Complex integration by parts on intervals and decaying lines

Write $u=r+is$, $v=p+iq$. Apply F1 to the four real pairs $(r,p),(s,q),(r,q),(s,p)$. Subtract the second identity from the first and add $i$ times the sum of the last two. Since $u^{\mathrm{\prime }}v=(r^{\mathrm{\prime }}p-s^{\mathrm{\prime }}q)+i(r^{\mathrm{\prime }}q+s^{\mathrm{\prime }}p)$ and $u{v}^{\mathrm{\prime }}=(r{p}^{\mathrm{\prime }}-s{q}^{\mathrm{\prime }})+i(r{q}^{\mathrm{\prime }}+s{p}^{\mathrm{\prime }})$, the result is the complex integration-by-parts identity. All integrands are continuous on the compact interval and hence bounded and Riemann integrable; F2 changes each real integral to a Lebesgue integral. Applying F3 to $r,s$ and the same F2 gives the complex FTC. This is the sole countable-choice use here.

For the whole-line assertion apply step 1.1 on $[-N,N]$. The truncated products converge pointwise to $u^{\mathrm{\prime }}v$ and $u{v}^{\mathrm{\prime }}$ and have integrable majorants $\mathrm{\mid }{u}^{\mathrm{\prime }}v\mathrm{\mid }$ and $\mathrm{\mid }u{v}^{\mathrm{\prime }}\mathrm{\mid }$. F4 gives convergence of both integrals; the boundary term $u(N)v(N)-u(-N)v(-N)$ tends to zero by the two assumed limits. Passing to the limit proves the assertion. No separate integrability of $u$ or $v$ is required.