Generating Functional in Momentum Space -- QFT

Question

Question attached:

inserting $\phi (x)= \int \frac{d^4k}{(2\pi)^2}\phi(x)e^{-i k_u x^u}$ and similar for $J(x) $ / $J(k)$ into the action and then integrating over $k$ gives:

Solution attached:

I AM STUCK on this part, completing the square ; so I see we get back the term $\frac{-h}{i}J(k)\phi(-k)$ however don't we also get an additional identiical term $J(-k)\phi (k)$?

What has happened to this?!

Many thanks in advance.

2. Relevant equations

see above

3. The attempt at a solution

see above

Answer 1

You pulled out a factor $1/2$, so you need to compensate by multiplying the $J(k)\phi(-k)$ term by $2$. Then use that $$ 2\int d^4k \, J(k)\phi(-k)=\int d^4k \, J(k)\phi(-k) + \int d^4k \, J(-k)\phi(k). $$