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For a real random variable with characteristic function $\phi (t)=\mathbb{E}[e^{itX}]$, the Gil-Pelaez inversion formula recovers the cumulative distribution function at every continuity point:At a jump the same expression gives $(F(x)+F(x-))/2$. The integral is interpreted as the indicated improper limit, not assumed absolutely convergent. The sign follows from the $e^{itX}$ convention: its imaginary part is $\mathbb{E}\sin (t(X-x))$. This is useful when a characteristic function is available algebraically but a density is not.^[ref]

J. Gil-Pelaez authored the 1951 Note on the inversion theorem by J. Gil-Pelaez, giving a way to recover a distribution function from its characteristic function. The primary journal byline supplies the verified initials.^[ref]

The inversion integrates an oscillatory expression involving the characteristic function. At a discontinuity, the formula naturally returns the midpoint of the two one-sided distribution values. Keeping this convention explicit matters for discrete laws as well as continuous ones: an inversion formula is not automatically a pointwise identity at every jump.^[ref]