I will be referring to the following document (page 16-17): http://www-thphys.physics.ox.ac.uk/people/JohnCardy/qft/qftcomplete.pdf
I would like to understand the expression of the generating function Z0[J] on page 17.
The document mentioned that I need to make a Wick rotation by letting τ=it and, as a result, I also need to let p0 --> ip0.
I observed that the expression for Z0[J] on page 17 is obtained by substituting p0=ip0 in the expression of Z0[J] on the bottom of page 16, which is the expression of the generating function in Euclidean space.
However, I'm wondering why didn't the document let τ=it in the expression of Z0[J] on page 17?