# Variables cv.complex – How to fix the multi-value function in the contour?

I'm sorry to ask such a shamefully simple question here. My question is about the integral contour of the multivalude function.
I want to calculate the Fourier transformation of a mutual value function $$f$$
$$G ( omega, mathbf {k}) = int dt d ^ {d-1} mathbf {x} f (t, mathbf {x}) e ^ {i omega t-i mathbf {k} cdot mathbf {x}}$$
with the condition of
$$omega> 0, , , , , , , , omega ^ 2- mathbf {k} ^ 2> 0$$
So that we can deform the contour from green to red.
Function $$f$$ is given by
$$f (t, mathbf {x}) = big ( frac {-1} {t ^ 2- mathbf {x} ^ 2-i epsilon t} big) ^ Delta$$
by $$t <0$$, $$f$$ It can be written as
$$f (t, mathbf {x}) = frac {e ^ {i pi Delta}} { big (t ^ 2-x ^ 2 big) ^ Delta}$$

by $$t> 0$$, $$f$$ It can be written as
$$f (t, mathbf {x}) = frac {e ^ {- i pi Delta}} { big (t ^ 2-x ^ 2 big) ^ Delta}$$

So I need to write the function on all four legs, $$1,2,3,4$$ as the picture shows. My question is, how do you combine the multi-value function in each contour? Thanks for any suggestions in advance.