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[tex]

\textrm{Ai}(x) = \frac{1}{2\pi} \int\limits_{-\infty}^{\infty} e^{i\big(\frac{t^3}{3} + tx\big)}dt

[/tex]

If we substitute in place of [itex]x[/itex] a complex parameter [itex]z[/itex] with [itex]\textrm{Im}(z)>0[/itex], the integral will converge on [itex][0,\infty[[/itex], but diverge on [itex]]-\infty,0][/itex]. With [itex]\textrm{Im}(z)<0[/itex] the integral will converge on [itex]]-\infty,0][/itex], but diverge on [itex][0,\infty[[/itex]. The Wikipedia page tells me that a complex analytic version of the Airy function exists, but apparently it cannot be defined simply by substituting a complex variable [itex]z[/itex] into the same integral formula that works for real variables [itex]x[/itex]. How is the analytic continuation of Airy function studied then?