How to implement "coordinate stretching"

Hello
I want to use the method of "complex coordinate stretching" in FEniCS. The transformation is:

enter image description here eq(!)

Where :
enter image description here eq(2)

enter image description here eq(3)

enter image description here eq(4)

This is the original equation I want to solve:
enter image description here eq(5)

But under the transformation explained above it converts to:

enter image description here eq(6)

I have already solved the eq(5) in FEniCS (and I know how to implement complex numbers in FEniCS) but I do not have any idea how to implement this transformation in order to solve the eq(6). In addition I am supposed to use this function for Z(k):
enter image description here eq(7)

Does anybody know how this method might be implemented in FEniCS?
Thanks!