## Applied Asymptotic Analysis

*Applied Asymptotic Analysis*, volume 75 of the Graduate Studies in Mathematics series published by the American Mathematical Society, is a textbook
intended for graduate students or advanced undergraduate students. It was originally developed as a text for the course Math 557, Methods of Applied Mathematics II: Asymptotic Analysis, which is part of the "core" of the Applied and Interdisciplinary Mathematics (AIM) graduate program at the University of Michigan.

### Update Log

Here is a current list of corrections to the text.

#### Chapter 2

- Section 2.1 (Review of basic methods), page 48. In the sum on the second line of the displayed equation following (2.2), the factor of \((n+a)\) in the denominator should instead be \((n+z)\). [AK]
- Section 2.1 (Review of basic methods), page 51. The index \(k\) of the product in equation (2.5) should start at \(k=1\) instead of \(k=0\). [JW]
- Section 2.2 (Exponential integrals and Watson's Lemma), page 55. The factor \(e^{-\lambda s/2}\) appearing on the second line of the displayed equation halfway down the page, in the following displayed equation, and inline on the following line should be replaced in each case with \(\lambda^{-1/2}e^{-\lambda s/2}\). [JB]
- Section 2.3 (Elementary generalizations of Watson's Lemma), page 58. Line 7 should read instead "Taking \(\sigma=2\), we then have the result". [SB]

#### Chapter 3

- Section 3.4 (Contributions from interior maxima), page 69. The first term in the numerator of the last displayed equation should be \(5R'''(t_{\mathrm{max}})^2\) instead of \(5R'''(t_\mathrm{max})\). [SB]
- Section 3.5 (Summary of generic leading-order behavior), page 72. Halfway down the page, in the displayed formula following the text "From (3.15) we thus have", the denominator of the summand should have \(\lambda^n\) instead of \(\lambda^{-n}\). [YZ]

#### Chapter 4

- Section 4.7 (Application: special functions), page 125. In part (c) of Exercise 4.9, "nonnegative" should be "nonpositive".
- Section 4.8 (The effect of branch points), page 125. In line -4 of the first paragraph of this section, the word "procedure" is misspelled. [AK]
- Section 4.8.2 (Application: Selection of particular solutions of linear differential equations admitting integral representations), page 146. In the first sentence following the second displayed equation on this page, the word "infinitesimally" is misspelled. [AK]

#### Chapter 5

- Section 5.1 (Introduction), page 149. In the last line of text on this page, \(t+\Delta t\) should instead read \(t_0+\Delta t\). [GM]
- Section 5.2 (Nonlocal Contributions), page 151. In the third footnote, an additional condition is required for a function \(f(x)\) to be absolutely continuous, namely that the fundamental theorem of calculus holds, i.e., \(f(x)=f(a)+\int_a^x f'(y)\,\mathrm{d}y\). [GY]
- Section 5.3.1 (Putting the exponent in normal form by a change of variables), page 157. In the first displayed equation there should be a factor of \(\mu_0(t(s))\) in the integrand. [YZ]
- Section 5.5.1 (Partial differential equations for linear dispersive waves), page 167. In Exercise 5.12, "coefficients \(A_\pm(k)\)" should be replaced with "Fourier transforms of \(f\) and \(g\)". [YZ]
- Section 5.6.4 (Rigorous semiclassical asymptotics using the method of stationary phase), page 178. In the exponent of equation (5.32) a semicolon has been written as a comma. The relevant factor should be \(e^{iI(\xi_k(x,t);x,t)/\hbar}\). [GM]
- Section 5.7 (Multidimensional Integrals), pages 184-185. A factor of \(2\pi\) has been accidentally omitted from all of the exponents on these two pages. Also, the second displayed equation on page 185 should have a factor of \(2\pi\) in the denominator of the right-hand side. Note also that to obtain the limiting result at the bottom of page 185 a more subtle convergence argument than one based on dominated convergence is probably required. [YZ]

#### Chapter 6

- Section 6.1.2 (Solutions viewed as analytic functions of the complex variable \(z\)), page 200. In the paragraph following the statement of Theorem 6.1, the statement "This combination satisfies the auxiliary conditions \(y(0)=\alpha\) and \(y'(0)=\beta\),..." should be replaced with "This combination satisfies the auxiliary conditions \(y(z_0)=\alpha\) and \(y'(z_0)=\beta\),...". [LL]
- Section 6.1.2 (Solutions viewed as analytic functions of the complex variable \(z\)), page 210. In the penultimate displayed formula about two thirds down the page, in the first line \((z-z^*)^{-\rho_j}\) should be instead \((z-z_*)^{-\rho_j}\).
- Section 6.3.2 (Existence of true solutions described by the formal asymptotic series. The Stokes phenomenon), pages 230-231. The starting values of the double sum indices \(j\) and \(k\) in the expression for \(G_N(z)\) given on page 230 should be zero instead of one. On line 5 of page 231, rather than "the dominant term..." it is better to complete the sentence by saying "the upper bound for \(e^{2\phi_0(z)}G_N(z)\) dominates all other terms on the right-hand side of (6.29)." [YZ]
- Section 6.3.2 (Existence of true solutions described by the formal asymptotic series. The Stokes phenomenon), pages 233-237. Some corrections are in order here related to the hypotheses required for the Contraction Mapping Principle to apply. I suggest the following concrete modifications. On page 233:
- In the statement of Theorem 6.2, replace "bounded" with "closed" on line 2 and replace the parenthetical remark with "(e.g. \(X=B\) or, for some \(g\in B\) and \(M\ge 0\), \(X=\{f\in B\;\;\text{such that}\;\|f-g\|\le M\}\))".
- In the final paragraph on this page, replace "bounded" with "closed" on line 3 and "open" with "closed" on line 4. On line 4 the given open interval should be written as a closed interval. Finally on line 6 the two inequalities should be written with \(\le\) rather than with \(<\).

- Section 6.3.2 (Existence of true solutions described by the formal asymptotic series. The Stokes phenomenon), pages 241-242. The last two sentences before Exercise 6.21 on page 241 beginning with "However, if \(\epsilon>0\) is small enough..." should be replaced with the following more accurate description: "However, if \(\epsilon>0\) is small enough, the region \(D_\ell(r,\epsilon)\) also contains ranges of directions along which \(y_\ell(z)\) is exponentially growing as \(z\to\infty\) and exactly two directions marking a transition between exponential growth and exponential decay of \(y_\ell(z)\). In these two distinguished directions \(y_\ell(z)\) may still grow or decay as \(z\to\infty\), but only at a slower rate since the dominant term in the exponent provides only oscillations." Similarly, on page 242, the last sentence of the paragraph beginning "Midway between each neighboring pair..." should be replaced with the more accurate statement "As \(z\to\infty\) along either of the Stokes rays in a given sector \(D_\ell^\pm(r,\epsilon)\), the formula (6.44) shows that the dominant term in the exponent of the solution \(y_\ell^\pm(z)\) provides only pure oscillations, although lower-order terms in the exponent and the power-law prefactor \(z^{u_{p+2}}\) can provide growth or decay."
- Section 6.3.3 (Another approach to the existence of true solutions and the Stokes phenomenon. Borel summation), page 247. In equation (6.53) and in the subsequent displayed equation, \(n(n+1)\) should be replaced by \(n(n-1)\). [YZ]
- Section 6.3.3 (Another approach to the existence of true solutions and the Stokes phenomenon. Borel summation), page 248. In the last paragraph the convergence condition \(|t|<80/121\) should read instead \(|t|<320/621\).
- Section 6.3.3 (Another approach to the existence of true solutions and the Stokes phenomenon. Borel summation), page 250. Just after the fourth displayed equation on the page, the half-plane of convergence should be \(\Re\{se^{i\theta}\}>0\). [MSh]
- Section 6.3.3 (Another approach to the existence of true solutions and the Stokes phenomenon. Borel summation), page 251. The sectors of the complex \(z\)-plane in which the final two displayed asymptotic expansions are valid should be swapped. [YZ]

#### Chapter 7

- Section 7.1.1 (Formal power series expansions), page 255. On line 5 of this section, "independent" is misspelled. [GM]
- Section 7.1.2 (Solving for \(y_n(x)\). Variation of parameters), page 259. Near the bottom of the page, there are in fact three wavelengths: the two given as well as \(2\pi/k\). [YZ]
- Section 7.2.2 (The special case of an asymptotic power series for \(f(x;\lambda)\)), page 274. Exercise 7.7 is not correctly formulated and should be omitted. Reference to this exercise in the footnotes at the bottom of page 280 should also be omitted. [YZ]
- Section 7.2.3 (Turning points), page 282. Part (d) of Exercise 7.8 should be modified as follows. The sentence preceding the displayed equations should read instead, "Then, by letting \(\lambda\to\infty\) with \(\lambda>0\), show that as long as \(e^{-i\pi/4}C+e^{i\pi/4}D\neq 0\), if \(x_- < 0\) is fixed...". Then after the displayed equations the sentence beginning "It follows that..." should be replaced by the sentences "It follows that in this case we have a well-posed connection problem. Show on the other hand that if \(e^{-i\pi/4}C+e^{i\pi/4}D=0\) (
i.e., if \(e^{-i\pi/4}C(\lambda)+e^{i\pi/4}D(\lambda)=o(1)\) as \(\lambda\to \infty\) with \(\lambda>0\)), the asymptotic behavior for \(x<0\) cannot be determined by the given information alone and we have an ill-posed connection problem." - Section 7.2.3 (Turning points), page 287. The correct definition of \(\nu^2\) displayed in the middle of this page should be \[ \nu^2:=\lambda^{-2}f_x(x_*(\lambda);\lambda)=f_0'(x_*)(1+O(\lambda^{-1}))>0.\] Thus \(\nu^2\) depends (weakly) on \(\lambda\) and the indicated inequality holds true for sufficiently large \(\lambda\). This weak dependence has no essential effect on the calculations that follow as they all involve the limit \(\lambda\to\infty\). [BL]
- Section 7.2.4 (Problems with more than one turning point. The Bohr-Sommerfeld quantization rule), page 303. In the first two displayed equations on this page, the integrand should be \((E-V(s))^{1/2}\) as the integration takes place between the turning points, in the domain where \(E>V(s)\). [BL]

#### Chapter 8

- Section 8.7 (Proving the validity of uniform approximations), pages 345-347. A sharper estimation that is also easier proceeds as follows. Starting with the last displayed inequality on page 344, one can bound the right-hand side above by the product of \(\|u(\cdot;\epsilon)\|\) and the \(L^1\) norm of \(R^{N,M}(\cdot;\epsilon)-S^{N,M}(\cdot;\epsilon)\). Then \(\epsilon\) times the square of (8.36) can be used to obtain a lower bound for the left-hand side of the same inequality. This results in an inequality of the form of (8.36) with the product of \((\beta-\alpha)^{3/2}\) and the \(L^2\) norm of \(R^{N,M}(\cdot;\epsilon)-S^{N,M}(\cdot;\epsilon)\) replaced by the product of \(\beta-\alpha\) and the \(L^1\) norm of the same. Finally, one arrives at (8.40) with the product of \((\beta-\alpha)^{3/2}\) and the \(L^2\) norm of \(R^{N,M}(\cdot;\epsilon)\) replaced by the product of \(\beta-\alpha\) and the \(L^1\) norm of the same. Then, Exercise 8.5 on page 334 should be modified to show that the \(L^1\) norm of \(R^{1,1}(\cdot;\epsilon)\) is \(O(\epsilon^2)\), and Exercise 8.12 on page 349 should be modified to yield an upper bound proportional by \(\epsilon^{-1}\) to the product of \(\beta-\alpha\) and the \(L^1\) norm of \(f(\cdot;\epsilon,t)\).

#### Chapter 9

- Section 9.1.2 (Degenerate theory), page 364. In Exercise 9.4, the matrix \(\mathbf{A}\) contains a typo in the first row and should instead be \[\mathbf{A}:=\begin{bmatrix}2 & 0 & 0\\1 & -2 & 1\\0 & 0 & 2\end{bmatrix}.\]
- Section 9.1.2 (Degenerate theory), page 364. In Exercise 9.5, the third sentence should begin "On the other hand, for each nonzero \(\epsilon\in\mathbb{C}\) with \(|\epsilon|\) sufficiently small..." (the word "nonzero" should be inserted).
- Section 9.2.2 (Periodic and antiperiodic solutions. Formal asymptotics), page 375. In equation (9.32) there is a missing error term of \(O(\epsilon^3)\).

#### Chapter 10

- Section 10.1.1 (Modulated wavetrains with dispersion and nonlinear effects. The cubic nonlinear Schrödinger equation), Exercise 10.2, page 406. In part (b), the resonant wavenumbers should be \(k=0\) and \(k=\pm 1/\sqrt{10}\). [JB]
- Section 10.1.2 (Spontaneous excitation of a mean flow), Exercise 10.5, page 416. In the displayed equation on line -3, the term \(-ikMA\) should read instead \(-kMA\).
- Section 10.1.3 (Multiple wave resonances), pages 418-420. For consistency with subsequent calculations, equation (10.40) should instead be written as \[\frac{\partial^2u}{\partial t^2}-2\frac{\partial^2u}{\partial x^2}+\frac{\partial^4u}{\partial x^4}+u=\epsilon u^2\] (so the nonlinear term appears on the right-hand side instead). Likewise, the displayed equation on line -9 of page 420 should instead be written as \[\frac{\partial^2u}{\partial t^2}-2\frac{\partial^2u}{\partial x^2}+\frac{\partial^4u}{\partial x^4} + u=\epsilon u^3.\] The subsequent calculations following from each of these are then correct as written. [MSt]
- Section 10.2.2 (Derivation of the cubic nonlinear Schrödinger equation), page 432. The expression for the coefficient of the nonlinear term in equation (10.60) is incorrect as written; the equation following (10.60) should instead read \[\beta:=-\frac{1}{4m\omega[V''(0)]^2}\left(m^2\omega^4V^{(IV)}(0)+4V''(0)[V'''(0)]^2\sin^2(\delta)\right).\] The text below is then correspondingly corrected to read "Thus, we see that whether we are in the stable or unstable case depends only on the sign of \(V^{(IV)}(0)\) and the size of \(m^2\omega^4V^{(IV)}(0)\) relative to \(4V''(0)[V'''(0)]^2\sin^2(\delta)\)." [N]
- Section 10.3.2 (Derivation of the Korteweg-de Vries equation), pages 446-447. A term has been omitted from equation (10.89). Equation (10.89) should instead read as \[\frac{\partial N}{\partial T}+G+\frac{\epsilon}{2}\left[\left(\frac{\partial N}{\partial X}\right)^2-\frac{\partial^3N}{\partial X^2\partial T}\right]=O(\epsilon^2)\] Making this correction, one finds that the final term on the left-hand side of each of equations (10.90) and (10.91) should be doubled. Thus, the final two equations in this section should read, respectively \[2\frac{\partial^2N}{\partial\xi\partial\tau}+\frac{1}{3}\frac{\partial^4N}{\partial\xi^4}+3\frac{\partial N}{\partial\xi}\frac{\partial^2N}{\partial\xi^2}=O(\epsilon^2)\] and \[\frac{\partial F}{\partial\tau}+\frac{3}{2}F\frac{\partial F}{\partial\xi}+\frac{1}{6}\frac{\partial^3 F}{\partial\xi^3}=0.\] Of course the factor of \(3/2\) could easily be absorbed by simply rescaling \(F\).

#### Thanks

All affiliations listed refer to the time the correction was pointed out.- [AK] Thanks to Aslan Kasimov (KAUST).
- [BL] Thanks to Bingying (Luby) Lu (University of Michigan).
- [GM] Thanks to Gary Marple (University of Michigan).
- [GY] Thanks to Giorgio Young (University of Michigan).
- [JB] Thanks to Jordan Bell (University of Toronto).
- [JW] Thanks to Jun-Chieh Wang (University of Michigan).
- [LL] Thanks to Louis Ly (University of Michigan).
- [MSh] Thanks to Mark Shoemaker (University of Utah).
- [MSt] Thanks to Marvin Strätz (University of Münster, Germany).
- [N] Thanks to Nathan (Bath University, UK).
- [SB] Thanks to Sergey Belov (Rice University).
- [YZ] Thanks to Yuchong Zhang (University of Michigan).