• # B1.19

State the convolution theorem for Laplace transforms.

The temperature $T(x, t)$ in a semi-infinite rod satisfies the heat equation

$\frac{\partial^{2} T}{\partial x^{2}}=\frac{1}{k} \frac{\partial T}{\partial t}, \quad x \geq 0, t \geq 0$

and the conditions $T(x, 0)=0$ for $x \geq 0, T(0, t)=f(t)$ for $t \geq 0$ and $T(x, t) \rightarrow 0$ as $x \rightarrow \infty$. Show that

$T(x, t)=\int_{0}^{t} f(\tau) G(x, t-\tau) d \tau$

where

$G(x, t)=\sqrt{\frac{x^{2}}{4 \pi k t^{3}}} e^{-x^{2} / 4 k t}$

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• # B2.19

(a) The Beta function is defined by

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

Show that

$\mathrm{B}(p, q)=\int_{1}^{\infty} x^{-p-q}(x-1)^{q-1} d x$

(b) The function $J(p, q)$ is defined by

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

where the integrand has a branch cut along the positive real axis. Just above the cut, $\arg t=0$. For $t>1$ just above the cut, arg $(1-t)=-\pi$. The contour $\gamma$ runs from $t=\infty e^{2 \pi i}$, round the origin in the negative sense, to $t=\infty$ (i.e. the contour is a reflection of the usual Hankel contour). What restriction must be placed on $p$ and $q$ for the integral to converge?

By evaluating $J(p, q)$ in two ways, show that

$\left(1-e^{2 \pi i p}\right) \mathrm{B}(p, q)+\left(e^{-\pi i(q-1)}-e^{\pi i(2 p+q-1)}\right) \mathrm{B}(1-p-q, q)=0,$

where $p$ and $q$ are any non-integer complex numbers.

Using the identity

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

deduce that

$\Gamma(p) \Gamma(1-p) \sin (\pi p)=\Gamma(p+q) \Gamma(1-p-q) \sin [\pi(1-p-q)]$

and hence that

$\pi=\Gamma(q) \Gamma(1-q) \sin [\pi(1-q)]$

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• # B3.19

The function $w(z)$ satisfies the third-order differential equation

$\frac{d^{3} w}{d z^{3}}-z w=0$

subject to the conditions $w(z) \rightarrow 0$ as $z \rightarrow \pm i \infty$ and $w(0)=1$. Obtain an integral representation for $w(z)$ of the form

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

and determine the function $f(t)$ and the contour $\gamma$.

Using the change of variable $t=z^{1 / 3} \tau$, or otherwise, compute the leading term in the asymptotic expansion of $w(z)$ as $z \rightarrow+\infty$.

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• # B4.19

Let $h(t)=i\left(t+t^{2}\right)$. Sketch the path of $\operatorname{Im}(h(t))=$ const. through the point $t=0$, and the path of $\operatorname{Im}(h(t))=$ const. through the point $t=1$.

By integrating along these paths, show that as $\lambda \rightarrow \infty$

$\int_{0}^{1} t^{-1 / 2} e^{i \lambda\left(t+t^{2}\right)} d t \sim \frac{c_{1}}{\lambda^{1 / 2}}+\frac{c_{2} e^{2 i \lambda}}{\lambda},$

where the constants $c_{1}$ and $c_{2}$ are to be computed.

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• # B1.19

By considering the integral

$\int_{C}\left(\frac{t}{1-t}\right)^{i} d t$

where $C$ is a large circle centred on the origin, show that

$B(1+i, 1-i)=\pi \operatorname{cosech} \pi$

where

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

By using $B(p, q)=\frac{\Gamma(p) \Gamma(q)}{\Gamma(p+q)}$, deduce that $\Gamma(i) \Gamma(-i)=\pi \operatorname{cosech} \pi$.

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• # B2.18

Let $\widehat{y}(p)$ be the Laplace transform of $y(t)$, where $y(t)$ satisfies

$y^{\prime}(t)=y(\pi-t)$

and

$y(0)=1 ; \quad y(\pi)=k ; \quad y(t)=0 \text { for } t<0 \text { and for } t>\pi$

Show that

$p \widehat{y}(p)+k e^{-\pi p}-1=e^{-\pi p} \widehat{y}(-p)$

and hence deduce that

$\widehat{y}(p)=\frac{(k+p)-(1+p k) e^{-\pi p}}{1+p^{2}}$

Use the inversion formula for Laplace transforms to find $y(t)$ for $t>\pi$ and deduce that a solution of the above boundary value problem exists only if $k=-1$. Hence find $y(t)$ for $0 \leqslant t \leqslant \pi$.

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• # B3.19

Let

$f(\lambda)=\int_{\gamma} e^{\lambda\left(t-t^{3} / 3\right)} d t, \quad \lambda \text { real and positive }$

where $\gamma$ is a path beginning at $\infty e^{-2 i \pi / 3}$ and ending at $+\infty$ (on the real axis). Identify the saddle points and sketch the paths of constant phase through these points.

Hence show that $f(\lambda) \sim e^{2 \lambda / 3} \sqrt{\pi / \lambda}$ as $\lambda \rightarrow \infty$.

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• # B4.19

By setting $w(z)=\int_{\gamma} f(t) e^{-z t} d t$, where $\gamma$ and $f(t)$ are to be suitably chosen, explain how to find integral representations of the solutions of the equation

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

where $k$ is a non-zero real constant and $z$ is complex. Discuss $\gamma$ in the particular case that $z$ is restricted to be real and positive and distinguish the different cases that arise according to the $\operatorname{sign}$ of $k$.

Show that in this particular case, by choosing $\gamma$ as a closed contour around the origin, it is possible to express a solution in the form

$w(z)=A \sum_{n=0}^{\infty} \frac{(z k)^{n+1}}{n !(n+1) !}$

where $A$ is a constant.

Show also that for $k>0$ there are solutions that satisfy

$w(z) \sim B z^{1 / 4} e^{-2 \sqrt{k z}} \quad \text { as } z \rightarrow \infty$

where $B$ is a constant.

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• # B1.19

State the Riemann-Lebesgue lemma as applied to the integral

$\int_{a}^{b} g(u) e^{i x u} d u$

where $g^{\prime}(u)$ is continuous and $a, b \in \mathbb{R}$.

Use this lemma to show that, as $x \rightarrow+\infty$,

$\int_{a}^{b}(u-a)^{\lambda-1} f(u) e^{i x u} d u \sim f(a) e^{i x a} e^{i \pi \lambda / 2} \Gamma(\lambda) x^{-\lambda}$

where $f(u)$ is holomorphic, $f(a) \neq 0$ and $0<\lambda<1$. You should explain each step of your argument, but detailed analysis is not required.

Hence find the leading order asymptotic behaviour as $x \rightarrow+\infty$ of

$\int_{0}^{1} \frac{e^{i x t^{2}}}{\left(1-t^{2}\right)^{\frac{1}{2}}} d t$

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• # B2.18

Show that

$\mathcal{P} \int_{-\infty}^{\infty} \frac{t^{z-1}}{t-a} d t=\pi i a^{z-1},$

where $a$ is real and positive, $0<\operatorname{Re}(z)<1$ and $\mathcal{P}$ denotes the Cauchy principal value; the principal branches of $t^{z}$ etc. are implied. Deduce that

$\int_{0}^{\infty} \frac{t^{z-1}}{t+a} d t=\pi a^{z-1} \operatorname{cosec} \pi z$

and that

$\mathcal{P} \int_{0}^{\infty} \frac{t^{z-1}}{t-a} d t=-\pi a^{z-1} \cot \pi z$

Use $(*)$ to show that, if $\operatorname{Im}(b)>0$, then

$\int_{0}^{\infty} \frac{t^{z-1}}{t-b} d t=-\pi b^{z-1}(\cot \pi z-i)$

What is the value of this integral if $\operatorname{Im}(b)<0$ ?

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• # B3.19

Show that the equation

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

has solutions of the form

$w(z)=\int_{\gamma}(t-1)^{(\lambda-1) / 2}(t+1)^{-(\lambda+1) / 2} e^{z t} d t$

Give examples of possible choices of $\gamma$ to provide two independent solutions, assuming $\operatorname{Re}(z)>0$. Distinguish between the cases $\operatorname{Re} \lambda>-1$ and $\operatorname{Re} \lambda<1$. Comment on the case $-1<\operatorname{Re} \lambda<1$ and on the case that $\lambda$ is an odd integer.

Show that, if $\operatorname{Re} \lambda<1$, there is a solution $w_{1}(z)$ that is bounded as $z \rightarrow+\infty$, and that, in this limit,

$w_{1}(z) \sim A e^{-z} z^{(\lambda-1) / 2}\left(1-\frac{(1-\lambda)^{2}}{8 z}\right),$

where $A$ is a constant.

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• # B4.19

Let

$I(\lambda, a)=\int_{-i \infty}^{i \infty} \frac{e^{\lambda\left(t^{3}-3 t\right)}}{t^{2}-a^{2}} d t$

where $\lambda$ is real, $a$ is real and non-zero, and the path of integration runs up the imaginary axis. Show that, if $a^{2}>1$,

$I(\lambda, a) \sim \frac{i e^{-2 \lambda}}{1-a^{2}} \sqrt{\frac{\pi}{3 \lambda}}$

as $\lambda \rightarrow+\infty$ and sketch the relevant steepest descent path.

What is the corresponding result if $a^{2}<1$ ?

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• # B1.19

State and prove the convolution theorem for Laplace transforms.

Use the convolution theorem to prove that the Beta function

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

may be written in terms of the Gamma function as

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

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• # B2.18

The Bessel function $J_{\nu}(z)$ is defined, for $|\arg z|<\pi / 2$, by

$J_{\nu}(z)=\frac{1}{2 \pi i} \int_{-\infty}^{\left(0^{+}\right)} \mathrm{e}^{\left(t-t^{-1}\right) z / 2} t^{-\nu-1} d t,$

where the path of integration is the Hankel contour and $t^{-\nu-1}$ is the principal branch.

Use the method of steepest descent to show that, as $z \rightarrow+\infty$,

$J_{\nu}(z) \sim(2 / \pi z)^{\frac{1}{2}} \cos (z-\pi \nu / 2-\pi / 4) .$

You should give a rough sketch of the steepest descent paths.

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• # B3.19

Consider the integral

$\int_{0}^{\infty} \frac{t^{z} \mathrm{e}^{-a t}}{1+t} d t$

where $t^{z}$ is the principal branch and $a$ is a positive constant. State the region of the complex $z$-plane in which the integral defines a holomorphic function.

Show how the analytic continuation of this function can be obtained by means of an alternative integral representation using the Hankel contour.

Hence show that the analytic continuation is holomorphic except for simple poles at $z=-1,-2, \ldots$, and that the residue at $z=-n$ is

$(-1)^{n-1} \sum_{r=0}^{n-1} \frac{a^{r}}{r !}$

Part II

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• # B4.19

Show that $\int_{0}^{\pi} \mathrm{e}^{i x \cos t} d t$ satisfies the differential equation

$x y^{\prime \prime}+y^{\prime}+x y=0$

and find a second solution, in the form of an integral, for $x>0$.

Show, by finding the asymptotic behaviour as $x \rightarrow+\infty$, that your two solutions are linearly independent.

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