Homework 09
1 Stirling approximation
Show using steepest descent that the leading asymptotics of
for , are given by the complex Stirling approximation
The Gamma function
after performing a change of variables by setting
The method of steepest descent involves analyzing the integrand’s behavior around its saddle points, where the first derivative of the exponent vanishes. For our integral, the exponent is
Near the saddle point
The integral can now be approximated as:
With the Gaussian integral evaluates to:
We get:
2 Airy function
Consider the time-independent Schrodinger equation for a linear potential:
Show that the “Airy function” defined by
solves the equation above.
Determine the saddle points of
in the plane and make a change of variables to the integral so that these saddle points are not moving. Determine the direction of steepest descent on the constant phase contours through these saddle points for each argument of complex . Find the steepest descent contour for
for for each . Determine the leading asymptotics for each .
2.1 Verifying the Airy Function as a Solution
Taking the Fourier transform on both sides of equation we get
For the L.H.S., we get
For the R.H.S. based on the derivative of the Fourier transform with respect to
Therefore,
We have
So the equation becomes
the only solution is
taking the inverse Fourier transform, we get a solution satisfying the differential equation. Hence we have the final solution as
2.2 Saddle Points
The exponent in the integral is
3 function
Compute the all-orders asymptotic series for
and then use Borel summation to evaluate the integral.
The expansion is based on the Taylor series of
Substituting this series into the integral and integrating term by term, we get:
Each term in this series can be integrated using standard methods, often involving gamma functions for even powers of
The Borel sum is then given by an integral of the Borel transform against an exponential:
where