@chat; how exactly did they use the fundamental theorem of calculus? I followed each step, but I fail to see how they used the thm to conclude that $\partial/\partial xU(p)=\operatorname{Re}f(p)$. I understand they took this limit;
$$
\lim_{h\to 0}\frac{1}{h}\int_0^h\operatorname{Re}f(P+s)ds,
$$
but in order to apply the fundamental thm of calculus, shouldn’t we know the antiderivative of $\operatorname{Re}f$?