1 Errata for “Principles of Applied Mathematics; Trans-

formation and Approximation”, second edition

Last update October 10, 2001

1. pg. 7; The law of cosines (line 15) should read ||x + y||^{2} = ||x||^{2} + 2||x|| · ||y|| cos θ + ||y||^{2}.

2. pg 12; line -3 of footnote. “vetor” should be “vector”.

3. pg. 15 line 3; M lies in a k-dimensional space,

4. pg. 25 line 14; the null space is spanned by (2, −1)^{T}.

5. pg 35; Caption to figure 1.8; “rotations” should be “reflections”.

6. pg. 49; Problem 1.3.c; n must be greater than 1.

7. pg. 65 (a) line 17; property (3) should be property (4)

(b) line 23; … = (||x|| + ||y||)^{2}.

8. pg. 67, last line; of the components

9. pg 72

(a) line -6; The coefficient for the scale factor for the Legendre polynomial should be 2^{n} rather than 2_{n}.

(b) line -3 ω(x) should be w(x).

10. pg 73 (a) lines 1, 4, 6, 8, 13; ω(x) should be w(x).

(b) line 8, w(x) = (1 − x)^{α}(1 + x)^{β}

(c) line 10, p_{n}^{αβ}(x) = \frac{(−1)^{n}}{2^{n}n!} (1 − x)^{−α}(1 + x)^{−β} · · ·.

(d) line 15, The normalization constant is not the same as in other places, such as

Hochstadt, The Functions of Mathematical Physics, pg. 41.

11. pg. 80, line 12; “basis functions” rather than “basis function”.

12. pg. 93, line 15, Ingrid

13. pg. 94, Exercise 8. (a): Use property 5 …, (b) Use property 6 …

14. pg. 96, Exercise 8 e), T_{n}(x) = \frac{(−1)^{n}2^{n}}{(2n)!} (1 − x^{2})^{1/2} · · ·.

15. pg. 109, line -11: In the string of (in)equalities, change |<Lu, v>| = ||Lu|| · ||v|| to ≤ instead of = (using the Schwartz inequality).

16. pg. 116, line -9; K_{n}u rather than Ku

17. pg. 125, line -3; I = ∫_{0}^{π/2} 2dθ = π.

18. pg. 126, (a) line 6; Integrand is missing an r.

(b) line 11; g_{r}(r) should be g_{n}(r)

(c) line 13; In denominator of integrand, \sqrt{ρ^{2} − x^{2}} should be \sqrt{ρ^{2} − z^{2}}

(d) line -1; This is not incorrect, although with the change of variables suggested, one finds I_{0} = \sqrt{a^{2} − b^{2}} ∫_{0}^{π} \frac{dθ}{a+b cos θ}, which because of symmetry of cos θ is the same as shown.

19. pg. 129, line 2; dy rather than dt

20. pg. 141, Example 1 should read ⟨H′, ϕ⟩ = − ∫_{0}^{∞} ϕ′(x)dx = ...

21. pg. 142, Example. As n → ∞, ...

22. pg. 150, line 2: a_{2}(x)h(x, x)f′(x)... (i.e. lower case h)

23. pg. 155, line 3: g^{∗}(x, y) should be g^{∗}(y, x), and J should be J_{y}

24. pg. 164, line -9: S_{n+1/2}(ξ − x) should be S_{n+1/2}(2π(ξ − x))

25. pg. 168, line 11; in definition of the Jacobi polynomials, (01)^{α} should be (−1)^{α}

26. pg. 171, Problem 1a. second line of definition should be for |x| ≥ 1.

27. pg. 175 Problem 4.3.9; −β should be replaced by = −β

28. pg. 180 line -2 Replace F + λG by F − λG (twice)

29. pg. 212, Caption to Figure 6.2, e^{(θ_{1}θ_{2})/2} replaced by e^{i(θ_{1}+θ_{2})/2}.

30. pg. 215 (a) line 11; f(z) = u(x, y) + iv(x, y)

(b) line 17; \frac{∂v}{∂z} should be \frac{∂v}{∂x}.

31. pg. 216 (a) line 2, ⋯ = u_{x}(x_{0}, y_{0}) + ⋯

(b) line -1 should read \frac{e^{z+Δz}−e^{z}}{Δz} = ⋯

32. pg. 223 line -3 the denominator is missing a factor of r, and should be 1 − 2r cos(ϕ − θ) + r^{2}.

33. pg. 235, line -7: ⋯ = −2πiρAV

34. pg. 242, line 7: ...with angle of attack α in the z-plane.

35. pg. 248, line -3; ⋯ = \frac{2πi}{2(z+i)} |_{z=i} ⋯

36. pg. 253, line -12; I(R) = ∫_{Γ(R)} e^{iaz}g(z)dz.

37. pg. 254, line 12, I = 2πi \frac{e^{iaz}}{z+i} |_{z=i}

38. pg. 259, line 6: = should be +

39. pg. 276 (a) Problem 6.2.9; The contour should be |z| = 1.

(b) Problem 6.3.5; The ranges for z should be strict inequalities, that is, Im z < 0

and Im z > 0.

40. pg. 286, line 17; = … |_{λ=k^{2}π^{2}}

41. pg. 298, line -8; … = \frac{1}{n!} \frac{d^{n}}{dz^{n}}

42. pg. 299, line -5 should begin = \frac{1}{2π} ∑_{k=−∞}^{∞} ⋯

43. pg. 308 line 4, singularities of F (s) ...

44. pg. 309, line 6, becomes Lu = −d^{2}u/dt^{2} − λu

45. pg. 310, line -2; (Δu)_{n} + λu_{n} = 0

46. pg. 311 (a) line 13 λ = 2 − ξ − 1/ξ.

(b) line -8; δ_{nj} = \frac{1}{2πi} ∫_{C′} z^{|n−j|−1}dz

47. pg. 314, line 2, “to be real” should be ”to be nonzero and real”.

48. pg. 316, last line; R_{R}(k) = −R_{L}(−k) is not correct. The correct statement is R_{R}(k)/T_{R}(k) = −R_{L}(−k)/T_{L}(−k) for real, nonzero k.

49. pg. 317, line 3: e^{ik_{0}} should be replaced by e^{ik_{0}x}.

50. pg. 328, last line, delete “α real”.

51. pg. 330, Exercise 11. Reference should be to Theorem 7.3.

52. pg. 361, line 6 ...if and only if J_{n}(\sqrt{λ}R) =

53. pg. 363, (8.34) … (sin ϕ\frac{∂U_{m}}{∂ϕ} ) …

54. pg. 382, line -4 should read “∇^{2}ϕ_{j} = v_{j} in Ω and n · ∇ϕ_{j} = 0 on ∂Ω, for j = 1, 2.”

55. pg. 386, line -4; u(R, t) = T_{1}

56. pg. 396, line -1 insert “to” before solve.

57. pg. 408, last line should end with “?”.

58. pg. 409, Problem 8.4.1, the equation should read u_{n_{t}} = \frac{1}{h^{2}} (u_{n+1} − 2u_{n} + u_{n−1}).

59. pg. 411, line 2: scattering transform should be inverse scattering transform.

60. pg. 412

(a) line 5; For reasons that are somewhat subtle, it is not sufficient to assume that q(x) is absolutely integrable. Instead, one must make a stronger assumption, such as ∫_{−∞}^{∞}(1 + |x|)|q(x)|dx < ∞.

(b) line -7, Im k ≥ 0 should be Im k > 0.

61. pg. 426 (a) line -12 \frac{dq_{n}}{dt} should be replaced by \frac{d^{2}q_{n}}{dt^{2}} .

(b) line -3 \frac{dv_{n}}{dt} should be replaced by \frac{d^{2}v_{n}}{dt^{2}} .

62. pg.441, line 3; In Table 10.1

63. pg. 463, prob 10.2.1; E_{n}(x) = ∫_{1}^{∞} \frac{e^{−xt}}{t^{n}} dt.

64. pg. 542, line 17 insert “(” (open parenthesis) before cardiac.

65. pg. 567, 1.1.2 should be 1.1.3.

66. pg. 568; The answer to 1.2.1 is not computed correctly. The matrices C and D are correct, but the representation of A should be

( \begin{matrix}\frac{53}{6} & −\frac{19}{3} & −4 \\ \frac{13}{12} & −\frac{5}{6} & −1 \\ \frac{49}{4} & −\frac{17}{2} & −5 \end{matrix} )  (1)

67. pg. 574

(a) 3.4.2; Answer (a) is the correct answer for (b). Answer (b) is the correct answer for some other unrelated problem.

(b) line -1: a_{n} = \frac{2n+1}{2}.

68. pg. 575, Problem 3.5.2; u(x) = f(x) + \frac{λ}{1−λ} + ∫_{0}^{1} f(t)dt = x + \frac{λ}{2(1−λ)} when f(x) = x, provided λ ≠ 1.

69. pg. 576 (a) line 5: (answer to 4.1.6) u(x) = δ(x) + c_{1}x + c_{2}

(b) line -6, (answer to 4.2.9) u(x) = ∫_{0}^{1} g(x, y)f(y)dy − λ ∫_{0}^{1} g(x, y)u(y)dy + ⋯ (c) line -4,(answer to 4.2.10) u(x) = ∫_{0}^{1} g(x, y)f(y)dy − λ ∫_{0}^{1} g(x, y)u(y)dy + ⋯ (d) line -2 (answer to 4.2.11): u(x) = ∫_{0}^{1} g(x, y)f(y)dy − λ ∫_{0}^{1} g(x, y)u(y)dy + ⋯

70. pg. 577 (a) line 2 (answer to 4.2.13) u(x) = cos x + λ ∫_{0}^{x} p(ξ) sin(x − ξ)u(ξ)dξ

(b) line 3 (answer to 4.2.14) u(x) = 1 + ∫_{0}^{x} ξ ln(\frac{x}{ξ})u(ξ)dξ

71. pg. 580

(a) Problem 6.1.3; (a) z = \frac{π}{2} + 2nπ − i ln(2 ± \sqrt{3}).

(b) z = (2n + 1)π − i ln(\sqrt{2} + 1), z = 2nπ − i ln(\sqrt{2} − 1).

(b) Problem 6.2.3; ∫_{C} z^{−1/3}dz = −3(2)^{1/3}e^{iπ/6}.

72. pg. 581, Problem 6.3.6; (b) F_{x} − F_{y} = −8ρπiA. (c)F − x − iF_{y} = ρπ(4γA − 8A^{2}i).

73. pg. 586 (a) Problem 8.1.13; u(r, θ) = −\frac{2}{π} ∫_{0}^{∞} …

(b) Problem 8.1.14; u(r, θ) = ∑_{n=−∞}^{∞} a_{n} …

74. pg. 589, Problem 10.2.2; ∑_{k=0}^{∞}(\frac{π}{x})^{2k}.

75. pg. 593, problem 12.1.8; ϕ_{τ} = \frac{1}{6}A^{2} instead of ϕ_{τ} = \frac{1}{6}^{2}. Feel free to let me know about any other errors you may find. I’ll add them to this list.