Asymptotic Analysis and Singular Perturbation Theory

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Example 3.10 The Gamma function Γ : (0, ∞) → R is defined by Γ(x) = ∞ e 0 −t t x−1 dt. Integration by parts shows that if n ∈ N, then Γ(n + 1) = n!. Thus, the Gamma function extends the factorial function to arbitrary positive real numbers. In fact, the Gamma function can be continued to an analytic function Γ : C \ {0, −1, −2, . . .} → C with simple poles at 0, −1, −2, . . .. Making the change of variables t = xs, we can write the integral representation of Γ as Γ(x) = x x ∞ 1 s exϕ(s) ds, where 0 ϕ(s) = −s + log s. The phase ϕ(s) has a nondegenerate maximum at s = 1, where ϕ(1) = −1, and ϕ ′′ (1) = −1. Using Laplace’s method, we find that Γ(x) ∼ 1/2 2π x x x e −x as x → ∞. In particular, setting x = n + 1, and using the fact that n + 1 lim n→∞ n we obtain Stirling’s formula for the factorial, n n! ∼ (2π) 1/2 n n+1/2 e −n = e, as n → ∞. This expansion of the Γ-function can be continued to higher orders to give: Γ(x) ∼ 1/2 2π x x x −x e 1 + a1 x + a2 x 2 + a3 + . . . x3 a1 = 1 12 , a2 = 1 288 , a3 = − 139 , . . . . 51, 840 44 as x → ∞,
3.5.1 Multiple integrals Proposition 3.11 Let A be a positive-definite n × n matrix. Then 1 − e 2 xT Ax (2π) dx = n/2 . | det A| 1/2 R n Proof. Since A is positive-definite (and hence symmetric) there is an orthogonal matrix S and a positive diagonal matrix D = diag[λ1, . . . , λn] such that A = S T DS. We make the change of variables y = Sx. Since S is orthogonal, we have | det S| = 1, so the Jacobian of this transformation is 1. We find that R n 1 − e 2 xT Ax dx = Now consider the multiple integral I(ε) = R n = = = R n n i=1 1 − e 2 yT Dy dy R 1 − e 2 λiy2 i dyi (2π) n/2 (λ1 . . . λn) 1/2 (2π) n/2 . | det A| 1/2 f(t)e ϕ(t)/ε dt. Suppose that ϕ : R n → R has a nondegenerate global maximum at t = c. Then ϕ(t) = ϕ(c) + 1 2 D2 ϕ(c) · (t − c, t − c) + O(|t − c| 3 ) as t → c. Hence, we expect that I(ε) ∼ R n 1 [ϕ(c)+ f(c)e 2 (t−c)T A(t−c)]/ε dt, where A is the matrix of D 2 ϕ(c), with components Aij = ∂2ϕ (c). ∂ti∂tj Using the previous proposition, we conclude that I(ε) ∼ (2π) n/2 |det D 2 ϕ(c)| 1/2 f(c)eϕ(c)/ε . 45
Page 1: Asymptotic Analysis and Singular Pe
Page 4 and 5: 4.1.1 Outer solution . . . . . . .
Page 6 and 7: Once we have constructed such an as
Page 8 and 9: For example, setting ε = 0 in (1.2
Page 10 and 11: Solving the first equation, we get
Page 12 and 13: and A ε : R n → R n is a linear
Page 14 and 15: Energy eigenstates are wavefunction
Page 16 and 17: and the constant α, with dimension
Page 18 and 19: The velocity gradient ∇u has dime
Page 20 and 21: Imposing the requirement that R d
Page 23 and 24: Chapter 2 Asymptotic Expansions In
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Page 27 and 28: where denotes the C n -norm. We def
Page 29 and 30: It follows that where Since we have
Page 31 and 32: 2.2.4 Nonuniform asymptotic expansi
Page 33 and 34: Chapter 3 Asymptotic Expansion of I
Page 35 and 36: we find that the optimal truncation
Page 37 and 38: and making the change of variable (
Page 39 and 40: 3.3 The method of stationary phase
Page 41 and 42: Applying this argument n times, we
Page 43 and 44: The standard normalization for the
Page 45 and 46: We assume for simplicity that the i
Page 47: where w = − t2/3v . 31/3 It follo
Page 51: When λ > 0, we can deform the inte
Page 54 and 55: plex C, and the complex breaks down
Page 56 and 57: with the substrate is balanced by t
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Page 60 and 61: Thus, the solution involves two ter
Page 62 and 63: where c is an arbitrary constant of
Page 64 and 65: (a) if a(x) > 0, then the boundary
Page 66 and 67: The inner solution near x = −1 is
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Page 72 and 73: We seek an asymptotic expansion of
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Page 77 and 78: Chapter 5 Method of Multiple Scales
Page 79 and 80: Using this expansion in the equatio
Page 81 and 82: where the overline denotes the aver
Page 83 and 84: 5.3 The method of averaging Conside
Page 85 and 86: where f : R n ×T → R n is a Lips
Page 87 and 88: 5.5 The WKB method for ODEs Suppose
Page 89: Thus, the amplitude of the oscillat

Asymptotic Analysis and Singular Perturbation Theory

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