In summer 2005, I attended the MSRI Mathematical Graphics Workshop at the Reed College. The purpose of projects created there was to show an example of using computer graphics in explaining certain mathematical concepts; to teach mathematical concepts using pictures instead of formulas.

Method of Steepest Descent (SD)

This applet demonstrates the validity of the SD method on a simple function with 2nd degree saddle for which blue and green curves are level curves and the SD direction is perpendicular to the blue hyperbolas.  By dragging the movable point (vertex of parabola) closer to the saddle point, the real and imaginary parts of the integrant are drawn as blue and red curves, respectively, along with the value of the real and imaginary parts of the integral given as numbers next to it. As a parabola degenerates into a SD line, oscillation vanishes.

 

  

 

This method is used for more complicated functions such as  Bessel function J0.
In order to solve the integral along the vertical part of the red line, we expand it to the Hankel contour, which is the whole red line, and then deform that contour into the SD curve which is the black line (perpendicular to the red level curves).
   

 
 
And few formulas, if badly needed:

We would like to find the asymptotic behavior of the family of integrals     as

If   and  are real functions, Laplace’s method and Watson’s lemma give full asymptotic behavior.

If  is complex, we obviously use Laplace on the imaginary and real parts separately.

If is purely imaginary,  the  method of stationary point gives us the leading term of the asymptotic behavior, as Rieman-Lebesque lemma shows that   as .

For thecomplex function, SD suggests that we fix the imaginary part and, as we travel along the SD curve, evaluate the integral.

We could have fixed the real part instead, but since the stationary point method is much weaker, we do not choose to do so.