# Find the path of steepest descent

1. Sep 18, 2012

### karlzr

1. The problem statement, all variables and given/known data
A function F(a) is defined by $$F(a)=-ie^{-i\pi a/2}\int_{\pi/2-i\infty}^{\pi/2+i\infty}e^{ia(e^{iz}+z)}dz$$
where the integration is along the vertical line ($Re(z)=\pi/2$).
(a) Show that the integral is convergent for real and positive values of a.
(b) Find the saddle point(s) of the exponent.
(c) Find the path of steepest descent.

2. The attempt at a solution
I can handle the first two questions. However, I don't quite understand the last one. How to find the descent along a particular path? What does the last question ask about exactly?