt{s}}$.

  [You may use the fact that $\int_{-\infty}^{\infty} e^{-b x^{2}} d x=\sqrt{\pi / b}$.]

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• 4.II.15C

  Complex Methods

  Part IB, 2008

  Let $H$ be the domain $\mathbb{C}-\{x+i y: x \leq 0, y=0\}$ (i.e., $\mathbb{C}$ cut along the negative $x$-axis). Show, by a suitable choice of branch, that the mapping

  $$z \mapsto w=-i \log z$$

  maps $H$ onto the strip $S=\{z=x+i y,-\pi<x<\pi\}$.

  How would a different choice of branch change the result?

  Let $G$ be the domain $\{z \in \mathbb{C}:|z|<1,|z+i|>\sqrt{2}\}$. Find an analytic transformation that maps $G$ to $S$, where $S$ is the strip defined above.

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• Paper 3, Section I, $5 \mathrm{D}$

  Complex Methods

  Part IB, 2009

  Use the residue calculus to evaluate (i) $\oint_{C} z e^{1 / z} d z$ and (ii) $\oint_{C} \frac{z d z}{1-4 z^{2}}$,

  where $C$ is the circle $|z|=1$.

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• Paper 4, Section II, D

  Complex Methods

  Part IB, 2009

  The function $u(x, y)$ satisfies Laplace's equation in the half-space $y \geqslant 0$, together with boundary conditions

  $$\begin{gathered} u(x, y) \rightarrow 0 \text { as } y \rightarrow \infty \text { for all } x \\ u(x, 0)=u_{0}(x), \text { where } x u_{0}(x) \rightarrow 0 \text { as }|x| \rightarrow \infty \end{gathered}$$

  Using Fourier transforms, show that

  $$u(x, y)=\int_{-\infty}^{\infty} u_{0}(t) v(x-t, y) d t$$

  where

  $$v(x, y)=\frac{y}{\pi\left(x^{2}+y^{2}\right)}$$

  Suppose that $u_{0}(x)=\left(x^{2}+a^{2}\right)^{-1}$. Using contour integration and the convolution theorem, or otherwise, show that

  $$u(x, y)=\frac{y+a}{a\left[x^{2}+(y+a)^{2}\right]}$$

  [You may assume the convolution theorem of Fourier transforms, i.e. that if $\tilde{f}(k), \tilde{g}(k)$ are the Fourier transforms of two functions $f(x), g(x)$, then $\tilde{f}(k) \tilde{g}(k)$ is the Fourier transform of $\int_{-\infty}^{\infty} f(t) g(x-t) d t$.]

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• Paper 3, Section I, A

  Complex Methods

  Part IB, 2010

  (a) Prove that the real and imaginary parts of a complex differentiable function are harmonic.

  (b) Find the most general harmonic polynomial of the form

  $$u(x, y)=a x^{3}+b x^{2} y+c x y^{2}+d y^{3}$$

  where $a, b, c, d, x$ and $y$ are real.

  (c) Write down a complex analytic function of $z=x+i y$ of which $u(x, y)$ is the real part.

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• Paper 4, Section II, A

  Complex Methods

  Part IB, 2010

  A linear system is described by the differential equation

  $$y^{\prime \prime \prime}(t)-y^{\prime \prime}(t)-2 y^{\prime}(t)+2 y(t)=f(t),$$

  with initial conditions

  $$y(0)=0, \quad y^{\prime}(0)=1, \quad y^{\prime \prime}(0)=1$$

  The Laplace transform of $f(t)$ is defined as

  $$\mathcal{L}[f(t)]=\tilde{f}(s)=\int_{0}^{\infty} e^{-s t} f(t) d t$$

  You may assume the following Laplace transforms,

  $$\begin{aligned} \mathcal{L}[y(t)] &=\tilde{y}(s) \\ \mathcal{L}\left[y^{\prime}(t)\right] &=s \tilde{y}(s)-y(0) \\ \mathcal{L}\left[y^{\prime \prime}(t)\right] &=s^{2} \tilde{y}(s)-s y(0)-y^{\prime}(0) \\ \mathcal{L}\left[y^{\prime \prime \prime}(t)\right] &=s^{3} \tilde{y}(s)-s^{2} y(0)-s y^{\prime}(0)-y^{\prime \prime}(0) \end{aligned}$$

  (a) Use Laplace transforms to determine the response, $y_{1}(t)$, of the system to the signal

  $$f(t)=-2$$

  (b) Determine the response, $y_{2}(t)$, given that its Laplace transform is

  $$\tilde{y}_{2}(s)=\frac{1}{s^{2}(s-1)^{2}} .$$

  (c) Given that

  $$y^{\prime \prime \prime}(t)-y^{\prime \prime}(t)-2 y^{\prime}(t)+2 y(t)=g(t)$$

  leads to the response with Laplace transform

  $$\tilde{y}(s)=\frac{1}{s^{2}(s-1)^{2}},$$

  determine $g(t)$.

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• Paper 3, Section I, D

  Complex Methods

  Part IB, 2011

  Write down the function $\psi(u, v)$ that satisfies

  $$\frac{\partial^{2} \psi}{\partial u^{2}}+\frac{\partial^{2} \psi}{\partial v^{2}}=0, \quad \psi\left(-\frac{1}{2}, v\right)=-1, \quad \psi\left(\frac{1}{2}, v\right)=1$$

  The circular arcs $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ in the complex $z$-plane are defined by

  $$|z+1|=1, z \neq 0 \text { and }|z-1|=1, z \neq 0,$$

  respectively. You may assume without proof that the mapping from the complex $z$-plane to the complex $\zeta$-plane defined by

  $$\zeta=\frac{1}{z}$$

  takes $\mathcal{C}_{1}$ to the line $u=-\frac{1}{2}$ and $\mathcal{C}_{2}$ to the line $u=\frac{1}{2}$, where $\zeta=u+i v$, and that the region $\mathcal{D}$ in the $z$-plane exterior to both the circles $|z+1|=1$ and $|z-1|=1$ maps to the region in the $\zeta$-plane given by $-\frac{1}{2}<u<\frac{1}{2}$.

  Use the above mapping to solve the problem

  $$\nabla^{2} \phi=0 \quad \text { in } \mathcal{D}, \quad \phi=-1 \text { on } \mathcal{C}_{1} \text { and } \phi=1 \text { on } \mathcal{C}_{2}$$

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• Paper 4, Section II, D

  Complex Methods

  Part IB, 2011

  State and prove the convolution theorem for Laplace transforms.

  Use Laplace transforms to solve

  $$2 f^{\prime}(t)-\int_{0}^{t}(t-\tau)^{2} f(\tau) d \tau=4 t H(t)$$

  with $f(0)=0$, where $H(t)$ is the Heaviside function. You may assume that the Laplace transform, $\widehat{f}(s)$, of $f(t)$ exists for Re $s$ sufficiently large.

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• Paper 3, Section I, A

  Complex Methods

  Part IB, 2012

  State the formula for the Laplace transform of a function $f(t)$, defined for $t \geqslant 0$.

  Let $f(t)$ be periodic with period $T$ (i.e. $f(t+T)=f(t)$ ). If $g(t)$ is defined to be equal to $f(t)$ in $[0, T]$ and zero elsewhere and its Laplace transform is $G(s)$, show that the Laplace transform of $f(t)$ is given by

  $$F(s)=\frac{G(s)}{1-e^{-s T}}$$

  Hence, or otherwise, find the inverse Laplace transform of

  $$F(s)=\frac{1}{s} \frac{1-e^{-s T / 2}}{1-e^{-s T}}$$

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• Paper 4, Section II, A

  Complex Methods

  Part IB, 2012

  State the convolution theorem for Fourier transforms.

  The function $\phi(x, y)$ satisfies

  $$\nabla^{2} \phi=0$$

  on the half-plane $y \geqslant 0$, subject to the boundary conditions

  $$\begin{gathered} \phi \rightarrow 0 \text { as } y \rightarrow \infty \text { for all } x \\ \phi(x, 0)= \begin{cases}1, & |x| \leqslant 1 \\ 0, & |x|>1\end{cases} \end{gathered}$$

  Using Fourier transforms, show that

  $$\phi(x, y)=\frac{y}{\pi} \int_{-1}^{1} \frac{1}{y^{2}+(x-t)^{2}} \mathrm{~d} t$$

  and hence that

  $$\phi(x, y)=\frac{1}{\pi}\left[\tan ^{-1}\left(\frac{1-x}{y}\right)+\tan ^{-1}\left(\frac{1+x}{y}\right)\right]$$

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• Paper 3, Section I, D

  Complex Methods

  Part IB, 2013

  Let $y(t)=0$ for $t<0$, and let $\lim _{t \rightarrow 0^{+}} y(t)=y_{0}$.

  (i) Find the Laplace transforms of $H(t)$ and $t H(t)$, where $H(t)$ is the Heaviside step function.

  (ii) Given that the Laplace transform of $y(t)$ is $\widehat{y}(s)$, find expressions for the Laplace transforms of $\dot{y}(t)$ and $y(t-1)$.

  (iii) Use Laplace transforms to solve the equation

  $$\dot{y}(t)-y(t-1)=H(t)-(t-1) H(t-1)$$

  in the case $y_{0}=0$.

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• Paper 4, Section II, D

  Complex Methods

  Part IB, 2013

  Let $C_{1}$ and $C_{2}$ be the circles $x^{2}+y^{2}=1$ and $5 x^{2}-4 x+5 y^{2}=0$, respectively, and let $D$ be the (finite) region between the circles. Use the conformal mapping

  $$w=\frac{z-2}{2 z-1}$$

  to solve the following problem:

  $$\nabla^{2} \phi=0 \text { in } D \text { with } \phi=1 \text { on } C_{1} \text { and } \phi=2 \text { on } C_{2}$$

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• Paper 3, Section I, B

  Complex Methods

  Part IB, 2014

  Find the most general cubic form

  $$u(x, y)=a x^{3}+b x^{2} y+c x y^{2}+d y^{3}$$

  which satisfies Laplace's equation, where $a, b, c$ and $d$ are all real. Hence find an analytic function $f(z)=f(x+i y)$ which has such a $u$ as its real part.

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• Paper 4, Section II, B

  Complex Methods

  Part IB, 2014

  Find the Laplace transforms of $t^{n}$ for $n$ a positive integer and $H(t-a)$ where $a>0$ and $H(t)$ is the Heaviside step function.

  Consider a semi-infinite string which is initially at rest and is fixed at one end. The string can support wave-like motions, and for $t>0$ it is allowed to fall under gravity. Therefore the deflection $y(x, t)$ from its initial location satisfies

  $$\frac{\partial^{2}}{\partial t^{2}} y=c^{2} \frac{\partial^{2}}{\partial x^{2}} y+g \quad \text { for } \quad x>0, t>0$$

  with

  $$y(0, t)=y(x, 0)=\frac{\partial}{\partial t} y(x, 0)=0 \quad \text { and } \quad y(x, t) \rightarrow \frac{g t^{2}}{2} \text { as } x \rightarrow \infty$$

  where $g$ is a constant. Use Laplace transforms to find $y(x, t)$.

  [The convolution theorem for Laplace transforms may be quoted without proof.]

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• Paper 3, Section I, B

  Complex Methods

  Part IB, 2015

  Find the Fourier transform of the function

  $$f(x)=\frac{1}{1+x^{2}}, \quad x \in \mathbb{R}$$

  using an appropriate contour integration. Hence find the Fourier transform of its derivative, $f^{\prime}(x)$, and evaluate the integral

  $$I=\int_{-\infty}^{\infty} \frac{4 x^{2}}{\left(1+x^{2}\right)^{4}} d x$$

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• Paper 4, Section II, B

  Complex Methods

  Part IB, 2015

  (i) State and prove the convolution theorem for Laplace transforms of two realvalued functions.

  (ii) Let the function $f(t), t \geqslant 0$, be equal to 1 for $0 \leqslant t \leqslant a$ and zero otherwise, where $a$ is a positive parameter. Calculate the Laplace transform of $f$. Hence deduce the Laplace transform of the convolution $g=f * f$. Invert this Laplace transform to obtain an explicit expression for $g(t)$.

  [Hint: You may use the notation $\left.(t-a)_{+}=H(t-a) \cdot(t-a) .\right]$

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• Paper 3, Section I, A

  Complex Methods

  Part IB, 2016

  The function $f(x)$ has Fourier transform

  $$\tilde{f}(k)=\int_{-\infty}^{\infty} f(x) e^{-i k x} d x=\frac{-2 k i}{p^{2}+k^{2}},$$

  where $p>0$ is a real constant. Using contour integration, calculate $f(x)$ for $x<0$. [Jordan's lemma and the residue theorem may be used without proof.]

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• Paper 4, Section II, A

  Complex Methods

  Part IB, 2016

  (a) Show that the Laplace transform of the Heaviside step function $H(t-a)$ is

  $$\int_{0}^{\infty} H(t-a) e^{-p t} d t=\frac{e^{-a p}}{p}$$

  for $a>0$.

  (b) Derive an expression for the Laplace transform of the second derivative of a function $f(t)$ in terms of the Laplace transform of $f(t)$ and the properties of $f(t)$ at $t=0$.

  (c) A bar of length $L$ has its end at $x=L$ fixed. The bar is initially at rest and straight. The end at $x=0$ is given a small fixed transverse displacement of magnitude $a$ at $t=0^{+}$. You may assume that the transverse displacement $y(x, t)$ of the bar satisfies the wave equation with some wave speed $c$, and so the tranverse displacement $y(x, t)$ is the solution to the problem:

  $$\begin{array}{cl} \frac{\partial^{2} y}{\partial t^{2}}=c^{2} \frac{\partial^{2} y}{\partial x^{2}} & \text { for } 0<x<L \text { and } t> \\ y(x, 0)=\frac{\partial y}{\partial t}(x, 0)=0 & \text { for } 0<x<L, \\ y(0, t)=a ; y(L, t)=0 & \text { for } t>0 . \end{array}$$

  (i) Show that the Laplace transform $Y(x, p)$ of $y(x, t)$, defined as

  $$Y(x, p)=\int_{0}^{\infty} y(x, t) e^{-p t} d t$$

  is given by

  $$Y(x, p)=\frac{a \sinh \left[\frac{p}{c}(L-x)\right]}{p \sinh \left[\frac{p L}{c}\right]}$$

  (ii) By use of the binomial theorem or otherwise, express $y(x, t)$ as an infinite series.

  (iii) Plot the transverse displacement of the midpoint of the bar $y(L / 2, t)$ against time.

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• Paper 3, Section I, A

  Complex Methods

  Part IB, 2017

  By using the Laplace transform, show that the solution to

  $$y^{\prime \prime}-4 y^{\prime}+3 y=t e^{-3 t},$$

  subject to the conditions $y(0)=0$ and $y^{\prime}(0)=1$, is given by

  $$y(t)=\frac{37}{72} e^{3 t}-\frac{17}{32} e^{t}+\left(\frac{5}{288}+\frac{1}{24} t\right) e^{-3 t}$$

  when $t \geqslant 0$.

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• Paper 4, Section II, A

  Complex Methods

  Part IB, 2017

  By using Fourier transforms and a conformal mapping

  $$w=\sin \left(\frac{\pi z}{a}\right)$$

  with $z=x+i y$ and $w=\xi+i \eta$, and a suitable real constant $a$, show that the solution to

  $$\begin{array}{rlrl} \nabla^{2} \phi & =0 & -2 \pi \leqslant x \leqslant 2 \pi, y \geqslant 0 \\ \phi(x, 0) & =f(x) & -2 \pi \leqslant x \leqslant 2 \pi \\ \phi(\pm 2 \pi, y) & =0 & y>0, \\ \phi(x, y) & \rightarrow 0 & y \rightarrow \infty,-2 \pi \leqslant x \leqslant 2 \pi \end{array}$$

  is given by

  $$\phi(\xi, \eta)=\frac{\eta}{\pi} \int_{-1}^{1} \frac{F\left(\xi^{\prime}\right)}{\eta^{2}+\left(\xi-\xi^{\prime}\right)^{2}} d \xi^{\prime}$$

  where $F\left(\xi^{\prime}\right)$ is to be determined.

  In the case of $f(x)=\sin \left(\frac{x}{4}\right)$, give $F\left(\xi^{\prime}\right)$ explicitly as a function of $\xi^{\prime}$. [You need not evaluate the integral.]

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• Paper 3, Section I, A

  Complex Methods

  Part IB, 2018

  (a) Let $f(z)=\left(z^{2}-1\right)^{1 / 2}$. Define the branch cut of $f(z)$ as $[-1,1]$ such that

  $$f(x)=+\sqrt{x^{2}-1} \quad x>1$$

  Show that $f(z)$ is an odd function.

  (b) Let $g(z)=\left[(z-2)\left(z^{2}-1\right)\right]^{1 / 2}$.

  (i) Show that $z=\infty$ is a branch point of $g(z)$.

  (ii) Define the branch cuts of $g(z)$ as $[-1,1] \cup[2, \infty)$ such that

  $$g(x)=e^{\pi i / 2} \sqrt{|x-2|\left|x^{2}-1\right|} \quad x \in(1,2) .$$

  Find $g\left(0_{\pm}\right)$, where $0_{+}$denotes $z=0$ just above the branch cut, and $0_{-}$denotes $z=0$ just below the branch cut.

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• Paper 4, Section II, A

  Complex Methods

  Part IB, 2018

  (a) Find the Laplace transform of

  $$y(t)=\frac{e^{-a^{2} / 4 t}}{\sqrt{\pi t}}$$

  for $a \in \mathbb{R}, a \neq 0$.

  [You may use without proof that

  $$\left.\int_{0}^{\infty} \exp \left(-c^{2} x^{2}-\frac{c^{2}}{x^{2}}\right) d x=\frac{\sqrt{\pi}}{2|c|} e^{-2 c^{2}} .\right]$$

  (b) By using the Laplace transform, show that the solution to

  $$\begin{aligned} \frac{\partial^{2} u}{\partial x^{2}} &=\frac{\partial u}{\partial t} \quad-\infty<x<\infty, \quad t>0 \\ u(x, 0) &=f(x) \\ u(x, t) \quad \text { bounded, } \end{aligned}$$

  can be written as

  $$u(x, t)=\int_{-\infty}^{\infty} K(|x-\xi|, t) f(\xi) d \xi$$

  for some $K(|x-\xi|, t)$ to be determined.

  [You may use without proof that a particular solution to

  $$y^{\prime \prime}(x)-s y(x)+f(x)=0$$

  is given by

  $$\left.y(x)=\frac{e^{-\sqrt{s} x}}{2 \sqrt{s}} \int_{0}^{x} e^{\sqrt{s} \xi} f(\xi) d \xi-\frac{e^{\sqrt{s} x}}{2 \sqrt{s}} \int_{0}^{x} e^{-\sqrt{s} \xi} f(\xi) d \xi .\right]$$

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• Paper 3, Section I, D

  Complex Methods

  Part IB, 2019

  By considering the transformation $w=i(1-z) /(1+z)$, find a solution to Laplace's equation $\nabla^{2} \phi=0$ inside the unit disc $D \subset \mathbb{C}$, subject to the boundary conditions

  $$\left.\phi\right|_{|z|=1}= \begin{cases}\phi_{0} & \text { for } \arg (z) \in(0, \pi) \\ -\phi_{0} & \text { for } \arg (z) \in(\pi, 2 \pi)\end{cases}$$

  where $\phi_{0}$ is constant. Give your answer in terms of $(x, y)=(\operatorname{Re} z, \operatorname{Im} z)$.

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• Paper 4, Section II, D

  Complex Methods

  Part IB, 2019

  (a) Using the Bromwich contour integral, find the inverse Laplace transform of $1 / s^{2}$.

  The temperature $u(r, t)$ of mercury in a spherical thermometer bulb $r \leqslant a$ obeys the radial heat equation

  $$\frac{\partial u}{\partial t}=\frac{1}{r} \frac{\partial^{2}}{\partial r^{2}}(r u)$$

  with unit diffusion constant. At $t=0$ the mercury is at a uniform temperature $u_{0}$ equal to that of the surrounding air. For $t>0$ the surrounding air temperature lowers such that at the edge of the thermometer bulb

  $$\left.\frac{1}{k} \frac{\partial u}{\partial r}\right|_{r=a}=u_{0}-u(a, t)-t$$

  where $k$ is a constant.

  (b) Find an explicit expression for $U(r, s)=\int_{0}^{\infty} e^{-s t} u(r, t) d t$.

  (c) Show that the temperature of the mercury at the centre of the thermometer bulb at late times is

  $$u(0, t) \approx u_{0}-t+\frac{a}{3 k}+\frac{a^{2}}{6}$$

  [You may assume that the late time behaviour of $u(r, t)$ is determined by the singular part of $U(r, s)$ at $s=0 .]$

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• Paper 3, Section I, B

  Complex Methods

  Part IB, 2021

  Find the value of $A$ for which the function

  $$\phi(x, y)=x \cosh y \sin x+A y \sinh y \cos x$$

  satisfies Laplace's equation. For this value of $A$, find a complex analytic function of which $\phi$ is the real part.

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• Paper 4, Section II, B

  Complex Methods

  Part IB, 2021

  Let $f(t)$ be defined for $t \geqslant 0$. Define the Laplace transform $\widehat{f}(s)$ of $f$. Find an expression for the Laplace transform of $\frac{d f}{d t}$ in terms of $\widehat{f}$.

  Three radioactive nuclei decay sequentially, so that the numbers $N_{i}(t)$ of the three types obey the equations

  $$\begin{aligned} \frac{d N_{1}}{d t} &=-\lambda_{1} N_{1} \\ \frac{d N_{2}}{d t} &=\lambda_{1} N_{1}-\lambda_{2} N_{2} \\ \frac{d N_{3}}{d t} &=\lambda_{2} N_{2}-\lambda_{3} N_{3} \end{aligned}$$

  where $\lambda_{3}>\lambda_{2}>\lambda_{1}>0$ are constants. Initially, at $t=0, N_{1}=N, N_{2}=0$ and $N_{3}=n$. Using Laplace transforms, find $N_{3}(t)$.

  By taking an appropriate limit, find $N_{3}(t)$ when $\lambda_{2}=\lambda_{1}=\lambda>0$ and $\lambda_{3}>\lambda$.

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