Asymptotic Methods

2021

• Paper 2, Section II, 32A

  Asymptotic Methods | Part II, 2021

  (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

  Asymptotic Methods | Part II, 2021

  (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

  Asymptotic Methods | Part II, 2021

  (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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2020

• Paper 2, Section II, D

  Asymptotic Methods | Part II, 2020

  (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

  Asymptotic Methods | Part II, 2020

  (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

  Asymptotic Methods | Part II, 2020

  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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2019

• Paper 2, Section II, A

  Asymptotic Methods | Part II, 2019

  (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

  Asymptotic Methods | Part II, 2019

  (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

  Asymptotic Methods | Part II, 2019

  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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2018

• Paper 2, Section II, B

  Asymptotic Methods | Part II, 2018

  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

  Asymptotic Methods | Part II, 2018

  (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

  Asymptotic Methods | Part II, 2018

  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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2017

• Paper 2, Section II, E

  Asymptotic Methods | Part II, 2017

  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

  Asymptotic Methods | Part II, 2017

  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

  Asymptotic Methods | Part II, 2017

  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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2016

• Paper 2, Section II, C

  Asymptotic Methods | Part II, 2016

  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

  Asymptotic Methods | Part II, 2016

  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

  Asymptotic Methods | Part II, 2016

  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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2015

• Paper 1, Section II, C

  Asymptotic Methods | Part II, 2015

  (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, $27 \mathrm{C}$

  Asymptotic Methods | Part II, 2015

  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

  Asymptotic Methods | Part II, 2015

  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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2014

• Paper 1, Section II, C

  Asymptotic Methods | Part II, 2014

  (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

  Asymptotic Methods | Part II, 2014

  (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 $a<t_{0}<b$.

  (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

  Asymptotic Methods | Part II, 2014

  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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2013

• Paper 1, Section II, B

  Asymptotic Methods | Part II, 2013

  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

  Asymptotic Methods | Part II, 2013

  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

  Asymptotic Methods | Part II, 2013

  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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2012

• Paper 1, Section II, B

  Asymptotic Methods | Part II, 2012

  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

  Asymptotic Methods | Part II, 2012

  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

  Asymptotic Methods | Part II, 2012

  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 $V(x)<E$ 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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2011

• Paper 1, Section II, A

  Asymptotic Methods | Part II, 2011

  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

  Asymptotic Methods | Part II, 2011

  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

  Asymptotic Methods | Part II, 2011

  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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2010

• Paper 1, Section II, C

  Asymptotic Methods | Part II, 2010

  For $\lambda>0$ let

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

  Assume that the function $f(x)$ is continuous on $0<x \leqslant b$, 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

  Asymptotic Methods | Part II, 2010

  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 $E_{n}<E_{n+1}$, behave as $E_{n}=\mathrm{O}\left(n^{2 / 3}\right)$ for large $n$.

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• Paper 4, Section II, C

  Asymptotic Methods | Part II, 2010

  (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 $a<c<b$. 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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2009

• Paper 1, Section II, A

  Asymptotic Methods | Part II, 2009

  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 $a<c<b$, 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

  Asymptotic Methods | Part II, 2009

  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

  Asymptotic Methods | Part II, 2009

  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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2008

• 1.II $. 30 \mathrm{~A}$

  Asymptotic Methods | Part II, 2008

  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 $. 30 \mathrm{~A}$

  Asymptotic Methods | Part II, 2008

  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 $a<t_{0}<b$; (b) $g^{\prime}(t)$ has more than one simple zero in the region $a<t<b$; and (c) $g^{\prime}(t)$ has only a simple zero at $t=b$.

  What is the order of magnitude of $I(\lambda)$ if $g^{\prime}(t)$ is non zero for $a \leqslant t \leqslant b$ ?

  Use the method of stationary phase to find the leading-order approximation for $\lambda \gg 1$ to

  $$J(\lambda)=\int_{0}^{1} \sin \left(\lambda\left(t^{3}-t\right)\right) d t$$

  [Hint:

  $$\left.\int_{-\infty}^{\infty} \exp \left(i u^{2}\right) d u=\sqrt{\pi} e^{i \pi / 4} .\right]$$

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• 4.II $. 31 \mathrm{~A}$

  Asymptotic Methods | Part II, 2008

  The Bessel equation of order $n$ is

  $$z^{2} y^{\prime \prime}+z y^{\prime}+\left(z^{2}-n^{2}\right) y=0 .$$

  Here, $n$ is taken to be an integer, with $n \geqslant 0$. The transformation $w(z)=z^{\frac{1}{2}} y(z)$ converts (1) to the form

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

  where

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

  Find two linearly independent solutions of the form

  $$w=e^{s z} \sum_{k=0}^{\infty} c_{k} z^{\rho-k}$$

  where $c_{k}$ are constants, with $c_{0} \neq 0$, and $s$ and $\rho$ are to be determined. Find recurrence relationships for the $c_{k}$.

  Find the first two terms of two linearly independent Liouville-Green solutions of (2) for $w(z)$ valid in a neighbourhood of $z=\infty$. Relate these solutions to those of the form (3).

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2007

• 1.II.30B

  Asymptotic Methods | Part II, 2007

  State Watson's lemma, describing the asymptotic behaviour of the integral

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

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

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

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

  Give an account of Laplace's method for finding asymptotic expansions of integrals of the form

  $$J(z)=\int_{-\infty}^{\infty} e^{-z p(t)} q(t) d t$$

  for large real $z$, where $p(t)$ is real for real $t$.

  Deduce the following asymptotic expansion of the contour integral

  $$\int_{-\infty-i \pi}^{\infty+i \pi} \exp (z \cosh t) d t=2^{1 / 2} i e^{z} \Gamma\left(\frac{1}{2}\right)\left[z^{-1 / 2}+\frac{1}{8} z^{-3 / 2}+O\left(z^{-5 / 2}\right)\right]$$

  as $z \rightarrow \infty$.

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• 3.II.30B

  Asymptotic Methods | Part II, 2007

  Explain the method of stationary phase for determining the behaviour of the integral

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

  for large $x$. Here, the function $f(u)$ is real and differentiable, and $a, b$ and $x$ are all real.

  Apply this method to show that the first term in the asymptotic behaviour of the function

  $$\Gamma(m+1)=\int_{0}^{\infty} d u u^{m} e^{-u}$$

  where $m=i n$ with $n>0$ and real, is

  $$\Gamma(i n+1) \sim \sqrt{2 \pi} e^{-i n} \exp \left[\left(i n+\frac{1}{2}\right)\left(\frac{i \pi}{2}+\log n\right)\right]$$

  as $n \rightarrow \infty$

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• 4.II.31B

  Asymptotic Methods | Part II, 2007

  Consider the time-independent Schrödinger equation

  $$\frac{d^{2} \psi}{d x^{2}}+\lambda^{2} q(x) \psi(x)=0$$

  where $\lambda \gg 1$ denotes $\hbar^{-1}$ and $q(x)$ denotes $2 m[E-V(x)]$. Suppose that

  and consider a bound state $\psi(x)$. Write down the possible Liouville-Green approximate solutions for $\psi(x)$ in each region, given that $\psi \rightarrow 0$ as $|x| \rightarrow \infty$.

  Assume that $q(x)$ may be approximated by $q^{\prime}(a)(x-a)$ near $x=a$, where $q^{\prime}(a)>0$, and by $q^{\prime}(b)(x-b)$ near $x=b$, where $q^{\prime}(b)<0$. The Airy function $\operatorname{Ai}(z)$ satisfies

  $$\frac{d^{2}(\mathrm{Ai})}{d z^{2}}-z(\mathrm{Ai})=0$$

  and has the asymptotic expansions

  $$\operatorname{Ai}(z) \sim \frac{1}{2} \pi^{-1 / 2} z^{-1 / 4} \exp \left(-\frac{2}{3} z^{3 / 2}\right) \quad \text { as } \quad z \rightarrow+\infty$$

  and

  $$\operatorname{Ai}(z) \sim \pi^{-1 / 2}|z|^{-1 / 4} \cos \left[\left(\frac{2}{3}|z|^{3 / 2}\right)-\frac{\pi}{4}\right] \quad \text { as } \quad z \rightarrow-\infty .$$

  Deduce that the energies $E$ of bound states are given approximately by the WKB condition:

  $$\lambda \int_{a}^{b} q^{1 / 2}(x) d x=\left(n+\frac{1}{2}\right) \pi \quad(n=0,1,2, \ldots)$$

  $$\begin{aligned} & q(x)>0 \quad \text { for } \quad a<x<b \\ & \text { and } q(x)<0 \text { for }-\infty<x<a \text { and } b<x<\infty \end{aligned}$$

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2006

• 1.II $. 30 \mathrm{~B}$

  Asymptotic Methods | Part II, 2006

  Two real functions $p(t), q(t)$ of a real variable $t$ are given on an interval $[0, b]$, where $b>0$. Suppose that $q(t)$ attains its minimum precisely at $t=0$, with $q^{\prime}(0)=0$, and that $q^{\prime \prime}(0)>0$. For a real argument $x$, define

  $$I(x)=\int_{0}^{b} p(t) e^{-x q(t)} d t$$

  Explain how to obtain the leading asymptotic behaviour of $I(x)$ as $x \rightarrow+\infty$ (Laplace's method).

  The modified Bessel function $I_{\nu}(x)$ is defined for $x>0$ by:

  $$I_{\nu}(x)=\frac{1}{\pi} \int_{0}^{\pi} e^{x \cos \theta} \cos (\nu \theta) d \theta-\frac{\sin (\nu \pi)}{\pi} \int_{0}^{\infty} e^{-x(\cosh t)-\nu t} d t .$$

  Show that

  $$I_{\nu}(x) \sim \frac{e^{x}}{\sqrt{2 \pi x}}$$

  as $x \rightarrow \infty$ with $\nu$ fixed.

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• 3.II $. 30$ B

  Asymptotic Methods | Part II, 2006

  The Airy function $\operatorname{Ai}(z)$ is defined by

  $$\operatorname{Ai}(z)=\frac{1}{2 \pi i} \int_{C} \exp \left(-\frac{1}{3} t^{3}+z t\right) d t$$

  where the contour $C$ begins at infinity along the ray $\arg (t)=4 \pi / 3$ and ends at infinity along the ray $\arg (t)=2 \pi / 3$. Restricting attention to the case where $z$ is real and positive, use the method of steepest descent to obtain the leading term in the asymptotic expansion for $\operatorname{Ai}(z)$ as $z \rightarrow \infty$ :

  $$\operatorname{Ai}(z) \sim \frac{\exp \left(-\frac{2}{3} z^{3 / 2}\right)}{2 \pi^{1 / 2} z^{1 / 4}}$$

  $\left[\right.$ Hint: put $\left.t=z^{1 / 2} \tau .\right]$

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• 4.II $. 31 \mathrm{~B}$

  Asymptotic Methods | Part II, 2006

  (a) Outline the Liouville-Green approximation to solutions $w(z)$ of the ordinary differential equation

  $$\frac{d^{2} w}{d z^{2}}=f(z) w$$

  in a neighbourhood of infinity, in the case that, near infinity, $f(z)$ has the convergent series expansion

  $$f(z)=\sum_{s=0}^{\infty} \frac{f_{s}}{z^{s}}$$

  with $f_{0} \neq 0$.

  In the case

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

  explain why you expect a basis of two asymptotic solutions $w_{1}(z), w_{2}(z)$, with

  $$\begin{aligned} &w_{1}(z) \sim z^{\frac{1}{2}} e^{z}\left(1+\frac{a_{1}}{z}+\frac{a_{2}}{z^{2}}+\cdots\right), \\ &w_{2}(z) \sim z^{-\frac{1}{2}} e^{-z}\left(1+\frac{b_{1}}{z}+\frac{b_{2}}{z^{2}}+\cdots\right), \end{aligned}$$

  as $z \rightarrow+\infty$, and show that $a_{1}=-\frac{9}{8}$.

  (b) Determine, at leading order in the large positive real parameter $\lambda$, an approximation to the solution $u(x)$ of the eigenvalue problem:

  $$u^{\prime \prime}(x)+\lambda^{2} g(x) u(x)=0 ; \quad u(0)=u(1)=0$$

  where $g(x)$ is greater than a positive constant for $x \in[0,1]$.

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2005

• 1.II $. 30 \mathrm{~A}$

  Asymptotic Methods | Part II, 2005

  Explain what is meant by an asymptotic power series about $x=a$ for a real function $f(x)$ of a real variable. Show that a convergent power series is also asymptotic.

  Show further that an asymptotic power series is unique (assuming that it exists).

  Let the function $f(t)$ be defined for $t \geqslant 0$ by

  $$f(t)=\frac{1}{\pi^{1 / 2}} \int_{0}^{\infty} \frac{e^{-x}}{x^{1 / 2}(1+2 x t)} d x$$

  By suitably expanding the denominator of the integrand, or otherwise, show that, as $t \rightarrow 0_{+}$,

  $$f(t) \sim \sum_{k=0}^{\infty}(-1)^{k} 1.3 \ldots(2 k-1) t^{k}$$

  and that the error, when the series is stopped after $n$ terms, does not exceed the absolute value of the $(n+1)$ th term of the series.

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• 3.II $. 30$

  Asymptotic Methods | Part II, 2005

  Explain, without proof, how to obtain an asymptotic expansion, as $x \rightarrow \infty$, of

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

  if it is known that $f(t)$ possesses an asymptotic power series as $t \rightarrow 0$.

  Indicate the modification required to obtain an asymptotic expansion, under suitable conditions, of

  $$\int_{-\infty}^{\infty} e^{-x t^{2}} f(t) d t$$

  Find an asymptotic expansion as $z \rightarrow \infty$ of the function defined by

  $$I(z)=\int_{-\infty}^{\infty} \frac{e^{-t^{2}}}{(z-t)} d t \quad(\operatorname{Im}(z)<0)$$

  and its analytic continuation to $\operatorname{Im}(z) \geqslant 0$. Where are the Stokes lines, that is, the critical lines separating the Stokes regions?

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• 4.II $. 31 \mathrm{~A}$

  Asymptotic Methods | Part II, 2005

  Consider the differential equation

  $$\frac{d^{2} w}{d x^{2}}=q(x) w$$

  where $q(x) \geqslant 0$ in an interval $(a, \infty)$. Given a solution $w(x)$ and a further smooth function $\xi(x)$, define

  $$W(x)=\left[\xi^{\prime}(x)\right]^{1 / 2} w(x) .$$

  Show that, when $\xi$ is regarded as the independent variable, the function $W(\xi)$ obeys the differential equation

  $$\frac{d^{2} W}{d \xi^{2}}=\left\{\dot{x}^{2} q(x)+\dot{x}^{1 / 2} \frac{d^{2}}{d \xi^{2}}\left[\dot{x}^{-1 / 2}\right]\right\} W$$

  where $\dot{x}$ denotes $d x / d \xi$.

  Taking the choice

  $$\xi(x)=\int q^{1 / 2}(x) d x$$

  show that equation $(*)$ becomes

  $$\frac{d^{2} W}{d \xi^{2}}=(1+\phi) W$$

  where

  $$\phi=-\frac{1}{q^{3 / 4}} \frac{d^{2}}{d x^{2}}\left(\frac{1}{q^{1 / 4}}\right)$$

  In the case that $\phi$ is negligible, deduce the Liouville-Green approximate solutions

  $$w_{\pm}=q^{-1 / 4} \exp \left(\pm \int q^{1 / 2} d x\right)$$

  Consider the Whittaker equation

  $$\frac{d^{2} w}{d x^{2}}=\left[\frac{1}{4}+\frac{s(s-1)}{x^{2}}\right] w$$

  where $s$ is a real constant. Show that the Liouville-Green approximation suggests the existence of solutions $w_{A, B}(x)$ with asymptotic behaviour of the form

  $$w_{A} \sim \exp (x / 2)\left(1+\sum_{n=1}^{\infty} a_{n} x^{-n}\right), \quad w_{B} \sim \exp (-x / 2)\left(1+\sum_{n=1}^{\infty} b_{n} x^{-n}\right)$$

  as $x \rightarrow \infty$.

  Given that these asymptotic series may be differentiated term-by-term, show that

  $$a_{n}=\frac{(-1)^{n}}{n !}(s-n)(s-n+1) \ldots(s+n-1) \text {. }$$

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