This question arises when I'm reading section "3.3.1 Minkowski Space" of page 16-17 of the following document: http://www-thphys.physics.ox.ac.uk/people/JohnCardy/qft/qftcomplete.pdf

On page 17, they took a functional derivative of Z[J] with respect to iJ to obtain an expression for G⁽⁰⁾(x₁,x₂). We're supposed to take derivatives with respect to J(x), but on page 17 the document took derivatives with respect to J(x'), where x₀=ix₀' (the subscript 0 indicates the first element of x; the other elements remain equivalent).

Is the results the same or did the document made a mistake?

Note: The definition of functional derivative the document is using is a delta function as the test function, as explained in section 4 of the following Wikipedia article: https://en.wikipedia.org/wiki/Functional_derivative#Using_the_delta_function_as_a_test_function

This question arises when I'm reading section "3.3.1 Minkowski Space" of page 16-17 of the following document: http://www-thphys.physics.ox.ac.uk/people/JohnCardy/qft/qftcomplete.pdf

On page 17, they took a functional derivative of $Z[J]$ with respect to $iJ$ to obtain an expression for $G_{(0)}(x_1,x_2)$. We're supposed to take derivatives with respect to $J(x)$, but on page 17 the document took derivatives with respect to $J(x')$, where $x_0=ix_0'$ (the subscript 0 indicates the first element of $x$; the other elements remain equivalent).

Is the results the same or did the document made a mistake?

Note: The definition of functional derivative the document is using is a delta function as the test function, as explained in section 4 of the following Wikipedia article: https://en.wikipedia.org/wiki/Functional_derivative#Using_the_delta_function_as_a_test_function