Asymptotics final
Answer as much as you can. The questions do not have equal weight. There are two pages.
1. For z 1, use Laplace’s method to find the leading-order approximation to the integral,
Z ∞
4
2
et−z(t −2t ) sin ωt dt,
0
where ω is a parameter that can take any value including one close to a multiple of π.
2. The function f (r) satisfies the equation,
(r + 1)
df
df
d2 f
+
+ rf 2
= 0,
dr 2
dr
dr
in r ≥ 1, with > 0 and the boundary conditions, f = 0 on r = 1 and f → 1 as r → ∞. Obtain
asymptotic expansions for f at fixed r and then at fixed ρ = r, as → 0, both in the asymptotic
sequence, 1, δ, δ 2 ,..., where δ() is a small parameter that you should select with an educated guess
(and is not a power of ). Match the two solutions for f ; if you use the asymptotic representation
of any integrals, derive the required formulae.
3. Using multiple scales, find the leading-order asymptotic approximation, valid for t = O( −1 ) to
the solution of the equations,
ẋ = y + x2 z,
ẏ = −x + y,
ż = x − z,
x(0) = 0,
y(0) = 1,
z(0) = 0.
4. Consider the eigenvalue problem,
y 00 + λ2 x2 (1 − x2 )y = 0,
0 ≤ x ≤ π,
y(0) = y(π) = 0 .
Using the WKB method, provide an approximation for the eigenvalue, λ, comparing your result
with the first six solutions obtained numerically: λ ≈ 5.86, 15.3, 24.7, 34.2, 43.6 and 53. Note that
√
w00 + λ2 xp−2 w = 0
=⇒
w(x) = x C1/p (2λxp/2 /p),
where Cν (z) is a Bessel function of order ν. For y 00 +f (x)y = 0, you may quote the WKB connection
formulae,
√
√
B
B
a = 2A + √
and
, b = 2A − √ ,
2
2
where
Z x
−1/2
2
0
0
y∼ω
(a cos θ + b sin θ), ω = f > 0, θ = ω(x ) dx ,
x∗
Z x
0
0
−1/2
−Φ
Φ
2
Ω(x ) dx ,
y∼Ω
(Ae + Be ), Ω = −f > 0, Φ = x∗
and x∗ is the turning point.
10
9
8
I3/2(k)
7
6
5
4
3
2
1
0
0.1
0.2
0.3
0.4
0.5
k
0.6
0.7
0.8
0.9
5. (a) For the integral,
Iβ (k) =
Z
1
0
x(1 + x) dx
,
(1 − kx2 )β
with β > 0, find the general term in an expansion for k 1. Sketch the result against k for
0 ≤ k ≤ 1 and β = 32 , using terms upto O(k 7 ) of the sequence, and compare the result with the
numerical computation shown in the figure; use a printout of the figure if needed!
(b) Compute Sn ( 34 ) for n = 0 to 7, where Sn (k) is the (n + 1)−term approximation of I 3 (k).
2
Use the Shanks transform with these eight numbers to generate five improved approximations of
I 3 ( 43 ). Iterate the Shanks transform to find even better approximations, and compare your result
2
with the numerically determined value, I 3 ( 34 ) ≈ 2.3877.
2
(c) Next, from the k 1 series solution find the nearest singularity to the origin and its type, α.
Returning to the integral, make a second approximation about the singularity, obtaining the first
two terms of the approximation I 3 ∼ J0 + Jα (1 − k)α . Add a sketch of the second approximation
2
to the plot.
(d) Use multiplicative and additive extraction to remove the nearest singularity in the smallk approximation for β = 23 , keeping terms upto and including order k 2 . Add both improved
approximations to your picture.
(e) Construct the (2,2) Padé approximant from the k 1 series solution, and add that to the
picture for β = 3/2. At what value of k does this approximant place the singularity?