I'm struggling to understand the difference between this two functions.
I have this condition: $P_j:=\mathbb{Q}(S_T>K):=\frac{1}{2}+\frac{1}{\pi}\int_{0}^{+\infty}Re[\frac{e^{iuK}f_j(u,x,v)}{iu}]\textrm{d}u$ with $Re$ the real part of integrand function, $i:=\sqrt{-1}$ and $f_j$ the characteristic function of $x:=\textrm{ln}(S)$ with density $P_j$. Well. From this…
…I know that characteristic function $\varphi$ is the F-transform. So, if the functions are equal, why here (page 31, Proposition 3.1, Formule 3.14) we are being told that there is the necessity to see the form of $f_j:=\phi_j$ (that, with a "guess and verify" approach, it is assumed that be the same as $e^{C_J(\tau,u)+D_j(\tau,u)+iux}$)? Don't we already know the form of the characteristic function $\phi_j$, namely that of Fourier transform?
I really thank anyone who wants to help me!