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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]

By J. Gil-Pelaez. Biometrika 38(3–4), 481–482 (1951). DOI: doi.org/10.1093/biomet/38.3-4.481

Locator: inversion identity on pp. 481–482. Publisher record: academic.oup.com/biomet/article-abstract/38/3-4/481/258591