• # Paper 2, Section II, 32A

(a) Let $x(t)$ and $\phi_{n}(t)$, for $n=0,1,2, \ldots$, be real-valued functions on $\mathbb{R}$.

(i) Define what it means for the sequence $\left\{\phi_{n}(t)\right\}_{n=0}^{\infty}$ to be an asymptotic sequence as $t \rightarrow \infty$.

(ii) Define what it means for $x(t)$ to have the asymptotic expansion

$x(t) \sim \sum_{n=0}^{\infty} a_{n} \phi_{n}(t) \quad \text { as } \quad t \rightarrow \infty$

(b) Use the method of stationary phase to calculate the leading-order asymptotic approximation as $x \rightarrow \infty$ of

$I(x)=\int_{0}^{1} \sin \left(x\left(2 t^{4}-t^{2}\right)\right) d t$

[You may assume that $\int_{-\infty}^{\infty} e^{i u^{2}} d u=\sqrt{\pi} e^{i \pi / 4}$.]

(c) Use Laplace's method to calculate the leading-order asymptotic approximation as $x \rightarrow \infty$ of

$J(x)=\int_{0}^{1} \sinh \left(x\left(2 t^{4}-t^{2}\right)\right) d t$

[In parts (b) and (c) you should include brief qualitative reasons for the origin of the leading-order contributions, but you do not need to give a formal justification.]

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• # Paper 3, Section II, 30A

(a) Carefully state Watson's lemma.

(b) Use the method of steepest descent and Watson's lemma to obtain an infinite asymptotic expansion of the function

$I(x)=\int_{-\infty}^{\infty} \frac{e^{-x\left(z^{2}-2 i z\right)}}{1-i z} d z \quad \text { as } \quad x \rightarrow \infty$

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• # Paper 4, Section II, A

(a) Classify the nature of the point at $\infty$ for the ordinary differential equation

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

(b) Find a transformation from $(*)$ to an equation of the form

$u^{\prime \prime}+q(x) u=0$

and determine $q(x)$.

(c) Given $u(x)$ satisfies ( $\dagger$, use the Liouville-Green method to find the first three terms in an asymptotic approximation as $x \rightarrow \infty$ for $u(x)$, verifying the consistency of any approximations made.

(d) Hence obtain corresponding asymptotic approximations as $x \rightarrow \infty$ of two linearly independent solutions $y(x)$ of $(*)$.

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• # Paper 2, Section II, D

(a) Let $\delta>0$ and $x_{0} \in \mathbb{R}$. Let $\left\{\phi_{n}(x)\right\}_{n=0}^{\infty}$ be a sequence of (real) functions that are nonzero for all $x$ with $0<\left|x-x_{0}\right|<\delta$, and let $\left\{a_{n}\right\}_{n=0}^{\infty}$ be a sequence of nonzero real numbers. For every $N=0,1,2, \ldots$, the function $f(x)$ satisfies

$f(x)-\sum_{n=0}^{N} a_{n} \phi_{n}(x)=o\left(\phi_{N}(x)\right), \quad \text { as } \quad x \rightarrow x_{0}$

(i) Show that $\phi_{n+1}(x)=o\left(\phi_{n}(x)\right)$, for all $n=0,1,2, \ldots$; i.e., $\left\{\phi_{n}(x)\right\}_{n=0}^{\infty}$ is an asymptotic sequence.

(ii) Show that for any $N=0,1,2, \ldots$, the functions $\phi_{0}(x), \phi_{1}(x), \ldots, \phi_{N}(x)$ are linearly independent on their domain of definition.

(b) Let

$I(\varepsilon)=\int_{0}^{\infty}(1+\varepsilon t)^{-2} e^{-(1+\varepsilon) t} d t, \quad \text { for } \varepsilon>0$

(i) Find an asymptotic expansion (not necessarily a power series) of $I(\varepsilon)$, as $\varepsilon \rightarrow 0^{+}$.

(ii) Find the first four terms of the expansion of $I(\varepsilon)$ into an asymptotic power series of $\varepsilon$, that is, with error $o\left(\varepsilon^{3}\right)$ as $\varepsilon \rightarrow 0^{+}$.

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• # Paper 3, Section II, D

(a) Find the leading order term of the asymptotic expansion, as $x \rightarrow \infty$, of the integral

$I(x)=\int_{0}^{3 \pi} e^{(t+x \cos t)} d t$

(b) Find the first two leading nonzero terms of the asymptotic expansion, as $x \rightarrow \infty$, of the integral

$J(x)=\int_{0}^{\pi}(1-\cos t) e^{-x \ln (1+t)} d t$

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• # Paper 4, Section II, A

Consider the differential equation

$\tag{†} y^{\prime \prime}-y^{\prime}-\frac{2(x+1)}{x^{2}} y=0$

(i) Classify what type of regularity/singularity equation $(†)$ has at $x=\infty$.

(ii) Find a transformation that maps equation ($†$) to an equation of the form

$u^{\prime \prime}+q(x) u=0$

(iii) Find the leading-order term of the asymptotic expansions of the solutions of equation $(*)$, as $x \rightarrow \infty$, using the Liouville-Green method.

(iv) Derive the leading-order term of the asymptotic expansion of the solutions $y$ of ($†$). Check that one of them is an exact solution for $(†)$.

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• # Paper 2, Section II, A

(a) Define formally what it means for a real valued function $f(x)$ to have an asymptotic expansion about $x_{0}$, given by

$f(x) \sim \sum_{n=0}^{\infty} f_{n}\left(x-x_{0}\right)^{n} \text { as } x \rightarrow x_{0}$

Use this definition to prove the following properties.

(i) If both $f(x)$ and $g(x)$ have asymptotic expansions about $x_{0}$, then $h(x)=f(x)+g(x)$ also has an asymptotic expansion about $x_{0} .$

(ii) If $f(x)$ has an asymptotic expansion about $x_{0}$ and is integrable, then

$\int_{x_{0}}^{x} f(\xi) d \xi \sim \sum_{n=0}^{\infty} \frac{f_{n}}{n+1}\left(x-x_{0}\right)^{n+1} \text { as } x \rightarrow x_{0}$

(b) Obtain, with justification, the first three terms in the asymptotic expansion as $x \rightarrow \infty$ of the complementary error function, $\operatorname{erfc}(x)$, defined as

$\operatorname{erfc}(x):=\frac{1}{\sqrt{2 \pi}} \int_{x}^{\infty} e^{-t^{2}} d t$

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• # Paper 3, Section II, A

(a) State Watson's lemma for the case when all the functions and variables involved are real, and use it to calculate the asymptotic approximation as $x \rightarrow \infty$ for the integral $I$, where

$I=\int_{0}^{\infty} e^{-x t} \sin \left(t^{2}\right) d t$

(b) The Bessel function $J_{\nu}(z)$ of the first kind of order $\nu$ has integral representation

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

where $\Gamma$ is the Gamma function, $\operatorname{Re}(\nu)>1 / 2$ and $z$ is in general a complex variable. The complex version of Watson's lemma is obtained by replacing $x$ with the complex variable $z$, and is valid for $|z| \rightarrow \infty$ and $|\arg (z)| \leqslant \pi / 2-\delta<\pi / 2$, for some $\delta$ such that $0<\delta<\pi / 2$. Use this version to derive an asymptotic expansion for $J_{\nu}(z)$ as $|z| \rightarrow \infty$. For what values of $\arg (z)$ is this approximation valid?

[Hint: You may find the substitution $t=2 \tau-1$ useful.]

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• # Paper 4, Section II, A

Consider, for small $\epsilon$, the equation

$\epsilon^{2} \frac{d^{2} \psi}{d x^{2}}-q(x) \psi=0$

Assume that $(*)$ has bounded solutions with two turning points $a, b$ where $b>a, q^{\prime}(b)>0$ and $q^{\prime}(a)<0$.

(a) Use the WKB approximation to derive the relationship

$\frac{1}{\epsilon} \int_{a}^{b}|q(\xi)|^{1 / 2} d \xi=\left(n+\frac{1}{2}\right) \pi \text { with } n=0,1,2, \cdots$

[You may quote without proof any standard results or formulae from WKB theory.]

(b) In suitable units, the radial Schrödinger equation for a spherically symmetric potential given by $V(r)=-V_{0} / r$, for constant $V_{0}$, can be recast in the standard form $(*)$ as:

$\frac{\hbar^{2}}{2 m} \frac{d^{2} \psi}{d x^{2}}+e^{2 x}\left[\lambda-V\left(e^{x}\right)-\frac{\hbar^{2}}{2 m}\left(l+\frac{1}{2}\right)^{2} e^{-2 x}\right] \psi=0$

where $r=e^{x}$ and $\epsilon=\hbar / \sqrt{2 m}$ is a small parameter.

Use result $(* *)$ to show that the energies of the bound states (i.e $\lambda=-|\lambda|<0)$ are approximated by the expression:

$E=-|\lambda|=-\frac{m}{2 \hbar^{2}} \frac{V_{0}^{2}}{(n+l+1)^{2}}$

[You may use the result

$\left.\int_{a}^{b} \frac{1}{r} \sqrt{(r-a)(b-r)} d r=(\pi / 2)[\sqrt{b}-\sqrt{a}]^{2} .\right]$

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• # Paper 2, Section II, B

Given that $\int_{-\infty}^{+\infty} e^{-u^{2}} d u=\sqrt{\pi}$ obtain the value of $\lim _{R \rightarrow+\infty} \int_{-R}^{+R} e^{-i t u^{2}} d u$ for real positive $t$. Also obtain the value of $\lim _{R \rightarrow+\infty} \int_{0}^{R} e^{-i t u^{3}} d u$, for real positive $t$, in terms of $\Gamma\left(\frac{4}{3}\right)=\int_{0}^{+\infty} e^{-u^{3}} d u .$

For $\alpha>0, x>0$, let

$Q_{\alpha}(x)=\frac{1}{\pi} \int_{0}^{\pi} \cos (x \sin \theta-\alpha \theta) d \theta$

Find the leading terms in the asymptotic expansions as $x \rightarrow+\infty$ of (i) $Q_{\alpha}(x)$ with $\alpha$ fixed, and (ii) of $Q_{x}(x)$.

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• # Paper 3, Section II, B

(a) Find the curves of steepest descent emanating from $t=0$ for the integral

$J_{x}(x)=\frac{1}{2 \pi i} \int_{C} e^{x(\sinh t-t)} d t$

for $x>0$ and determine the angles at which they meet at $t=0$, and their asymptotes at infinity.

(b) An integral representation for the Bessel function $K_{\nu}(x)$ for real $x>0$ is

$K_{\nu}(x)=\frac{1}{2} \int_{-\infty}^{+\infty} e^{\nu h(t)} d t \quad, \quad h(t)=t-\left(\frac{x}{\nu}\right) \cosh t$

Show that, as $\nu \rightarrow+\infty$, with $x$ fixed,

$K_{\nu}(x) \sim\left(\frac{\pi}{2 \nu}\right)^{\frac{1}{2}}\left(\frac{2 \nu}{e x}\right)^{\nu}$

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• # Paper 4, Section II, B

Show that

$I_{0}(x)=\frac{1}{\pi} \int_{0}^{\pi} e^{x \cos \theta} d \theta$

is a solution to the equation

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

and obtain the first two terms in the asymptotic expansion of $I_{0}(x)$ as $x \rightarrow+\infty$.

For $x>0$, define a new dependent variable $w(x)=x^{\frac{1}{2}} y(x)$, and show that if $y$ solves the preceding equation then

$w^{\prime \prime}+\left(\frac{1}{4 x^{2}}-1\right) w=0 .$

Obtain the Liouville-Green approximate solutions to this equation for large positive $x$, and compare with your asymptotic expansion for $I_{0}(x)$ at the leading order.

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• # Paper 2, Section II, E

Consider the function

$f_{\nu}(x) \equiv \frac{1}{2 \pi} \int_{C} \exp [-i x \sin z+i \nu z] d z$

where the contour $C$ is the boundary of the half-strip $\{z:-\pi<\operatorname{Re} z<\pi$ and $\operatorname{Im} z>0\}$, taken anti-clockwise.

Use integration by parts and the method of stationary phase to:

(i) Obtain the leading term for $f_{\nu}(x)$ coming from the vertical lines $z=\pm \pi+i y(0<$ $y<+\infty)$ for large $x>0$.

(ii) Show that the leading term in the asymptotic expansion of the function $f_{\nu}(x)$ for large positive $x$ is

$\sqrt{\frac{2}{\pi x}} \cos \left(x-\frac{1}{2} \nu \pi-\frac{\pi}{4}\right)$

and obtain an estimate for the remainder as $O\left(x^{-a}\right)$ for some $a$ to be determined.

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• # Paper 3, Section II, E

Consider the integral representation for the modified Bessel function

$I_{0}(x)=\frac{1}{2 \pi i} \oint_{C} t^{-1} \exp \left[\frac{i x}{2}\left(t-\frac{1}{t}\right)\right] d t$

where $C$ is a simple closed contour containing the origin, taken anti-clockwise.

Use the method of steepest descent to determine the full asymptotic expansion of $I_{0}(x)$ for large real positive $x .$

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• # Paper 4, Section II, E

Consider solutions to the equation

$\frac{d^{2} y}{d x^{2}}=\left(\frac{1}{4}+\frac{\mu^{2}-\frac{1}{4}}{x^{2}}\right) y$

of the form

$y(x)=\exp \left[S_{0}(x)+S_{1}(x)+S_{2}(x)+\ldots\right]$

with the assumption that, for large positive $x$, the function $S_{j}(x)$ is small compared to $S_{j-1}(x)$ for all $j=1,2 \ldots$

Obtain equations for the $S_{j}(x), j=0,1,2 \ldots$, which are formally equivalent to ( $)$. Solve explicitly for $S_{0}$ and $S_{1}$. Show that it is consistent to assume that $S_{j}(x)=c_{j} x^{-(j-1)}$ for some constants $c_{j}$. Give a recursion relation for the $c_{j}$.

Deduce that there exist two linearly independent solutions to $(\star)$ with asymptotic expansions as $x \rightarrow+\infty$ of the form

$y_{\pm}(x) \sim e^{\pm x / 2}\left(1+\sum_{j=1}^{\infty} A_{j}^{\pm} x^{-j}\right)$

Determine a recursion relation for the $A_{j}^{\pm}$. Compute $A_{1}^{\pm}$and $A_{2}^{\pm}$.

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• # Paper 2, Section II, C

What is meant by the asymptotic relation

$f(z) \sim g(z) \quad \text { as } \quad z \rightarrow z_{0}, \operatorname{Arg}\left(z-z_{0}\right) \in\left(\theta_{0}, \theta_{1}\right) ?$

Show that

$\sinh \left(z^{-1}\right) \sim \frac{1}{2} \exp \left(z^{-1}\right) \quad \text { as } \quad z \rightarrow 0, \operatorname{Arg} z \in(-\pi / 2, \pi / 2),$

and find the corresponding result in the sector $\operatorname{Arg} z \in(\pi / 2,3 \pi / 2)$.

What is meant by the asymptotic expansion

$f(z) \sim \sum_{j=0}^{\infty} c_{j}\left(z-z_{0}\right)^{j} \quad \text { as } \quad z \rightarrow z_{0}, \operatorname{Arg}\left(z-z_{0}\right) \in\left(\theta_{0}, \theta_{1}\right) ?$

Show that the coefficients $\left\{c_{j}\right\}_{j=0}^{\infty}$ are determined uniquely by $f$. Show that if $f$ is analytic at $z_{0}$, then its Taylor series is an asymptotic expansion for $f$ as $z \rightarrow z_{0}\left(\right.$ for any $\left.\operatorname{Arg}\left(z-z_{0}\right)\right)$.

Show that

$u(x, t)=\int_{-\infty}^{\infty} \exp \left(-i k^{2} t+i k x\right) f(k) d k$

defines a solution of the equation $i \partial_{t} u+\partial_{x}^{2} u=0$ for any smooth and rapidly decreasing function $f$. Use the method of stationary phase to calculate the leading-order behaviour of $u(\lambda t, t)$ as $t \rightarrow+\infty$, for fixed $\lambda$.

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• # Paper 3, Section II, C

Consider the integral

$I(x)=\int_{0}^{1} \frac{1}{\sqrt{t(1-t)}} \exp [i x f(t)] d t$

for real $x>0$, where $f(t)=t^{2}+t$. Find and sketch, in the complex $t$-plane, the paths of steepest descent through the endpoints $t=0$ and $t=1$ and through any saddle point(s). Obtain the leading order term in the asymptotic expansion of $I(x)$ for large positive $x$. What is the order of the next term in the expansion? Justify your answer.

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• # Paper 4, Section II, C

Consider the equation

$\epsilon^{2} \frac{d^{2} y}{d x^{2}}=Q(x) y$

where $\epsilon>0$ is a small parameter and $Q(x)$ is smooth. Search for solutions of the form

$y(x)=\exp \left[\frac{1}{\epsilon}\left(S_{0}(x)+\epsilon S_{1}(x)+\epsilon^{2} S_{2}(x)+\cdots\right)\right],$

and, by equating powers of $\epsilon$, obtain a collection of equations for the $\left\{S_{j}(x)\right\}_{j=0}^{\infty}$ which is formally equivalent to (1). By solving explicitly for $S_{0}$ and $S_{1}$ derive the Liouville- Green approximate solutions $y^{L G}(x)$ to (1).

For the case $Q(x)=-V(x)$, where $V(x) \geqslant V_{0}$ and $V_{0}$ is a positive constant, consider the eigenvalue problem

$\frac{d^{2} y}{d x^{2}}+E V(x) y=0, \quad y(0)=y(\pi)=0$

Show that any eigenvalue $E$ is necessarily positive. Solve the eigenvalue problem exactly when $V(x)=V_{0}$.

Obtain Liouville-Green approximate eigenfunctions $y_{n}^{L G}(x)$ for (2) with $E \gg 1$, and give the corresponding Liouville Green approximation to the eigenvalues $E_{n}^{L G}$. Compare your results to the exact eigenvalues and eigenfunctions in the case $V(x)=V_{0}$, and comment on this.

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• # Paper 1, Section II, C

(a) State the integral expression for the gamma function $\Gamma(z)$, for $\operatorname{Re}(z)>0$, and express the integral

$\int_{0}^{\infty} t^{\gamma-1} e^{i t} d t, \quad 0<\gamma<1$

in terms of $\Gamma(\gamma)$. Explain why the constraints on $\gamma$ are necessary.

(b) Show that

$\int_{0}^{\infty} \frac{e^{-k t^{2}}}{\left(t^{2}+t\right)^{\frac{1}{4}}} d t \sim \sum_{m=0}^{\infty} \frac{a_{m}}{k^{\alpha+\beta m}}, \quad k \rightarrow \infty$

for some constants $a_{m}, \alpha$ and $\beta$. Determine the constants $\alpha$ and $\beta$, and express $a_{m}$ in terms of the gamma function.

State without proof the basic result needed for the rigorous justification of the above asymptotic formula.

[You may use the identity:

$\left.(1+z)^{\alpha}=\sum_{m=0}^{\infty} c_{m} z^{m}, \quad c_{m}=\frac{\Gamma(\alpha+1)}{m ! \Gamma(\alpha+1-m)}, \quad|z|<1 .\right]$

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• # Paper 3, Section II, $27 \mathrm{C}$

Show that

$\int_{0}^{1} e^{i k t^{3}} d t=I_{1}-I_{2}, \quad k>0$

where $I_{1}$ is an integral from 0 to $\infty$ along the line $\arg (z)=\frac{\pi}{6}$ and $I_{2}$ is an integral from 1 to $\infty$ along a steepest-descent contour $C$ which you should determine.

By employing in the integrals $I_{1}$ and $I_{2}$ the changes of variables $u=-i z^{3}$ and $u=-i\left(z^{3}-1\right)$, respectively, compute the first two terms of the large $k$ asymptotic expansion of the integral above.

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• # Paper 4, Section II, C

Consider the ordinary differential equation

$\frac{d^{2} u}{d z^{2}}+f(z) \frac{d u}{d z}+g(z) u=0$

where

$f(z) \sim \sum_{m=0}^{\infty} \frac{f_{m}}{z^{m}}, \quad g(z) \sim \sum_{m=0}^{\infty} \frac{g_{m}}{z^{m}}, \quad z \rightarrow \infty$

and $f_{m}, g_{m}$ are constants. Look for solutions in the asymptotic form

$u(z)=e^{\lambda z} z^{\mu}\left[1+\frac{a}{z}+\frac{b}{z^{2}}+O\left(\frac{1}{z^{3}}\right)\right], \quad z \rightarrow \infty$

and determine $\lambda$ in terms of $\left(f_{0}, g_{0}\right)$, as well as $\mu$ in terms of $\left(\lambda, f_{0}, f_{1}, g_{1}\right)$.

Deduce that the Bessel equation

$\frac{d^{2} u}{d z^{2}}+\frac{1}{z} \frac{d u}{d z}+\left(1-\frac{\nu^{2}}{z^{2}}\right) u=0$

where $\nu$ is a complex constant, has two solutions of the form

\begin{aligned} &u^{(1)}(z)=\frac{e^{i z}}{z^{1 / 2}}\left[1+\frac{a^{(1)}}{z}+O\left(\frac{1}{z^{2}}\right)\right], \quad z \rightarrow \infty \\ &u^{(2)}(z)=\frac{e^{-i z}}{z^{1 / 2}}\left[1+\frac{a^{(2)}}{z}+O\left(\frac{1}{z^{2}}\right)\right], \quad z \rightarrow \infty \end{aligned}

and determine $a^{(1)}$ and $a^{(2)}$ in terms of $\nu .$

Can the above asymptotic expansions be valid for all $\arg (z)$, or are they valid only in certain domains of the complex $z$-plane? Justify your answer briefly.

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• # Paper 1, Section II, C

(a) Consider the integral

$I(k)=\int_{0}^{\infty} f(t) e^{-k t} d t, \quad k>0$

Suppose that $f(t)$ possesses an asymptotic expansion for $t \rightarrow 0^{+}$of the form

$f(t) \sim t^{\alpha} \sum_{n=0}^{\infty} a_{n} t^{\beta n}, \quad \alpha>-1, \quad \beta>0$

where $a_{n}$ are constants. Derive an asymptotic expansion for $I(k)$ as $k \rightarrow \infty$ in the form

$I(k) \sim \sum_{n=0}^{\infty} \frac{A_{n}}{k^{\gamma+\beta n}}$

giving expressions for $A_{n}$ and $\gamma$ in terms of $\alpha, \beta, n$ and the gamma function. Hence establish the asymptotic approximation as $k \rightarrow \infty$

$I_{1}(k)=\int_{0}^{1} e^{k t} t^{-a}\left(1-t^{2}\right)^{-b} d t \sim 2^{-b} \Gamma(1-b) e^{k} k^{b-1}\left(1+\frac{(a+b / 2)(1-b)}{k}\right)$

where $a<1, b<1$.

(b) Using Laplace's method, or otherwise, find the leading-order asymptotic approximation as $k \rightarrow \infty$ for

$I_{2}(k)=\int_{0}^{\infty} e^{-\left(2 k^{2} / t+t^{2} / k\right)} d t$

[You may assume that $\Gamma(z)=\int_{0}^{\infty} t^{z-1} e^{-t} d t$ for $\operatorname{Re} z>0$,

$\text { and that } \left.\int_{-\infty}^{\infty} e^{-q t^{2}} d t=\sqrt{\pi / q} \text { for } q>0 .\right]$

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• # Paper 3, Section II, C

(a) Find the Stokes ray for the function $f(z)$ as $z \rightarrow 0$ with $0<\arg z<\pi$, where

$f(z)=\sinh \left(z^{-1}\right)$

(b) Describe how the leading-order asymptotic behaviour as $x \rightarrow \infty$ of

$I(x)=\int_{a}^{b} f(t) e^{i x g(t)} d t$

may be found by the method of stationary phase, where $f$ and $g$ are real functions and the integral is taken along the real line. You should consider the cases for which:

(i) $g^{\prime}(t)$ is non-zero in $[a, b)$ and has a simple zero at $t=b$.

(ii) $g^{\prime}(t)$ is non-zero apart from having one simple zero at $t=t_{0}$, where $a.

(iii) $g^{\prime}(t)$ has more than one simple zero in $(a, b)$ with $g^{\prime}(a) \neq 0$ and $g^{\prime}(b) \neq 0$.

Use the method of stationary phase to find the leading-order asymptotic form as $x \rightarrow \infty$ of

$J(x)=\int_{0}^{1} \cos \left(x\left(t^{4}-t^{2}\right)\right) d t$

[You may assume that $\left.\int_{-\infty}^{\infty} e^{i u^{2}} d u=\sqrt{\pi} e^{i \pi / 4} .\right]$

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• # Paper 4, Section II, C

Derive the leading-order Liouville Green (or WKBJ) solution for $\epsilon \ll 1$ to the ordinary differential equation

$\epsilon^{2} \frac{d^{2} f}{d y^{2}}+\Phi(y) f=0$

where $\Phi(y)>0$.

The function $f(y ; \epsilon)$ satisfies the ordinary differential equation

$\epsilon^{2} \frac{d^{2} f}{d y^{2}}+\left(1+\frac{1}{y}-\frac{2 \epsilon^{2}}{y^{2}}\right) f=0$

subject to the boundary condition $f^{\prime \prime}(0)=2$. Show that the Liouville-Green solution of (1) for $\epsilon \ll 1$ takes the asymptotic forms

where $\alpha_{1}, \alpha_{2}, B$ and $\theta_{2}$ are constants.

$\left[\right.$ Hint: You may assume that $\left.\int_{0}^{y} \sqrt{1+u^{-1}} d u=\sqrt{y(1+y)}+\sinh ^{-1} \sqrt{y} \cdot\right]$

Explain, showing the relevant change of variables, why the leading-order asymptotic behaviour for $0 \leqslant y \ll 1$ can be obtained from the reduced equation

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

The unique solution to $(2)$ with $f^{\prime \prime}(0)=2$ is $f=x^{1 / 2} J_{3}\left(2 x^{1 / 2}\right)$, where the Bessel function $J_{3}(z)$ is known to have the asymptotic form

$J_{3}(z) \sim\left(\frac{2}{\pi z}\right)^{1 / 2} \cos \left(z-\frac{7 \pi}{4}\right) \text { as } z \rightarrow \infty .$

Hence find the values of $\alpha_{1}$ and $\alpha_{2}$.

\begin{aligned} & f \sim \alpha_{1} y^{\frac{1}{4}} \exp (2 i \sqrt{y} / \epsilon)+\alpha_{2} y^{\frac{1}{4}} \exp (-2 i \sqrt{y} / \epsilon) \quad \text { for } \quad \epsilon^{2} \ll y \ll 1 \\ & \text { and } \quad f \sim B \cos \left[\theta_{2}+(y+\log \sqrt{y}) / \epsilon\right] \quad \text { for } \quad y \gg 1, \end{aligned}

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• # Paper 1, Section II, B

Suppose $\alpha>0$. Define what it means to say that

$F(x) \sim \frac{1}{\alpha x} \sum_{n=0}^{\infty} n !\left(\frac{-1}{\alpha x}\right)^{n}$

is an asymptotic expansion of $F(x)$ as $x \rightarrow \infty$. Show that $F(x)$ has no other asymptotic expansion in inverse powers of $x$ as $x \rightarrow \infty$.

To estimate the value of $F(x)$ for large $x$, one may use an optimal truncation of the asymptotic expansion. Explain what is meant by this, and show that the error is an exponentially small quantity in $x$.

Derive an integral respresentation for a function $F(x)$ with the above asymptotic expansion.

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• # Paper 3, Section II, B

Let

$I(x)=\int_{0}^{\pi} f(t) e^{i x \psi(t)} d t$

where $f(t)$ and $\psi(t)$ are smooth, and $\psi^{\prime}(t) \neq 0$ for $t>0 ;$ also $f(0) \neq 0$, $\psi(0)=a$, $\psi^{\prime}(0)=\psi^{\prime \prime}(0)=0$ and $\psi^{\prime \prime \prime}(0)=6 b>0$. Show that, as $x \rightarrow+\infty$,

$I(x) \sim f(0) e^{i(x a+\pi / 6)}\left(\frac{1}{27 b x}\right)^{1 / 3} \Gamma(1 / 3) .$

Consider the Bessel function

$J_{n}(x)=\frac{1}{\pi} \int_{0}^{\pi} \cos (n t-x \sin t) d t$

Show that, as $n \rightarrow+\infty$,

$J_{n}(n) \sim \frac{\Gamma(1 / 3)}{\pi} \frac{1}{(48)^{1 / 6}} \frac{1}{n^{1 / 3}}$

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• # Paper 4, Section II, B

Show that the equation

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

has an irregular singular point at infinity. Using the Liouville-Green method, show that one solution has the asymptotic expansion

$y(x) \sim \frac{1}{x} e^{x}\left(1+\frac{1}{2 x}+\ldots\right)$

as $x \rightarrow \infty$

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• # Paper 1, Section II, B

What precisely is meant by the statement that

$f(x) \sim \sum_{n=0}^{\infty} d_{n} x^{n}$

as $x \rightarrow 0 ?$

Consider the Stieltjes integral

$I(x)=\int_{1}^{\infty} \frac{\rho(t)}{1+x t} d t$

where $\rho(t)$ is bounded and decays rapidly as $t \rightarrow \infty$, and $x>0$. Find an asymptotic series for $I(x)$ of the form $(*)$, as $x \rightarrow 0$, and prove that it has the asymptotic property.

In the case that $\rho(t)=e^{-t}$, show that the coefficients $d_{n}$ satisfy the recurrence relation

$d_{n}=(-1)^{n} \frac{1}{e}-n d_{n-1} \quad(n \geqslant 1)$

and that $d_{0}=\frac{1}{e}$. Hence find the first three terms in the asymptotic series.

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• # Paper 3, Section II, B

Find the two leading terms in the asymptotic expansion of the Laplace integral

$I(x)=\int_{0}^{1} f(t) e^{x t^{4}} d t$

as $x \rightarrow \infty$, where $f(t)$ is smooth and positive on $[0,1]$.

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• # Paper 4, Section II, B

The stationary Schrödinger equation in one dimension has the form

$\epsilon^{2} \frac{d^{2} \psi}{d x^{2}}=-(E-V(x)) \psi$

where $\epsilon$ can be assumed to be small. Using the Liouville-Green method, show that two approximate solutions in a region where $V(x) are

$\psi(x) \sim \frac{1}{(E-V(x))^{1 / 4}} \exp \left\{\pm \frac{i}{\epsilon} \int_{c}^{x}\left(E-V\left(x^{\prime}\right)\right)^{1 / 2} d x^{\prime}\right\}$

where $c$ is suitably chosen.

Without deriving connection formulae in detail, describe how one obtains the condition

$\frac{1}{\epsilon} \int_{a}^{b}\left(E-V\left(x^{\prime}\right)\right)^{1 / 2} d x^{\prime}=\left(n+\frac{1}{2}\right) \pi$

for the approximate energies $E$ of bound states in a smooth potential well. State the appropriate values of $a, b$ and $n$.

Estimate the range of $n$ for which $(*)$ gives a good approximation to the true bound state energies in the cases

(i) $V(x)=|x|$,

(ii) $V(x)=x^{2}+\lambda x^{6}$ with $\lambda$ small and positive,

(iii) $V(x)=x^{2}-\lambda x^{6}$ with $\lambda$ small and positive.

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• # Paper 1, Section II, A

A function $f(n)$, defined for positive integer $n$, has an asymptotic expansion for large $n$ of the following form:

$f(n) \sim \sum_{k=0}^{\infty} a_{k} \frac{1}{n^{2 k}}, \quad n \rightarrow \infty$

What precisely does this mean?

Show that the integral

$I(n)=\int_{0}^{2 \pi} \frac{\cos n t}{1+t^{2}} d t$

has an asymptotic expansion of the form $(*)$. [The Riemann-Lebesgue lemma may be used without proof.] Evaluate the coefficients $a_{0}, a_{1}$ and $a_{2}$.

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• # Paper 3, Section II, A

Let

$I_{0}=\int_{C_{0}} e^{x \phi(z)} d z \text {, }$

where $\phi(z)$ is a complex analytic function and $C_{0}$ is a steepest descent contour from a simple saddle point of $\phi(z)$ at $z_{0}$. Establish the following leading asymptotic approximation, for large real $x$ :

$I_{0} \sim i \sqrt{\frac{\pi}{2 \phi^{\prime \prime}\left(z_{0}\right) x}} e^{x \phi\left(z_{0}\right)}$

Let $n$ be a positive integer, and let

$I=\int_{C} e^{-t^{2}-2 n \ln t} d t$

where $C$ is a contour in the upper half $t$-plane connecting $t=-\infty$ to $t=\infty$, and $\ln t$ is real on the positive $t$-axis with a branch cut along the negative $t$-axis. Using the method of steepest descent, find the leading asymptotic approximation to $I$ for large $n$.

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• # Paper 4, Section II, A

Determine the range of the integer $n$ for which the equation

$\frac{d^{2} y}{d z^{2}}=z^{n} y$

has an essential singularity at $z=\infty$.

Use the Liouville-Green method to find the leading asymptotic approximation to two independent solutions of

$\frac{d^{2} y}{d z^{2}}=z^{3} y$

for large $|z|$. Find the Stokes lines for these approximate solutions. For what range of $\arg z$ is the approximate solution which decays exponentially along the positive $z$-axis an asymptotic approximation to an exact solution with this exponential decay?

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• # Paper 1, Section II, C

For $\lambda>0$ let

$I(\lambda)=\int_{0}^{b} f(x) \mathrm{e}^{-\lambda x} d x, \quad \text { with } \quad 0

Assume that the function $f(x)$ is continuous on $0, and that

$f(x) \sim x^{\alpha} \sum_{n=0}^{\infty} a_{n} x^{n \beta}$

as $x \rightarrow 0_{+}$, where $\alpha>-1$ and $\beta>0$.

(a) Explain briefly why in this case straightforward partial integrations in general cannot be applied for determining the asymptotic behaviour of $I(\lambda)$ as $\lambda \rightarrow \infty$.

(b) Derive with proof an asymptotic expansion for $I(\lambda)$ as $\lambda \rightarrow \infty$.

(c) For the function

$B(s, t)=\int_{0}^{1} u^{s-1}(1-u)^{t-1} d u, \quad s, t>0$

obtain, using the substitution $u=e^{-x}$, the first two terms in an asymptotic expansion as $s \rightarrow \infty$. What happens as $t \rightarrow \infty$ ?

[Hint: The following formula may be useful

$\Gamma(y)=\int_{0}^{\infty} x^{y-1} \mathrm{e}^{-x} d t, \quad \text { for } \quad x>0$

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• # Paper 3, Section II, C

Consider the ordinary differential equation

$y^{\prime \prime}=(|x|-E) y$

subject to the boundary conditions $y(\pm \infty)=0$. Write down the general form of the Liouville-Green solutions for this problem for $E>0$ and show that asymptotically the eigenvalues $E_{n}, n \in \mathbb{N}$ and $E_{n}, behave as $E_{n}=\mathrm{O}\left(n^{2 / 3}\right)$ for large $n$.

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• # Paper 4, Section II, C

(a) Consider for $\lambda>0$ the Laplace type integral

$I(\lambda)=\int_{a}^{b} f(t) \mathrm{e}^{-\lambda \phi(t)} d t$

for some finite $a, b \in \mathbb{R}$ and smooth, real-valued functions $f(t), \phi(t)$. Assume that the function $\phi(t)$ has a single minimum at $t=c$ with $a. Give an account of Laplace's method for finding the leading order asymptotic behaviour of $I(\lambda)$ as $\lambda \rightarrow \infty$ and briefly discuss the difference if instead $c=a$ or $c=b$, i.e. when the minimum is attained at the boundary.

(b) Determine the leading order asymptotic behaviour of

$I(\lambda)=\int_{-2}^{1} \cos t \mathrm{e}^{-\lambda t^{2}} d t$

as $\lambda \rightarrow \infty$

(c) Determine also the leading order asymptotic behaviour when cos $t$ is replaced by $\sin t$ in $(*)$.

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• # Paper 1, Section II, A

Consider the integral

$I(\lambda)=\int_{0}^{A} \mathrm{e}^{-\lambda t} f(t) d t, \quad A>0$

in the limit $\lambda \rightarrow \infty$, given that $f(t)$ has the asymptotic expansion

$f(t) \sim \sum_{n=0}^{\infty} a_{n} t^{n \beta}$

as $t \rightarrow 0_{+}$, where $\beta>0$. State Watson's lemma.

Now consider the integral

$J(\lambda)=\int_{a}^{b} \mathrm{e}^{\lambda \phi(t)} F(t) d t$

where $\lambda \gg 1$ and the real function $\phi(t)$ has a unique maximum in the interval $[a, b]$ at $c$, with $a, such that

$\phi^{\prime}(c)=0, \phi^{\prime \prime}(c)<0$

By making a monotonic change of variable from $t$ to a suitable variable $\zeta$ (Laplace's method), or otherwise, deduce the existence of an asymptotic expansion for $J(\lambda)$ as $\lambda \rightarrow \infty$. Derive the leading term

$J(\lambda) \sim \mathrm{e}^{\lambda \phi(c)} F(c)\left(\frac{2 \pi}{\lambda\left|\phi^{\prime \prime}(c)\right|}\right)^{\frac{1}{2}}$

The gamma function is defined for $x>0$ by

$\Gamma(x+1)=\int_{0}^{\infty} \exp (x \log t-t) d t$

By means of the substitution $t=x s$, or otherwise, deduce Stirling's formula

$\Gamma(x+1) \sim x^{\left(x+\frac{1}{2}\right)} \mathrm{e}^{-x} \sqrt{2 \pi}\left(1+\frac{1}{12 x}+\cdots\right)$

as $x \rightarrow \infty$

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• # Paper 3, Section II, A

Consider the contour-integral representation

$J_{0}(x)=\operatorname{Re} \frac{1}{i \pi} \int_{C} e^{i x \cosh t} d t$

of the Bessel function $J_{0}$ for real $x$, where $C$ is any contour from $-\infty-\frac{i \pi}{2}$ to $+\infty+\frac{i \pi}{2}$.

Writing $t=u+i v$, give in terms of the real quantities $u, v$ the equation of the steepest-descent contour from $-\infty-\frac{i \pi}{2}$ to $+\infty+\frac{i \pi}{2}$ which passes through $t=0$.

Deduce the leading term in the asymptotic expansion of $J_{0}(x)$, valid as $x \rightarrow \infty$

$J_{0}(x) \sim \sqrt{\frac{2}{\pi x}} \cos \left(x-\frac{\pi}{4}\right)$

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• # Paper 4, Section II, A

The differential equation

$f^{\prime \prime}=Q(x) f$

has a singular point at $x=\infty$. Assuming that $Q(x)>0$, write down the Liouville Green lowest approximations $f_{\pm}(x)$ for $x \rightarrow \infty$, with $f_{-}(x) \rightarrow 0$.

The Airy function $\operatorname{Ai}(x)$ satisfies $(*)$ with

$Q(x)=x$

and $\operatorname{Ai}(x) \rightarrow 0$ as $x \rightarrow \infty$. Writing

$\operatorname{Ai}(x)=w(x) f_{-}(x)$

show that $w(x)$ obeys

$x^{2} w^{\prime \prime}-\left(2 x^{5 / 2}+\frac{1}{2} x\right) w^{\prime}+\frac{5}{16} w=0$

Derive the expansion

$w \sim c\left(1-\frac{5}{48} x^{-3 / 2}\right) \quad \text { as } \quad x \rightarrow \infty$

where $c$ is a constant.

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• # 1.II $. 30 \mathrm{~A}$

Obtain an expression for the $n$th term of an asymptotic expansion, valid as $\lambda \rightarrow \infty$, for the integral

$I(\lambda)=\int_{0}^{1} t^{2 \alpha} e^{-\lambda\left(t^{2}+t^{3}\right)} d t \quad(\alpha>-1 / 2) .$

Estimate the value of $n$ for the term of least magnitude.

Obtain the first two terms of an asymptotic expansion, valid as $\lambda \rightarrow \infty$, for the integral

$J(\lambda)=\int_{0}^{1} t^{2 \alpha} e^{-\lambda\left(t^{2}-t^{3}\right)} d t \quad(-1 / 2<\alpha<0)$

[Hint:

$\left.\Gamma(z)=\int_{0}^{\infty} t^{z-1} e^{-t} d t .\right]$

[Stirling's formula may be quoted.]

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• # 3.II $. 30 \mathrm{~A}$

Describe how the leading-order approximation may be found by the method of stationary phase of

$I(\lambda)=\int_{a}^{b} f(t) \exp (i \lambda g(t)) d t$

for $\lambda \gg 1$, where $\lambda, f$ and $g$ are real. You should consider the cases for which: (a) $g^{\prime}(t)$ has one simple zero at $t=t_{0}$, where $a; (b) $g^{\prime}(t)$ has more than one simple zero in the region $a; and (c) $g^{\prime}(t)$ has only a simple zero at $t=b$