Problems, set 7. PHYSICS 212A: Mathematical Physics
1. Using integration by parts, find asymptotic series of the integral for x→ ∞,a > 0:
F(x, a) = Z∞
0
e−xt
t+adt
Write down the nth term of the series, ϕn(x), (numerate term of the series from n= 0).
Obtain the remainder term Rn(x) explicitly and find the upper bound on its magnitude.
Show that |Rn(x)/ϕn(x)| → 0 as x→ ∞.
2. Expansion under integral as a modified Laplace method.
The complimentary error function erfc(x) can be re-written as:
erfc(x) = 2
√πZ∞
x
exp(−t2)dt =2
√πexp(−x2)Z∞
0exp(−2tx) exp(−t2)dt
The “straightforward” Laplace’s method cannot be applied since the maximum of the function
in exponential is outside of the range of integration [find where], so the function is maximal
at the end point; also the slope of the exponent function is not zero at this end point.
Nevertheless, one can apply the ideology of the Laplace’s method here since the contribution
of the integrand tends to be confined to the neighborhood of the end point as x→+∞.
Expanding exp(−t2) in a power series around the end point, integrate the above expression
term by term to obtain an asymptotic expansion for large x. Write 4 non-zero terms. Obtain
explicitly the nth term of such an expansion, (with the first term in expansion corresponding
to n= 0). Write down explicitly the convergence test. Show whether this asymptotic series
is converging or not.
3. M&W: 3-35, leading term of the expansion only.
4. Use method of stationary phase to find leading behavior, valid as x→+∞, of the integral
I(x) = Z1
0exp µix µ1−sin t
t¶¶ dt
Hint. After expanding the function in exponential about its extremum extend the upper limit
of integration to ∞.
5. Using the saddle-point method, find leading behavior of the integral valid for u→+∞
I(u) = Z∞
−∞
dx
2πsin(x2+ux)
Hint. Change sine to exponent, then x→z. Identify saddle point(s) of the function in
exponential. Bend the contour so it goes through the saddle point in the “correct” way.
6. ∗[Extra credit problem. Part of the credit will be given for completing part (a)].
Use the Laplace’s method to obtain (a) the leading term only, (b) the first three terms of the
asymptotic expansion of the integral, for n→+∞
In=Z1
−1(1 −x2)ndx
