Asymptotic Analysis and Singular Perturbation Theory

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Example 2.16 Consider the function f : C → C defined by f(z) = sinh z 2 = ez2 Let |z| → ∞ with arg z fixed, and define Ω1 = {z ∈ C | −π/4 < arg z < π/4} , − e−z2 . 2 Ω2 = {z ∈ C | π/4 < arg z < 3π/4 or −3π/4 < arg z < −π/4} . If z ∈ Ω1, then Re z2 > 0 and ez2 ez2 ≪ e−z2. Hence f(z) ∼ ≫ e−z2, whereas if z ∈ Ω2, then Re z2 < 0 and 1 2ez2 1 2e−z2 as |z| → ∞ in Ω1, as |z| → ∞ in Ω2. The lines arg z = ±π/4, ±3π/4 where the asymptotic expansion changes form are called anti-Stokes lines. The terms ez2 and e−z2 switch from being dominant to subdominant as z crosses an anti-Stokes lines. The lines arg z = 0, π, ±π/2 where the subdominant term is as small as possible relative to the dominant term are called Stokes lines. This example concerns a simple explicit function, but a similar behavior occurs for solutions of ODEs with essential singularities in the complex plane, such as error functions, Airy functions, and Bessel functions. Example 2.17 The error function can be extended to an entire function erf : C → C with an essential singularity at z = ∞. It has the following asymptotic expansions in different wedges: ⎧ ⎨ 1 − exp(−z erf z ∼ ⎩ 2 )/(z √ π) as z → ∞ with z ∈ Ω1, −1 − exp(−z2 )/(z √ π) as z → ∞ with z ∈ Ω2, − exp(−z2 )/(z √ π) as z → ∞ with z ∈ Ω3. where Ω1 = {z ∈ C | −π/4 < arg z < π/4} , Ω2 = {z ∈ C | 3π/4 < arg z < 5π/4} , Ω3 = {z ∈ C | π/4 < arg z < 3π/4 or 5π/4 < arg z < 7π/4} . Often one wants to determine the asymptotic behavior of such a function in one wedge given its behavior in another wedge. This is called a connection problem (see Section 3.4 for the case of the Airy function). The apparently discontinuous change in the form of the asymptotic expansion of the solutions of an ODE across an anti-Stokes line can be understood using exponential asymptotics as the result of a continuous, but rapid, change in the coefficient of the subdominant terms across the Stokes line [2]. 28
Chapter 3 Asymptotic Expansion of Integrals In this chapter, we give some examples of asymptotic expansions of integrals. We do not attempt to give a complete discussion of this subject (see [4], [21] for more information). 3.1 Euler’s integral Consider the following integral (Euler, 1754): I(x) = ∞ 0 e−t dt, (3.1) 1 + xt where x ≥ 0. First, we proceed formally. We use the power series expansion 1 1 + xt = 1 − xt + x2 t 2 + . . . + (−1) n x n t n + . . . (3.2) inside the integral in (3.1), and integrate the result term-by-term. Using the integral we get ∞ t 0 n e −t dx = n!, I(x) ∼ 1 − x + 2!x 2 + . . . + (−1) n n!x n + . . . . (3.3) The coefficients in this power series grow factorially, and the terms diverge as n → ∞. Thus, the series does not converge for any x = 0. On the other hand, the following proposition shows that the series is an asymptotic expansion of I(x) as x → 0 + , and the the error between a partial sum and the integral is less than the first term neglected in the asymptotic series. The proof also illustrates the use of integration by parts in deriving an asymptotic expansion. 29
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
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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
Page 25 and 26: If, as is usually the case, the gau
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: 2.2.4 Nonuniform asymptotic expansi
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
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Page 47 and 48: where w = − t2/3v . 31/3 It follo
Page 49 and 50: 3.5.1 Multiple integrals Propositio
Page 51: When λ > 0, we can deform the inte
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Page 56 and 57: with the substrate is balanced by t
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Page 64 and 65: (a) if a(x) > 0, then the boundary
Page 66 and 67: The inner solution near x = −1 is
Page 68 and 69: If the fluid interface is a graph z
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Page 72 and 73: We seek an asymptotic expansion of
Page 74 and 75: It follows that dψ dX = −U0 sec
Page 77 and 78: Chapter 5 Method of Multiple Scales
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5.3 The method of averaging Conside
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where f : R n ×T → R n is a Lips
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5.5 The WKB method for ODEs Suppose
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Thus, the amplitude of the oscillat
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Asymptotic Analysis and Singular Perturbation Theory

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