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.
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