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Im having some trouble with the discussion about the pseudo-probabilities in Gatheral's book. In chapter 2, it reads

Taking the inverse transform using equation (2.8) and performing the complex integration carefully gives the final form of the pseudo-probabilities $P_j$ in the form of an integral of a real-valued function. $$P_j (x, v, \tau ) = \dfrac{1}{2} + \dfrac{1}{\pi} \int^\infty_0 du \, \text{Re} \left\lbrace \dfrac{ \exp \left(C_j (u, \tau ) \bar{v} + D_j (u, \tau ) v + i u x\right)}{ iu} \right\rbrace $$

I have seen the derivation using the Gil Pelaez theorem and get some of the logic. However, if I try to solve this following Gatheral's idea (performing the complex integration directly), I don't see how to proceed. Note that the starting point here is

$$P_j (x, v, \tau ) = \dfrac{1}{2\pi} \int^\infty_0 du \, e^{iux}\dfrac{ \exp \left(C_j (u, \tau ) \bar{v} + D_j (u, \tau ) v\right)}{ iu} $$ since the value of the $\tilde{P}_j$ function at $\tau = 0$ has been set to $\tilde{P}(u, v, 0) = \frac{1}{iu}$. This already seems "funny" to me: doesn't the inverse of the Heaviside function include a delta term to take care of its value at $u=0$?.

Does anyone have a rough idea? Do you know where I could find a proof for this? Thanks in advance!


Edit: A good answer to this has been given previously in https://math.stackexchange.com/a/960560/1222817

The boundary condition should have been $$\tilde{P}_j (u, v, 0) = \text{p.v.}\dfrac{1}{iu} + \pi \delta(u),$$ from where the $1/2$ factor in the expression for $P_j(x, v, \tau)$ follows straight forward.

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  • $\begingroup$ $P_j(x,v,0)$ is just the Heaviside function. It is it's transform $\tilde{P}_j$ that is given by ,$\tilde{P}_j(u,v,0) = \frac{1}{iu}$. $\endgroup$
    – Achrbot
    Jan 29 at 12:48
  • $\begingroup$ Thanks for noticing, it was just a typo. What I am asking (one of the questions) is, is that really the FT of the Heaviside function? What about the behavior at $u=0$? $\endgroup$
    – KT8
    Jan 29 at 13:00
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    $\begingroup$ Aah, I now see your conundrum, since the defining integral of $\tilde{P}_j$ doesn't converge for real u (I don't know why Gatheral doesn't elaborate). You can get around this by inverting $\tilde{P}_j$ along a strip in the complex plane, as in Lewis (2001). $\tilde{P}_j$ (as defined in Gatheral) is well-defined for $Im(u) < 0$, and this shouldn't meaningfully affect the other derivations. $\endgroup$
    – Achrbot
    Jan 29 at 13:36

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