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Was ever studied a function $f$ which solves $J_0(f(x))=x$? Integral representations, natural domains of existence and whatever.

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    $\begingroup$ For the record: the differential equation is given here, but apparently not much more is said about $J_0^{-1}$ there. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Dec 6, 2022 at 20:22

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Function $J_0$ belongs to the so-called Laguerre-Polya class. The "natural domain" of the inverse function of an entire function of this class is a simply connected Riemann surface whose structure has been described by Gerald MacLane in the paper Concerning the uniformization of certain Riemann surfaces allied to the inverse-cosine and inverse-gamma surfaces, Trans. Amer. Math. Soc. 62 (1947) 99–113.

For a modern exposition, of this work, you may consult https://arxiv.org/abs/1109.1464.

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