Find the path of steepest descent

[karlzr]

Homework Statement

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?

 

[lippyka]

(a) To show that the integral is convergent for real and positive values of a, we need to show that the integrand is bounded. We can do this by using the absolute value of the integrand:|e^{ia(e^{iz}+z)}| = e^{-aIm(e^{iz}+z)}Now, since Re(z)=\pi/2, we have that Im(e^{iz}+z) \geq 0, so e^{-aIm(e^{iz}+z)} \leq 1. Since the integrand is bounded, the integral is convergent for real and positive values of a.(b) The saddle point(s) of the exponent can be found by setting its derivative to zero, i.e.\frac{d}{dz}ia(e^{iz}+z)=0This yields the equatione^{iz}+z=0which has the solution z_0=\pi/2+i\ln(-1).(c) I don't understand how to find the path of steepest descent.