Calculating a modified version of integral representation of the Hankel's function of the first kind

Question

Firstly I have put all the formula together:

All of the following formulation

Considering that the Hankel function of the first kind can be expressed as follows:

Hankel function of the first kind

in which the path of integration is from -Infinity to 0 then zero to PIi then from PIi to PI*i+Infinity.

How can I calculate the following integral on the same path:

Integral to be calculated

where:

definition of A

and:

definition of Delta

Sorry for the pictures. I do not know how to use MathJax.