PHYS 201 Mathematical Physics, Fall 2015, Homework 5
Due date: Thursday, November 5th, 2015
1. Use integration by parts and show that the asymptotic expansion as x → ∞ of the
upper incomplete Gamma function Γ(1/2, x) is
Z x
Γ(1/2, x) =
t−1/2 e−t dt
0
1
e−x
+ ...
∼ Γ(1/2) − √ 1 −
2x
x
2.
a. Use Laplace’s method to verify the first two or three terms of the asymptotic expansion as x → ∞ of the modified Bessel function K0 (x)
Z ∞
K0 (x) =
(s2 − 1)−1/2 e−xs ds
1
−x
∞
[Γ(n + 12 )]2
e X
n
∼ √
(−1) n+1/2
x n=0
2
n!Γ( 21 )xn
R a>0
This is an application of Watson’s lemma where the integral 0 h(t)e−xt dt is approximated as x → ∞ by expanding h(t) in a power series about 0. (Hint: First
shift the limits of the integral to 0 and ∞.) corrected, Nov 4th
b. Use similar ideas as part (a) to find the first two terms of the asymptotic expansion
as x → ∞ of
Z π/2
2
f (x) =
e−x sin t dt
(1)
0
c. Show the following leading behavior as x → ∞ using Laplace’s method:
Z ∞
x f (x) =
exp − t − √ dt
t
0
2/3
∼ π 1/2 22/3 3−1/2 x1/3 e−3(x/2)
(Hint: Direct application of Laplace’s method does not work (why?) Use the transformation t = sx2/3 and then apply Laplace’s method.)
3. Use the method of stationary phase to find the leading behavior of the following
integrals as x → ∞
1
i.
R1
ii.
R1
0
2
eixt cosh t2 dt
−1
sin[x(t − sin t)] sinh(t)dt
4. Use the method of steepest descent to find the first two terms of the asymptotic
expansion for the Airy function I(x) as x → ∞ where
Z ∞
t3
I(x) =
exp[ix( + t)]dt
3
−∞
See the full problem description in Exercise 2 of Section 6-3 of Carrier et al for hints.
2