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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

  1. 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]
  2. 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

  1. 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]
  2. 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 5

  1. 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]
  2. 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]
  3. 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

  1. 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]
  2. 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 \(<\).
    On page 234 the five inequalities written with \(<\) on lines 2, 5, and 7 (one on line 7, the second displayed equation on this page, occuring in a subscript) should all be written instead with \(\le\). Finally, on page 237, in the seventh line from the bottom of the page, "bounded subset" should be replaced by "closed subset".
  3. 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]
  4. 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 \(|t|<320/621\).
  5. 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

  1. 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]
  2. 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]

Chapter 10

  1. 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]
  2. 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]
  3. 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 \(F\).

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